Descriptive Statistics - Jones & Bartlett Learning

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6

Descriptive Statistics

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Learning Objectives The principal goal of this chapter is to explain what © Jones & Bartlett descriptive statistics are and how theyLearning, can be usedLLC to NOT distribution. FOR SALEConfidence OR DISTRIBUTION examine a normal intervals are also discussed. This chapter will prepare you to:

• Explain the purpose of descriptive statistics • Compute measures of central tendency © Jones & Bartlett Learning, LLC © Jones & Bartlett Lea • Compute measures of variability NOT FOR SALE OR DIS NOT FOR SALE OR DISTRIBUTION • Understand and choose the best central tendency and variability statistic for different levels of measurement • Describe the normal distribution and associated © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC statistics and probabilities • Apply concepts of interval estimates describe NOT FOR SALE OR DISTRIBUTION NOT FORand SALE OR DISTRIBUTION methods for determining sample size • Apply understanding of central tendency and variability to nursing practice

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Key Terms NOT FOR SALE OR DISTRIBUTION Bimodal distribution

Median

Central tendency

Mode

© Jones & Bartlett Learning, LLC © Jones & Bartlett Lea Confidence interval Multimodal distribution NOT FOR SALE OR DIS NOT FOR SALEDegrees OR DISTRIBUTION of freedom (df) Normal distribution Descriptive statistics

Point estimates

Interquartile range

Range

Mean

Skewness/kurtosis

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DESCRIPTIVE STATISTICS

© Jones & Bartlett Learning, LLC NOTStandard FOR SALE OR DISTRIBUTION deviation Variance Standard normal distribution

Variation

Unimodal distribution

Z-scores/standardized scores

Variability © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION ●

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INTRODUCTION We have seen how nurses in practice can present data in a variety of

s & Bartlett Learning, formats LLC such as graphs, charts, © and Jones & These Bartlett Learning, LLC tables. graphical formats are R SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION useful for presenting data because they allow the reader to understand

the data visually. However, we lose some detail in the data when it is displayed graphically, especially around the distribution of data that are measured at the interval and ratio level (continuous variables). When data are measured atLLC the interval or ratio level, it im© Jones & the Bartlett Learning, ©isJones & Bartlett Lea portant to present the distribution of data in terms of central tendency NOT FOR SALE OR DIS NOT FOR SALE OR DISTRIBUTION (i.e., the average case) and variability (i.e., the range and spread of the data from the center). For example, Figure 6-1 shows a histogram of incomes of recent graduates in Family Nurse Practitioner (FNP) programs. Questions we might ask about graduates are, “What would be © Jones & Bartlett LLC measurement value if © Bartlett the Learning, typical or average oneJones person & was selected Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION at random from this group?” and “How far from the average are data values spread?” These are difficult questions to answer with visual displays such as graphs, charts, and tables. We need numerical measures of central tendency and variability so that we can understand the disbasis. s & Bartlett Learning, tribution LLC of the data on an objective © Jones & Bartlett Learning, LLC These numeric measurements of central tendency and variability are R SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION examples of descriptive statistics and they help us to explain the data more accurately and in greater detail than graphical display. However, it is always good to begin with graphical displays of the data to visually inspect the distribution; you should then confirm what was seen in the © Jones & Bartlett Learning, LLC © Jones & Bartlett Lea graphical displays with numeric descriptive statistics. Data can be distributed in many different ways depending uponFOR SALE OR DIS NOT NOT FOR SALE OR DISTRIBUTION where the average is located and how the data values differ. The center of the distribution can be located in the middle, but it may be shifted to the left or right. The data can present with a high peak, where most of the data values are close to each other, but they may be far different © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC from each other. Many statistical procedures we will discuss in later

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Introduction

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Histogram of incomes of recent graduates in family nurse practitioner (FNP) programs.

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© Jones & Bartlett Learning, LLC Mean = 46041.6 Std.FOR Dev. = 12864.517 NOT SALE OR DISTRIBUTION N = 343

Frequency

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30

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20

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0 s & Bartlett Learning, LLC 0 R SALE OR DISTRIBUTION

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© Jones60000 & Bartlett Learning, 20000 40000 80000 100000LLC NOT FOR SALE DISTRIBUTION Incomes of Recent Graduates in FamilyOR Nurse Practitioner (FNP) Programs

© Jones &assume Bartlett LLCnormal distribution, in©which Jones & Bartlett Lea chapters thatLearning, the data follow percentages dataDISTRIBUTION values are equal from the center of the distribuNOT FOR SALE OR DIS NOTtheFOR SALEofOR tion. Therefore, it is important to understand the characteristics of the normal distribution. We will present how to compute measures of central tendency and variability and how to interpret them correctly to describe the data. Bartlett Learning, LLC © Jones & Bartlett Learning, We will also explain how descriptive statistics are used to understand

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DESCRIPTIVE STATISTICS

© Jones & Bartlett Learning, LLC © Jones & Bartlett Lea NOT FOR SALE OR DIS NOTnormal FOR distributions. SALE OR DISTRIBUTION Three common measures of central tendency— mode, median, and mean—will be explained first. Then, the three common measures of variability, range, variance, and standard deviation, are discussed, followed by the characteristics of normal distributionLearning, and confidence intervals. Let us begin with example of the use Learning, LLC © Jones & Bartlett LLC © an Jones & Bartlett descriptive statistics from the real world. NOT FOR SALE OR DISTRIBUTION NOT FOR SALE of OR DISTRIBUTION ●

Case Study

s & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC Dr. Huey-Ming Tzeng (2011) reported results from a study designed to R SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION explore the perceptions of patients and their visitors on the importance of and response time to call lights on general medical/surgical units in a Veterans Administration Hospital. Dr. Tzeng noted that there is an

©established Jones &relationship Bartlett Learning, LLC © Jones & Bartlett Lea between the use of call lights and the incidence NOT NOT FOR SALE OR DISTRIBUTION of falls in acute care settings. The more patients use their call lights, the FOR SALE OR DIS less likely falls are to occur. Dr. Tzeng was interested in finding out the reasons for and nature of patient- and family-initiated call lights, call light use, and response time to call lights. Such a study could provide

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC a better understanding of call light use and support interventions to NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION encourage call light use and improve response times. Dr. Tzeng used descriptive statistics, measures of central tendency, and measures of variation to describe the results from the study. For example,

Dr. LLC Tzeng found that, on average,©patients used call light 3.66 timesLLC s & Bartlett Learning, Jones & their Bartlett Learning, R SALE OR DISTRIBUTION SALE OR DISTRIBUTION per day with a standard deviation NOT of 2.96.FOR We would understand that means a typical patient on a medical/surgical unit in this hospital used their call light about 3.66 times and most of these patients (68%) used their call light between 0.70 times and 6.62 times. Descriptive statistics, such as the

© Jones & Bartlett Learning, LLC © Jones & Bartlett Lea arithmetic mean and the standard deviation, help us to understand both NOT FOR SALE OR DIS NOT FOR SALE OR DISTRIBUTION the typical case and the range of cases. Findings like this can be used by the researcher, advanced practice nurse, and nurse executive for a variety of purposes: evidence of the need for further research, support for improving practice, or assigningLLC resources to manage a problem support & a solution. © Jones & Bartlett Learning, ©orJones Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION ●

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Measures of Central Tendency

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MEASURES OF CENTRAL TENDENCY

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The following are some of the example statements we can find in a newspaper and/or published journal articles. © Jones & Bartlett Learning, Bartlett Learning, LLC

© Jones & LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION The average annual premium for employer-sponsored health insur-

ance in 2011 are $5,429 for single coverage and $15,703 for family coverage (Kaiser Family Foundation, Health Research & Educational Trust, 2011). rating for& theBartlett study sample was 5.2 onLLC s & Bartlett Learning, LLCThe average job satisfaction © Jones Learning, a 7-point scale (Kovner, Brewer, Fairchild, Poornima, Kim, & Djukic, R SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION 2007). The average blood pressure for all patients at the beginning of the study was 159/94 mmHg (Kershner, 2011).

All of these statements have used aLLC single number to describe the data, © Jones & Bartlett Learning, © Jones & Bartlett Lea it helps us inOR understanding the data in terms of “average.” ThereFOR SALE OR DIS NOT NOTand FOR SALE DISTRIBUTION are multiple ways of computing and presenting averages, but we will describe the three most commonly used measures of central tendency: mode, median, and mean.

© Jones & Bartlett Learning, LLC The Mode NOT FOR SALE OR DISTRIBUTION

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The mode is simply the most frequently occurring number in a given data set. For example, let us take a look at the following data set of seven systolic blood pressure (SBP) measurements:

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120 114 116 117 114SALE 121 OR 124DISTRIBUTION NOT FOR

Notice that 114 appears twice, where the other measurements appear only one time. Therefore, the SBP measurement of 114 will be the mode in this data set since it is the most frequently occurring value. This distribution is called unimodal only © Jones & Bartlett Learning, LLCdistribution since there©is Jones & Bartlett Lea one mode. Note, however, that it is possible to have more than one NOT FOR SALE OR DIS NOT FOR SALE OR DISTRIBUTION mode in a given data set. To explain, let us take a look at the following data set: 117 120 114 116 117 114 121 124

© Jones & Bartlett © they Jones Bartlett ThisLearning, data set has LLC two modes, 114 and 117, since each& appear twice Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION where the others appear only once. When a data set has two modes, © Jones & Bartlett Learning, LLC. NOT FOR SALE OR DISTRIBUTION.

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© Jones & Bartlett Learning, LLC © Jones & Bartlett Lea NOT SALE OR DIS NOTit FOR SALE OR DISTRIBUTION is known as bimodal distribution; a multimodal distribution is FOR a distribution with more than two modes in a data set. As you probably noticed by now, the mode is useful primarily for variables measured at the nominal level since it is merely the most frequently occurring ForJones example, we have Learning, LLC © Jones & Bartlett Learning, LLCnumber in the data set. © & ifBartlett the following numbers to the sex ofNOT participants, 1 for men NOT FOR SALE assigned OR DISTRIBUTION FOR SALE OR DISTRIBUTION and 2 for women, and out of a sample of 100 there are 75 women, the mode is 2. The mode will not be useful with continuous levels of measurement, or as the data set gets larger.

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The Mean

The arithmetic mean (often called the average) is the sum of all data values in a data set divided by the number of data values and is shown Jones & Bartlett Learning, LLC © Jones in the following equation:

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Sum of all data values number of data values

Mean 

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The mean involves the minor mathematical operations of addition and division, and so is not an appropriate measure central & tendency for Learning, LLC © Jones & Bartlett Learning, LLC ©ofJones Bartlett nominal levels of measurement. For example,NOT it is impossible to find NOT FOR SALE OR DISTRIBUTION FOR SALE OR DISTRIBUTION the mean for the variable political affiliations, with categories of Republican, Independent, and Democratic. The meaning of the mean will only makes sense when a variable’s measurements can be quantifiable, such as in interval and ratio levels of measurement. s & Bartlett Learning, LLC © Jones Learning, LLC Let us consider the following data set & of Bartlett sodium content level meaR SALE OR DISTRIBUTION sured in milligrams per liter:NOT FOR SALE OR DISTRIBUTION 20

18

16

22

27

11

For this data set, the mean will be

© Jones & Bartlett(20 Learning, © Jones & Bartlett Lea  18  16LLC  22  27  11)  19 Mean  NOT FOR SALE OR DIS NOT FOR SALE OR DISTRIBUTION 6 We have computed a mean of 19 for a group of 6 sodium content levels. How should we interpret this finding? Remember, the mean is the average score in the data set. Therefore, the mean of 19 tells us that Bartlett Learning, LLC © Jones & Bartlett Learning, there is, on average, 19 mg of sodium per liter in the data set.

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Measures of Central Tendency

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The Median

The median is the exact middle value in a distribution, which divides the data set into two exact halves. Let us consider the following data Bartlett © Jones & Bartlett Learning, set, Learning, which consistLLC of five income levels for registered nurses:

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35,000 39,500 42,000 47,500 52,000

In this data set, the value of 42,000 is the median, since it divides the data set into exact two halves with an equal number of values below s & Bartlett Learning, and LLCabove it. © Jones & Bartlett Learning, LLC Notice the data set is ordered from the smallest to the largest data R SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION value. However, correctly finding the middle value may be difficult and misleading if the data values are not ordered consecutively. Consider the following data set:

47,500 39,500 32,000 52,500 42,000 © Jones & Bartlett Learning, LLC © Jones & Bartlett Lea will not make OR senseDISTRIBUTION to choose 32,000 and report it as the median, NOT FOR SALE OR DIS NOTIt FOR SALE since it is the smallest data value in this data set. Therefore, ordering the data from the smallest to the largest (or vice versa) is the first and the most important step in finding the median of any given data set. After ordering, it is easy to see that the median for this data set should be 39,500. © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC Notice also that the previous two data files had odd numbers of NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION data values. Finding the median in a data set with an odd numbers of values is easy since you will end up with an equal number of data values above and below the median. However, it is less straightforward to find the median when there are an even numbers of data values in data set. Let us take a look the following data set: Bartlett Learning, the LLC ©atJones & Bartlett Learning, LLC

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The data values represent age in years of six individuals and there is no such number that divides this data set into two exact halves. Theoretically, such a number should be between 32 and 35, leaving three data © Jones & Bartlett Learning, LLC © Jones & Bartlett Lea values above and below it. However, such a number does not actually NOT FOR SALE OR DIS NOTexist FOR SALE OR in the data set. In DISTRIBUTION this case, you will sum the two middle numbers, 32 and 35, and divide the sum by 2. You are basically computing the average of those two middle values as the median, which is:

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32  35  33.5 2

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© Jones & Bartlett Learning, LLC © Jones & Bartlett Lea NOT NOTThis FOR SALE OR DISTRIBUTION value of 33.5 as the median makes sense since we have an equalFOR SALE OR DIS number of data values above and below it.

Choosing a Measure of Central Tendency

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE We ORhave DISTRIBUTION NOT FOR SALE OR DISTRIBUTION discussed three types of central tendency—the mode, the

mean, and the median—and examined how they differ in terms of finding the center of a data distribution. The next legitimate question to ask may be “When do we use which measure?” The mode is simply the most frequently occurringLearning, data valuesLLC in s & Bartlett Learning, LLC © Jones & Bartlett the data set. Therefore, it isNOT mainly useful for the nominal level of R SALE OR DISTRIBUTION FOR SALE OR DISTRIBUTION measurement. Both median and mean are useful when the variable being measured can be quantified. However, one important thing to note here is that the mean is extremely sensitive to unusual cases. To explain this further, let us consider the following data sets:

© Jones & Bartlett Learning, LLC Data set DISTRIBUTION #1: 108 112 116 120 124 NOT FOR SALE OR

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Data set #2: 108 112 116 120 205 In both data sets, the median is 116, as it is the number that divides the data set into two exact halves. However, you will notice that the © Jones & Bartlett Learning, LLCin both data sets. For the©first Jones & Bartlett mean is not identical data set, the mean Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION is equal to 108  112  116  120  124  116 5 where the mean of the second data set is equal to

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108  112  116  FOR 120 SALE 205 OR DISTRIBUTION NOT  132.5 5

Notice how the mean of the second data set has been influenced by the presence of an unusual case in the data set. If we were to say the mean © Jones Learning, LLC © Jones & Bartlett Lea is equal&toBartlett 132.5 for the second data set and it represents a typical case, will not make much sense because the majority of data values NOTareFOR SALE OR DIS NOTthis FOR SALE OR DISTRIBUTION less than 120. Therefore, the mean should not be used when unusual, or outlying, data values are present in the data set, as the mean tends to be extremely sensitive to the unusual values. Rather, the median should be reported in this case. This is why the average housing price is Bartlett Learning, LLC © Jones & Bartlett Learning, LLC always reported with the median, since even one million-dollar house

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© Jones & Bartlett Learning, LLC © Jones & Bartlett Lea NOT NOTcan FOR SALE OR DISTRIBUTION distort the average housing price when most of the houses are inFOR SALE OR DIS $200,000–$350,000 range.

MEASURES OF VARIABILITY © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC ● NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION

Measures of central tendency allow us to know the typical value in the data set. However, we know that when we measure a variable, there will be differences between and among the values in the data set. For example, if we were measuring systolic blood pressure among a group s & Bartlett Learning, of LLC © Jones & Bartlett Learning, LLC research participants, we would expect that there would be a range of R SALE OR DISTRIBUTION NOT FORwe SALE DISTRIBUTION values between individuals. Furthermore, wouldOR expect similar variation on systolic blood pressure measurements in any given individual participant. In other words, some level of variation among data values in any data set is expected. Given this expected variation, we might ask, “How accurate isLearning, the measureLLC of central tendency?” The computed © Jones & Bartlett © Jones & Bartlett Lea measure of central tendency will be most accurate when the dataNOT valuesFOR SALE OR DIS NOT FOR SALE OR DISTRIBUTION vary only a little, but accuracy of the mean declines as the variation in data values increases. Measures of variability provide information about the spread of scores and indicate how well a measure of central tendency represents the “middle/average” value in the data set. There are multiple © Jones & Bartlett Learning, LLC & Bartlett ways of computing and presenting variability,© butJones we describe the four Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION that are most commonly used: range, interquartile range, variance, and standard deviation.

Range s & Bartlett Learning, LLC R SALE OR DISTRIBUTION

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Range is the difference between the largest and the smallest values in the data set. For example, suppose a researcher measured patients’ level of pain after vascular surgery on a scale of 1 to 10. These data are shown below.

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The first step is to sort the data from the smallest to the largest values, as it will make our job of finding these two values easy. After sorting, the range of this data set is 9  2  7. is simple to calculate. However, we should be cautious about Bartlett Range Learning, LLC © Jones & Bartlett Learning, using range as a measure of variability. As seen in the previous example,

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© Jones & Bartlett Learning, LLC © Jones & Bartlett Lea NOT NOTthe FOR SALE OR DISTRIBUTION range is calculated simply by subtracting the smallest value fromFOR SALE OR DIS the highest value. In addition, it allows us to understand what the collected data set looks like. However, the range is a very crude measure of variability as it only uses the highest and lowest values in computation. Therefore, itLLC does not accurately capture© information how Learning, LLC © Jones & Bartlett Learning, Jones &about Bartlett in the set differ if the data set contains anFOR unusual value(s). NOT FOR SALE data OR values DISTRIBUTION NOT SALE OR DISTRIBUTION Consider the following data set. 3

4

2

3

3

4

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9

This data set is still a collection of pain level measurements of pa-

s & Bartlett Learning, tients LLC who went under vascular © Jones & Bartlett Learning, LLC surgery, but notice that the value of R SALE OR DISTRIBUTION NOT FOR SALE 9 seems unusual in this data set. Here, the rangeOR is 9 DISTRIBUTION  2  7 after

sorting. Does this make sense? Most of the values are between 2 and 4 and claiming the variability is 7 does not really make sense in the context of this data set. It is clear that the range is extremely sensitive to the & unusual dataLearning, values. To get around this problem sometimes © Jones Bartlett LLC © Jones & Bartlett Lea researchers will simply report the range as the lowest and NOT highestFOR SALE OR DIS NOT FOR SALE OR DISTRIBUTION values, “reports of pain intensity ranged from three to nine”, rather than computing a range.

© Jones & Bartlett Learning, LLC Interquartile Range NOT FOR SALE OR DISTRIBUTION

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Interquartile range is the difference between the 75th percentile and 25th percentile. As we saw in chapter 5, the percentile is a measure of location and tells us how many data values fall below a certain of observations. Therefore, the 25th percentile is the data s & Bartlett Learning, percentage LLC © Jones & Bartlett Learning, LLC value that the bottom 25% falls below and the 75th percentile is the R SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION data value that the bottom 75% falls below. In results, the interquartile range is less sensitive to an unusual case(s) in the data set as it does not use the smallest and the largest value. For example, suppose the number of patient falls per week at a local nursing home have © Jones & Bartlett Learning, LLC © Jones & been measured.

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Note that the data set has already been sorted from the smallest to the largest. It is easier to find the median first and then to find 25th and 75th percentiles, since it less straightforward to directly identify the Bartlett Learning, LLC © Jones & Bartlett Learning, percentiles.

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© Jones & Bartlett Learning, LLC © Jones & Bartlett Lea NOT FOR SALE OR DIS NOT FOR SALE OR DISTRIBUTION The median of this data set is 3, since 3 is the exact middle that divides this data set into two exact halves. From the median, the 25th percentile is equal to 2 and the 75th percentile is equal to 4, as they divide the lower and upper halves of the data set into two exact halves, respectively. The LLC interquartile range is then the between the Learning, LLC © Jones & Bartlett Learning, ©difference Jones & Bartlett percentile and the 75th percentile, which is 4 FOR 2  2. NOT FOR SALE 25th OR DISTRIBUTION NOT SALE OR DISTRIBUTION Let us now consider the next data set 1

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24

you can see, it is the same data set as before, except the highest s & Bartlett Learning, As LLC © Jones & Bartlett Learning, LLC value, 24, which seems to be an unusual value. Notice that the interR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION quartile range is still 4  2  2 and is not affected by the unusual data value. Therefore, interquartile range is not as sensitive to unusual or outlying values as the standard range.

© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION Variance and Standard Deviation

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While range provides a rough estimate of the variability of a data set, it does not use all of the data values in computation and is very sensitive to an unusual value in the data set. Interquartile range is& anBartlett improve- Learning, LLC © Jones & Bartlett Learning, LLC © Jones ment, but still does not account for every data value in the set. On the NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION other hand, the next two measures of variability, variance and standard deviation, use all of the data values in the set in computation and may capture information about variability more precisely than the range or the interquartile range. As standard deviation is simply the square root variance, we will explain variance first.& Bartlett Learning, LLC s & Bartlett Learning, of LLC © Jones Variance is the average amount that dataSALE values differ from the mean R SALE OR DISTRIBUTION NOT FOR OR DISTRIBUTION and is computed with the following formula: Population Variance 

g (X  m) 2 N

© Jones & Bartlett Learning, LLC g (X  X) 2 Variance  NOT FOR SALE ORSample DISTRIBUTION n1

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In this equation we compute the difference between each raw value and the mean (X  X), square it, sum ( g ) those values and then divide by the total number of values in the data set (n). Note that the Bartlett Learning, LLC © Jones & Bartlett Learning, denominator will be changed to n  1 when working with samples.

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Box ●

Degrees of Freedom

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC Calculations of variance and many other statistics require an estimate of the range NOT FOR SALE OR DISTRIBUTION NOTdegrees FOR of SALE OR DISTRIBUTION of variability, known as degrees of freedom. From a sample, freedom

are always equal to n  1. Here is an analogy that might help: Envision a beverage holder from any fast food restaurant—most of these hold four drinks. In this case, the degrees of freedom would be equal to 4  1, or 3. As each section of the holder is occupied by a drink, there is a chance of varying what section of the holder any s & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC given drink is placed, top left or top right for example, until three of the sections R SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION are filled; at this point, there is only one section left where a drink may be placed and no variation is possible. Each statistical test or calculation has a variation of degrees of freedom. Watch for these throughout the book.

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Consider the following data set of toddler weights in an outpatient clinic to explain how to compute the variance, assuming that the data values were taken from a population:

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The computation steps are shown in Table 6-1. Computed variance for this data set is 5.84. What does this mean? In fact, we cannot use this as a measure of variability. Let us asweight losses measured in pounds s & Bartlett Learning, sume LLC that the values represent © Jones & Bartlett Learning, LLC taken from five subjects. Because the deviation of each observation R SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION from the mean has been squared, the unit for the variance is now in (pound)2. What does (pound)2 mean? If we were to say that data values differ from the mean on average about 5.84 (pound)2, would this claim make sense? Probably not, since there is no such a unit 2. © Jones & Bartlett Learning, LLC © Jones & Bartlett Lea as a (pound) 2 will Why do we then the square of the deviation if the (unit) NOT FOR SALE OR DIS NOT FOR SALE ORtake DISTRIBUTION not make sense to interpret at the end? The answer is simple: If you do not square the deviation and sum each deviation, it will always add up to zero no matter what data set you work with. We suggest you to try this with small data sets you can find in this textbook or other

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Measures of Variability

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Variance 

Step 4

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g (X  X) 2 29.2  N 5  5.84

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113

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78  15.6 5 x

X 17 12 14 16 19

g X  78

Step 2

6-1

Table



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Step 1

How to Compute the Variance

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XX 1.4 3.6 1.6 0.4 3.4

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g (X  X)2  29.2

(X  X)2 1.96 12.96 2.56 0.16 11.56

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Step 3

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© Jones & Bartlett Learning, LLC © Jones & Bartlett Lea NOTsources. FOR SALE OR DISTRIBUTION How can we then talk about variability if the measureNOT of vari-FOR SALE OR DIS

ability comes out to be equal to zero? This is why we take square of the deviation to compute the variance first and then take square root of it to compute the standard deviation, bringing us back to the original unitLearning, of measurement. © Jones & Bartlett LLC © Jones & Bartlett Learning, LLC WeDISTRIBUTION get the standard deviation of 2.42 by taking of 5.84; NOT FOR SALE OR NOTsquare FORroot SALE OR DISTRIBUTION we can then say that the data values differ from the mean (15.60 lbs.) on an average of about 2.42 pounds. We can interpret this finding to mean that, on average, about two thirds of the weights fall between 13.18 and 18.02 pounds. This makes more sense when you look at the s & Bartlett Learning, data LLCset, compared to the variance. © Jones & Bartlett Learning, LLC Note that the mean and standard R SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION deviation should always be reported together!

Choosing a Measure of Variability

© Jones & Bartlett Learning, LLC © Jones & Bartlett Lea NOT FOR SALE OR DIS NOTWeFOR SALE OR DISTRIBUTION have shown you how to compute three measures of variability—

range, interquatile range, and variation and standard deviation—and how they differ. Like the measures of central tendency, the next legitimate question to ask is, “When do we use which?” should use the range only as a crude©measure, it is ex- Learning, LLC © Jones & Bartlett You Learning, LLC Jones since & Bartlett tremely sensitive to unusual values in the data set. Interquartile rangeOR is DISTRIBUTION NOT FOR SALE OR DISTRIBUTION NOT FOR SALE not as sensitive to unusual data values, where standard deviation is very sensitive to unusual values. Therefore, the interquartile range should be used with the median when the data contain unusual data values. However, the standard deviation should be used with the mean when Bartlett Learning, the LLC © Jones data are free of unusual data values. & Bartlett Learning, LLC

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Obtaining Measures of Central Tendency and Variability in SPSS

© Jones & Bartlett Learning, LLC © Jones & Bartlett Lea NOTofFOR SALE OR DIS NOTThere FOR DISTRIBUTION areSALE severalOR places in SPSS where you can request measures

central tendency and variability. To obtain these measures, go to Analyze  Descriptive statistics. In the next menu, choose “Frequencies” (Figure 6-2). a variable(s) Figure&6-3. Of the Learning, Bartlett Move Learning, LLC of interest, as shown©inJones Bartlett three buttons on the right side of the window, select “Statistics” (see

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Measures of Variability

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© Jones & Bartlett Learning, LLC © Jones & Bartlett Lea NOT NOTFigure FOR6-4). SALE OR DISTRIBUTION You can select measures of both central tendency andFOR SALE OR DIS variability to obtain the measures to suit your needs. The same measures can be obtained by choosing “Descriptives” or “Explore” under Analyze  Descriptive pull-down menu. Note also thatLearning, these measures of central tendency and variability can&beBartlett obtained Learning, LLC © Jones & Bartlett LLC © Jones within windows for several other statistical procedures. NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION



Figure 6-3

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Descriptive statistics helps us understand whether the distribution of a continuously measured variable is normal. Figure 6-5 is an example © Jones & Bartlett Learning, LLC © Jones & Bartlett normal distribution of a variable, age. Some notable characteristics of Learning, LLC NOT FOR SALE OR DISTRIBUTION normal distribution are summarized below. NOT FOR SALE OR DISTRIBUTION

Box ●

Characteristics of Normal Distribution s & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC R SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION

• It is bell-shaped and symmetric. • The area under a normal curve is equal to 1.00 or 100%. • 68% of observations fall within one standard deviation from the mean in directions. © both Jones & Bartlett Learning, LLC © Jones & Bartlett Lea • 95% of observations fall within two standard deviations from the mean in NOT FOR SALE OR DIS NOT FOR SALE OR DISTRIBUTION both directions. • 99.7% of observations fall within three standard deviations from the mean in both directions. • Many normal distributions exist with different means and standard deviations. Bartlett Learning, LLC © Jones & Bartlett Learning, LLC

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Normal Distribution

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© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION 1

2

3

4

5

6

7

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9 10 11 12 Age (Years)

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16 17

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19

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE ORWhen DISTRIBUTION FOR SALE OR DISTRIBUTION a normal distribution is said to beNOT symmetrical, it means

that the area on both sides of the distribution from the mean is equal; in other words, 50% of the data values in the set are smaller than the mean and the other 50% are larger than the mean. In a normal distrihighest peak of the distribution and s & Bartlett Learning, bution, LLC the mean is located at©the Jones & Bartlett Learning, LLC the spread of a normal distribution can be presented in terms of the R SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION standard deviation. No data will ever be exactly/perfectly normally distributed in reality. If so, how do we know whether or not a collected data set is normally distributed? We can begin with a visual display of the data in © Jones & Bartlett © Jones & Bartlett Lea a histogram to see ifLearning, the data set LLC is normally distributed. However, a check, alone, may not be sufficient to know whether the data NOTareFOR SALE OR DIS NOTvisual FOR SALE OR DISTRIBUTION normally distributed. There are statistical measures, skewness and kurtosis, which, along with a histogram, allow us to determine whether the set is normally distributed. Skewness is a measure of whether the set is symmetrical or off-center, which means probabilities on both © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC sides of the distribution are not the same. Kurtosis is a measure of

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© Jones & Bartlett Learning, LLC 68% NOT FOR SALE OR DISTRIBUTION 95% 99.7%

© Jones & Bartlett Learning, LLC μ − 3σ μ − 2σ μ−σ μ μ+σ NOT FOR SALE OR DISTRIBUTION

μ + 2σ

μ + 3σ

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how peaked a distribution is. A distribution is said to be “normal”

© Jones & Bartlett Learning, LLCof skewness and kurtosis©fallJones &1 Bartlett when both measures between and 1 Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION range and non-normal if both measures fall either below 1 or above

1. Note that these measures can be selected in the same window as measures of central tendency and variability, which we just discussed. Figure 6-6 shows how much percentage of the set falls within how many standard deviations away from the mean. If a variable follows a s & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC normal distribution, these rules can be applied to understand the disR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION tribution of the variable in terms of the mean and the standard deviation. In addition, different normal distributions can be found when the mean and the standard deviation are defined as shown in Figure 6-7 and Figure 6-8. © Jones Learning, © Jones & Bartlett Lea Why&doBartlett we then care about thisLLC normal distribution so much? The important is that many human characteristics fall NOT into anFOR SALE OR DIS NOTmost FOR SALE reason OR DISTRIBUTION approximately normal distribution and that the measurement scores are assumed to be normally distributed when running most statistical analyses. Therefore, the statistical results you get at the end may not be trustworthy if the variable is not normally distributed. We will discuss © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC this more in Chapter 8.

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Normal Distribution

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Normal distributions with different means.

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75

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79 © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION

Let us consider an example where a student looks at their final exam scores in their statistics and research courses. The student scored 79 out of 100 on the final exam in statistics on © Jones & Bartlett Learning, LLC course and 40 out of©60 Jones & Bartlett Lea the final exam in the research course. Can the student conclude that NOT FOR SALE OR DIS NOT FOR SALE OR DISTRIBUTION their performance was better in statistics because of the higher score in the statistics course than the research course? Before making such a conclusion, the student will need to examine the distribution of scores on the two final exams. Let us assume that the final exam in statistics

© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION Figure 6-8

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Normal distributions with different standard deviations.

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75

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© Jones & Bartlett Learning, LLC © Jones & Bartlett Lea NOT NOThad FOR SALE OR DISTRIBUTION a mean of 75 with a standard deviation of 3, and the final examFOR SALE OR DIS

in research had a mean of 40 with a standard deviation of 2.5. It seems that the student did better than the average in both classes, but it is still difficult to judge in which course the student performed better. ThisLearning, question cannot different&normal dis- Learning, LLC © Jones & Bartlett LLCbe directly answered using © Jones Bartlett because they have different means NOT and standard deviations NOT FOR SALE tributions OR DISTRIBUTION FOR SALE OR DISTRIBUTION (i.e., they are not on identical scale, which is necessary to make direct comparisons). We need to somehow put these two different distributions on the same scale so that we can make a legitimate comparison of the stus & Bartlett Learning, dent’s LLC performance; a standard © Jones & Bartlett Learning, LLC normal distribution is the solution. R SALE OR DISTRIBUTION NOT FOR SALE OR all DISTRIBUTION By definition, a normal distribution is one in which scores have been put on the same scale (standardized). These standardized scores (also known as z-scores) represent how far below or above the mean a given score falls and allows us to determine percentile/probabilities associated with a given score. © Jones & Bartlett Learning, LLC © Jones & Bartlett Lea Figure 6-9 shows a graphical transition from a general normal dis-FOR SALE OR DIS NOT NOT FOR SALE OR DISTRIBUTION tribution to a standard normal distribution. Characteristics of the standard normal distribution are summarized below. To compute a z-score, you will need two pieces of information about a distribution: the mean and the standard deviation. Z-scores (stan© Jones & Bartlett Learning, © Jones & Bartlett dardized scores)LLC are computed using the following equation and cal- Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE culated such that positive values indicate how far above the meanOR a DISTRIBUTION score falls and negative values indicate how far below the mean a score falls. Whether positive or negative, larger z-scores mean that scores are

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Transition from a general normal distribution to a standard normal distribution. Z=

X−μ

=

75 − 75

=0

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σ = 3.2

75 © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION

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σ=1

0 © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION

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Normal Distribution

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Box ●

Characteristics of the Standard Normal Distribution

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC • The standard normal distribution has a mean of 0 and standard deviation NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION of 1. • The area under the standard normal curve is equal to 1 or 100% • Z-scores have associated probabilities, which are fixed and known.

s & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC R SALE OR DISTRIBUTION NOT FOR SALE OR that DISTRIBUTION far away from the mean and smaller z-scores means scores are close to the mean. Z

Xm s

© Jones & Bartlett Learning, LLC © Jones & Bartlett Lea population mean (m) is subtracted from the raw scoreFOR SALE OR DIS NOT NOTWhere FORthe SALE OR DISTRIBUTION

and divided by the population standard deviation (s). When do you think z-scores will be computed with positive or negative sign? Z-scores will be positive when a student performs better than the mean on a test—the numerator of the equation above will be posi© Jones & Bartlett LLC Jones & Bartlett tiveLearning, and be above the mean. On the other©hand, z-scores will be Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION negative when as student performs below the mean. Let us consider an example test, again a statistics final exam, with a mean of 78 and standard deviation of 3. Suppose Brian has a final exam score of 84. His z-score will be

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Z

X ©mJones 84 &78Bartlett Learning, LLC 2  sNOT FOR 3 SALE OR DISTRIBUTION

What does Brian’s z-score of 2 mean in terms of his performance relative to the average person who took this statistics final exam? First, we can see that Brian did perform better than the average person on © Jones & Bartlett Learning, LLC & Bartlett Lea this final exam. Second, his z-score of 2 tells us that his score©isJones two deviations the average score of 78 since a standard NOT FOR SALE OR DIS NOTstandard FOR SALE ORabove DISTRIBUTION normal distribution has a standard deviation of 1. However, this second point about Brian’s score does not really make perfect sense to us yet. From Figure 6-10, we can see that Brian seems to perform better than a number of students in his class. However, we still do not know Bartlett Learning, LLC © Jones & Bartlett Learning, LLC exactly how much better he did. To find out the exact percentile rank of

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Brian’s z-score.

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© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION Brian’s Score

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© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION 0

2

© Jones Bartlett Learning, © 6-11. Jones & Bartlett Lea another&student, Sam, we need toLLC use a z table, shown in Figure NOTareFOR SALE OR DIS NOTSteps FOR OR DISTRIBUTION in SALE using the z table to find a corresponding percentile rank summarized below. Let us consider another example that will help us understand how to find the corresponding probability for a given score. The sodium intakes for a group of obese patients at a local known to Learning, LLC © Jones & Bartlett Learning, LLC © hospital Jones are & Bartlett have a mean of 4,500 mg/day and a standard deviation of /150 NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION mg/day. Assuming that the sodium intake is normally distributed, let us find the probability that a randomly selected obese patient will have a sodium intake level below 4,275 mg/day. First, we need

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Using the z Table to Find a Corresponding Percentile Rank of a Score

1. Convert a score to corresponding z-score. © Jones & Bartlett Learning, LLC © Jones & Bartlett Lea 2. Locate the row in the z table for a z-score of 2.00. Note that the z-scores in NOT FOR SALE OR DIS NOT FOR SALE OR DISTRIBUTION the first column are shown in only the first decimal. Locate also the column for .00 so that you get 2.00 when you add 2.0 and .00. 3. Sam’s z-score of 2.00 gives probabilities of .9772 to the left. 4. Therefore, Sam’s final exam score of 2.00 corresponds to the 98th percentile. Sam did better than Bartlett Learning, LLC98% of students in the class. © Jones & Bartlett

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Figure 6-11 ©zJones & Bartlett LLC Jones & Bartlett Lea table. Source: Gerstman,Learning, B. (2008). Basic biostatistics: Statistics for public © health practice. Sudbury, MA:OR JonesDISTRIBUTION and Barlett. NOT FOR SALE OR DIS NOT FOR SALE



© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION

© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION

Table Entry

s & Bartlett Learning, LLC z R SALE OR DISTRIBUTION

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0

z tenths

.00

.01

.02

.03

hundredths .04 .05

.06

.07

.08

.09

3.4 3.3 3.2 3.1 3.0 2.9 2.8 2.7 2.6 2.5 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

.0003 .0005 .0007 .0010 .0013 .0019 .0026 .0035 .0047 .0062 .0082 .0107 .0139 .0179 .0228 .0287 .0359 .0446 .0548 .0668 .0808 .0968 .1151 .1357 .1587 .1841 .2119 .2420 .2743 .3085 .3446 .3821 .4207 .4602 .5000

.0003 .0005 .0007 .0009 .0013 .0018 .0025 .0034 .0045 .0060 .0080 .0104 .0136 .0174 .0222 .0281 .0351 .0436 .0537 .0655 .0793 .0951 .1131 .1335 .1562 .1814 .2090 .2389 .2709 .3050 .3409 .3783 .4168 .4562 .4960

.0003 .0005 .0006 .0009 .0013 .0018 .0024 .0033 .0044 .0059 .0078 .0102 .0132 .0170 .0217 .0274 .0344 .0427 .0526 .0643 .0778 .0934 .1112 .1314 .1539 .1788 .2061 .2358 .2676 .3015 .3372 .3745 .4129 .4522 .4920

.0003 .0004 .0006 .0009 .0012 .0017 .0023 .0032 .0043 .0057 .0075 .0099 .0129 .0166 .0212 .0268 .0336 .0418 .0516 .0630 .0764 .0918 .1093 .1292 .1515 .1762 .2033 .2327 .2643 .2981 .3336 .3707 .4090 .4483 .4880

.0003 .0004 .0006 .0008 .0012 .0016 .0023 .0031 .0041 .0055 .0073 .0096 .0125 .0162 .0207 .0262 .0329 .0409 .0505 .0618 .0749 .0901 .1075 .1271 .1492 .1736 .2005 .2296 .2611 .2946 .3300 .3669 .4052 .4443 .4840

.0003 .0004 .0006 .0008 .0011 .0015 .0021 .0029 .0039 .0052 .0069 .0091 .0119 .0154 .0197 .0250 .0314 .0392 .0485 .0594 .0721 .0869 .1038 .1230 .1446 .1685 .1949 .2236 .2546 .2877 .3228 .3594 .3974 .4364 .4761

.0003 .0004 .0005 .0008 .0011 .0015 .0021 .0028 .0038 .0051 .0068 .0089 .0116 .0150 .0192 .0244 .0307 .0384 .0475 .0582 .0708 .0853 .1020 .1210 .1423 .1660 .1922 .2206 .2514 .2843 .3192 .3557 .3936 .4325 .4721

.0003 .0004 .0005 .0007 .0010 .0014 .0020 .0027 .0037 .0049 .0066 .0087 .0113 .0146 .0188 .0239 .0301 .0375 .0465 .0571 .0694 .0838 .1003 .1190 .1401 .1635 .1894 .2177 .2483 .2810 .3156 .3520 .3897 .4286 .4681

.0002 .0003 .0005 .0007 .0010 .0014 .0019 .0026 .0036 .0048 .0064 .0084 .0110 .0143 .0183 .0233 .0294 .0367 .0455 .0559 .0681 .0823 .0985 .1170 .1379 .1611 .1867 .2148 .2451 .2776 .3121 .3483 .3859 .4247 .4641

.0003 .0004 .0006 .0008 .0011 .0016 .0022 .0030 .0040 .0054 .0071 .0094 .0122 .0158 .0202 .0256 .0322 .0401 .0495 .0606 .0735 .0885 .1056 .1251 .1469 .1711 .1977 .2266 .2578 .2912 .3264 .3632 .4013 .4404 .4801

© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION

© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION

s & Bartlett Learning, LLC R SALE OR DISTRIBUTION

© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION

© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION

© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION

© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION

© Jones & Bartlett Lea NOT FOR SALE OR DIS

Cumulative probabilities computed with Microsoft Excel 9.0 NORMSDIST function.

© Jones & Bartlett Lea NOT FOR SALE OR DIS

© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION

© Jones & Bartlett Learning, LLC. NOT FOR SALE OR DISTRIBUTION.

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Figure 6-11 ©Continued Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION



Table Entry © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION

s & Bartlett Learning, LLC R SALE OR DISTRIBUTION z tenths 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4

.00 .5000 .5398 .5793 .6179 .6554 .6915 .7257 .7580 .7881 .8159 .8413 .8643 .8849 .9032 .9192 .9332 .9452 .9554 .9641 .9713 .9772 .9821 .9861 .9893 .9918 .9938 .9953 .9965 .9974 .9981 .9987 .9990 .9993 .9995 .9997

.01 .5040 .5438 .5832 .6217 .6591 .6950 .7291 .7611 .7910 .8186 .8438 .8665 .8869 .9049 .9207 .9345 .9463 .9564 .9649 .9719 .9778 .9826 .9864 .9896 .9920 .9940 .9955 .9966 .9975 .9982 .9987 .9991 .9993 .9995 .9997

0

z

.02 .5080 .5478 .5871 .6255 .6628 .6985 .7324 .7642 .7939 .8212 .8461 .8686 .8888 .9066 .9222 .9357 .9474 .9573 .9656 .9726 .9783 .9830 .9868 .9898 .9922 .9941 .9956 .9967 .9976 .9982 .9987 .9991 .9994 .9995 .9997

.03 .5120 .5517 .5910 .6293 .6664 .7019 .7357 .7673 .7967 .8238 .8485 .8708 .8907 .9082 .9236 .9370 .9484 .9582 .9664 .9732 .9788 .9834 .9871 .9901 .9925 .9943 .9957 .9968 .9977 .9983 .9988 .9991 .9994 .9996 .9997

© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION

© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION hundredths .04 .05 .5160 .5199 .5557 .5596 .5948 .5987 .6331 .6368 .6700 .6736 .7054 .7088 .7389 .7422 .7704 .7734 .7995 .8023 .8264 .8289 .8508 .8531 .8729 .8749 .8925 .8944 .9099 .9115 .9251 .9265 .9382 .9394 .9495 .9505 .9591 .9599 .9671 .9678 .9738 .9744 .9793 .9798 .9838 .9842 .9875 .9878 .9904 .9906 .9927 .9929 .9945 .9946 .9959 .9960 .9969 .9970 .9977 .9978 .9984 .9984 .9988 .9989 .9992 .9992 .9994 .9994 .9996 .9996 .9997 .9997

© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION

© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION

s & Bartlett Learning, LLC R SALE OR DISTRIBUTION

.06 .5239 .5636 .6026 .6406 .6772 .7123 .7454 .7764 .8051 .8315 .8554 .8770 .8962 .9131 .9279 .9406 .9515 .9608 .9686 .9750 .9803 .9846 .9881 .9909 .9931 .9948 .9961 .9971 .9979 .9985 .9989 .9992 .9994 .9996 .9997

.07 .5279 .5675 .6064 .6443 .6808 .7157 .7486 .7794 .8078 .8340 .8577 .8790 .8980 .9147 .9292 .9418 .9525 .9616 .9693 .9756 .9808 .9850 .9884 .9911 .9932 .9949 .9962 .9972 .9979 .9985 .9989 .9992 .9995 .9996 .9997

.08 .5319 .5714 .6103 .6480 .6844 .7190 .7517 .7823 .8106 .8365 .8599 .8810 .8997 .9162 .9306 .9429 .9535 .9625 .9699 .9761 .9812 .9854 .9887 .9913 .9934 .9951 .9963 .9973 .9980 .9986 .9990 .9993 .9995 .9996 .9997

.09 .5359 .5753 .6141 .6517 .6879 .7224 .7549 .7852 .8133 .8389 .8621 .8830 .9015 .9177 .9319 .9441 .9545 .9633 .9706 .9767 .9817 .9857 .9890 .9916 .9936 .9952 .9964 .9974 .9981 .9986 .9990 .9993 .9995 .9997 .9998

© Jones & Bartlett Lea NOT FOR SALE OR DIS

© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION

© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION

© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION

© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION

© Jones & Bartlett Lea NOT FOR SALE OR DIS

© Jones & Bartlett Lea NOT FOR SALE OR DIS

© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION

124 © Jones & Bartlett Learning, LLC. NOT FOR SALE OR DISTRIBUTION.

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Normal Distribution

125

© Jones & Bartlett Learning, LLC © Jones & Bartlett Lea NOTforFOR SALE OR DIS NOTtoFOR SALE OR DISTRIBUTION convert this value into the z-score. The corresponding z-score 4,275 mg/day will be Z

Xm 4275  4500  1.5  s 150

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC the row in the z table for a z-score of 1.5 the column for DISTRIBUTION NOT FOR SALE Locating OR DISTRIBUTION NOTand FOR SALE OR

.00, you should get a probability of .0668. Therefore, the probability that a randomly selected obese patient will take in below 4,275 mg/ day will be 6.68%. How about the probability that a randomly selected obese patient will have between 4,350 mg/day and 4,725 mg/day? s & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC Notice here that we have two scores to transform. The corresponding R SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION z-score of lower level, 4,350 mg/day, will be Z

Xm 4350  4500  1  s 150

© Jones Bartlett LLC and the&upper level, Learning, 4,725 mg/day, will be NOT FOR SALE OR XDISTRIBUTION m 4725  4500 Z

s



150

© Jones & Bartlett Lea NOT FOR SALE OR DIS  1.5

Therefore, we are looking at the area under the normal curve between 1 and 1.5 standard deviations, as shown in Figure 6-12. The prob© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC ability to the left of 1.5 is .9332 and the probability to the left of 1 NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION is .1587. To get the probability between 1 and 1.5, we will subtract

Figure 6-12 s & Bartlett Learning, Jones & deviations. Bartlett Learning, LLC TheLLC normal curve between 1 and© 1.5 standard R SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION ●

© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION

−1 © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION

0

© Jones & Bartlett Lea NOT FOR SALE OR DIS

©1.5Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION

© Jones & Bartlett Learning, LLC. NOT FOR SALE OR DISTRIBUTION.

s & Bartlett Learning, LLC R SALE OR DISTRIBUTION

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s & Bartlett Learning, LLC OR SALE OR DISTRIBUTION 126

CHAPTER SIX



DESCRIPTIVE STATISTICS

© Jones & Bartlett Learning, LLC © Jones & Bartlett Lea NOT NOT.1598 FOR SALE OR DISTRIBUTION from .9332 and should get .7866. Therefore, the probability thatFOR SALE OR DIS a randomly selected obese patient will have a sodium intake between 4,350 mg/day and 4,725 mg/day will be 78.66%. Finding the corresponding probabilities for a given score can be tricky, so we recommend you work on as many as examples as you can, including those Learning, LLC © Jones & Bartlett Learning, LLC © Jones & Bartlett at the end of this chapter. NOT FOR SALE included OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION As a final closing note about the standard normal distribution, recall that the following are true when a variable is normally distributed:

• 68% of observations fall within one standard deviation from s & Bartlett Learning, LLCthe mean in both directions © Jones & Bartlett Learning, LLC • 95% of observations fallNOT within two standard deviations from R SALE OR DISTRIBUTION FOR SALE OR DISTRIBUTION the mean in both directions • 99.7% of observations fall within three standard deviation from the mean in both directions.

This means that 68%Learning, of the z-scores will fall between 1 and 1, 95% © Jones & Bartlett LLC © Jones & Bartlett Lea the z-scores fallDISTRIBUTION between 2 and 2, and 99.7% of the NOT z-scoresFOR SALE OR DIS NOTofFOR SALEwill OR will fall between 3 and 3, since the standard normal distribution has a mean of 0 and a standard deviation of 1. This is important because any z-score that is greater than 3 or less than 3 can be treated as an unusual.

© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION ●

CONFIDENCE INTERVAL

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Up to this point, all of the estimates we calculated were with a single central tendency and variability were a s & Bartlett Learning, number. LLC Measures of both © Jones & Bartlett Learning, LLC single number and allowed us to say that those measures are the avR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION erage measurements and the spread of values on average of a given variable, respectively. These are called point estimates. However, we may not be lucky enough to hit exactly what the actual average will be in the population, since we are likely to use a sample taken from the © Jones & Bartlett LLC © Jones & Bartlett Lea population. In otherLearning, words, we will never be sure that our estimates accurately values in the population as a whole, asNOT shownFOR SALE OR DIS NOTwill FOR SALEreflect OR DISTRIBUTION in Figure 6-13. To deal with this problem, we can create boundaries that we think the population mean will fall between, instead of computing a single estimate from a sample; these boundaries are called confidence in© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC tervals. It is another way of answering an important question, “How

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NOT FOR SALE OR DISTRIBUTION

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Confidence Interval

© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION Figure 6-13

127

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Different sample means from a population.

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x3 = 107

x2 = 115

s & Bartlett Learning, LLC R SALE OR DISTRIBUTION x1 = 112

© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION x4 = 114

© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION Population

© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION

x5 = 121

© Jones & Bartlett Lea NOT FOR SALE OR DIS

well does the sample statistic represent the unknown population parameter?” © Jones & Bartlett Confidence Learning,intervals LLC use confidence levels in ©the Jones & Bartlett computation. Con- Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION fidence level is determined by the researcher and reflects how accurate you want to be in computing a confidence interval as a percentage. There are three confidence levels that you can choose from: 90%, 95%, and 99% (although the 95% confidence level seems to be the most mean? Let us say that s & Bartlett Learning, popular LLC choice). What does confidence © Jones interval & Bartlett Learning, LLC you chose a 95% confidence level to compute a confidence interval; R SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION this means that if you were to compute 100 confidence intervals, 95 of those confidence intervals will contain the population parameter and 5 of those will not. Another way of thinking about it is to say that should we calculate 100 confidence intervals, 5 of those would likely © Jones Bartlett Learning, LLC © Jones & Bartlett Lea not be & accurate. There are different equations for different parameters the computation confidence intervals, but we will introduce onlyFOR SALE OR DIS NOT NOTinFOR SALE ORofDISTRIBUTION one here for a population mean and focus on how to interpret the computed confidence interval. Let us assume that you are a health researcher and would like to investigate the average number of hours nursing students at a local © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC university spent per week studying for statistics. Number of hours is

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NOT FOR SALE OR DISTRIBUTION

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CHAPTER SIX



DESCRIPTIVE STATISTICS

© Jones & Bartlett Learning, LLC © Jones & Bartlett Lea NOTmeasured FOR SALE OR DISTRIBUTION on ratio level of measurement and we are lookingNOT at theFOR SALE OR DIS mean hours. Since we need to compute a confidence interval for the mean, we will use the following equation: S

S

x  za/2  m  x  za/2 © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC 1n 1n NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION

Where x is the sample mean, za/2 is the corresponding z-value for a/2 where a is equal to 1-confidence level, s is the sample standard deviation, and n is the sample size. Let us assume that we obtained a sample of 30 nursing students and s & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC the distribution of number of hours they study for statistics per week R SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION had a mean of 8 and standard deviation of 2. We would like to compute a 90% confidence interval where za/2  1.645. Our a is .10 since we are using 90% confidence level and a/2 is .05. We will find that the corresponding z-score for the probability of .05 inside the z table is © Jones &1.645. Bartlett Learning, LLC interval will be: © Jones & equal to Then, the 90% confidence

NOT FOR SALE OR DISTRIBUTION 8  1.645

2 2  m  8  1.645 150 150

Bartlett Lea NOT FOR SALE OR DIS

7.5347  m  8.4653 We Learning, can concludeLLC from this finding that 90% of time the mean will Learning, LLC © Jones & Bartlett ©the Jones & Bartlett fall between 7.53 and 8.47 hours of studyingNOT for statistics. NOT FOR SALE OR DISTRIBUTION FOR SALE OR DISTRIBUTION Consider now that you would like to compute a 95% confidence interval for the same example above. Our za/2 is 1.96 since our a/2 is .025 for a 95% confidence level and the 95% confidence interval will be:

s & Bartlett Learning, LLC R SALE OR DISTRIBUTION

8  1.96

2© Jones & Bartlett 2 Learning, LLC  m  8  1.96 150 150 NOT FOR SALE OR DISTRIBUTION

7.4456  m  8.5544

In this case we can conclude that 95% of the time the mean hours of studying for statistics fall between 7.45 and 8.55. © Jones Bartlett LLC for the same example©above? Jones & Bartlett Lea How&about a 99%Learning, confidence interval NOT NOTOur FOR OR DISTRIBUTION za/2 SALE is 2.58 since our a/2 is .005 for a 99% confidence level andFOR SALE OR DIS the 99% confidence interval will be: 8  2.58

2 2  m  8  2.58 150 150

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC 7.2702  m  8.7298 NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION

© Jones & Bartlett Learning, LLC. NOT FOR SALE OR DISTRIBUTION.

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Summary

129

© Jones & Bartlett Learning, LLC © Jones & Bartlett Lea NOT NOTInFOR SALE OR DISTRIBUTION this example we can conclude that 99% of the time the mean hoursFOR SALE OR DIS that students spend studying for statistics is between 7.27 and 8.73. As you look at these three confidence intervals, you will notice that the confidence interval gets wider as your desired confidence level increases. This makes theJones confidence interval, Learning, LLC © Jones & Bartlett Learning, LLC sense since the wider © & Bartlett you are sure that the interval will NOT include the population NOT FOR SALE the ORmore DISTRIBUTION FOR SALE OR DISTRIBUTION parameter. The trade-off is as confidence level increases, the likelihood of confidence interval including a true population parameter increases.

s & Bartlett Learning, LLC R SALE OR DISTRIBUTION SUMMARY

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Descriptive statistics, such as measures of central tendency and variability, help us to understand typical cases in a sample and the distribution of a variable more clearly. Measures the © Jones & Bartlett Learning, LLCof central tendency include © Jones & Bartlett Lea mode, the median, and the mean and these provide us with an idea NOTofFOR SALE OR DIS NOT FOR SALE OR DISTRIBUTION what may be the typical/average data value in the data set. The mode should be used only for categorical data as it basically counts the frequencies. The median should be reported when an unusual data value is present in the data set. Otherwise, the mean should be reported as it © Jones & Bartlett Learning, LLCpreferable characteristics. © Jones & Bartlett Learning, LLC possesses statistically NOT FOR SALE ORMeasures DISTRIBUTION FOR SALE OR DISTRIBUTION of variability include the range,NOT the interquartile range, the variance, and the standard deviation and they provide us an idea of the accuracy of the measures of central tendency. The range should be used as a crude measure of variability as it is extremely sensitive to presence of unusual data values. The interquartile range should be s & Bartlett Learning, the LLC © Jones & Bartlett Learning, LLC reported when an unusual or outlying data value is present in the data R SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION set. Otherwise, the standard deviation should be reported as it possesses statistically preferable characteristics. A normal distribution is a very important probability distribution, which can represent many human characteristics, such as height, © Jones Bartlett Learning, LLC and kurtosis can be used © Jones & Bartlett Lea weight,&and blood pressure. Skewness to variable is normally distributed; values should NOTbeFOR SALE OR DIS NOTassess FORwhether SALEaOR DISTRIBUTION between 1 and 1 standard deviations to be normal. It is important that variables of interest be normally distributed as most statistical analyses assume a normal distribution. a variable is normally distributed, 68% of observations will © Jones & Bartlett When Learning, LLC © Jones & Bartlett Learning, LLC fall within one standard deviation from the mean, 95% of observations

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NOT FOR SALE OR DISTRIBUTION

© Jones & Bartlett Learning, LLC. NOT FOR SALE OR DISTRIBUTION.

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CHAPTER SIX



DESCRIPTIVE STATISTICS

© Jones & Bartlett Learning, LLC © Jones & Bartlett Lea NOTofFOR SALE OR DIS NOTwill FOR SALE OR DISTRIBUTION fall within two standard deviations from the mean, and 99.7%

observations will fall within three standard deviations from the mean. Any value that falls outside of the three standard deviation range can be treated as an unusual value for the data set. are aLLC good example of how we © canJones compute standard- Learning, LLC © Jones & BartlettZ-scores Learning, & Bartlett scores to determine where any given score(s) fall inSALE a normal NOT FOR SALE ized OR DISTRIBUTION NOT FOR OR DISTRIBUTION distribution. We can use standardized scores to make comparisons between a single score, such as on a standardized test, with all scores. Instead of estimating an unknown population parameter with a s & Bartlett Learning, single LLC number or point estimate, © Jones & Bartlett Learning, LLC one can create an interval, called a R SALE OR DISTRIBUTION NOT FOR ORtoDISTRIBUTION confidence interval, as a different way of SALE answering the question, “How well does the sample statistic represent an unknown population parameter?” Confidence intervals are interpreted as the interval that will include the true parameter with a given confidence level, either 90%, 95%, or 99%. Learning, As the percentage goes © Jones & Bartlett LLCof the confidence interval © Jones & Bartlett Lea up (increased confidence that the mean falls within that range) NOTtheFOR SALE OR DIS NOT FOR SALE OR DISTRIBUTION likelihood of confidence interval including a true population parameter increases.

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC Critical Questions and Activities NOT FOR SALE OR Thinking DISTRIBUTION NOT FOR SALE OR DISTRIBUTION

1. What is the purpose of computing descriptive statistics? Why should we look at them along with graphical displays of a data set? s & Bartlett Learning, LLC © Jones & Bartlett Learning, 2. Which measure of central tendency and variability should LLC be R SALE OR DISTRIBUTIONreported when an unusual NOTdata FOR SALE OR in DISTRIBUTION value is present the data set? Explain. 3. The 95% confidence interval for sodium content level in 32 nursing home patients is (4,250 mg/day, 4,750 mg/day). What does this confidence interval tell us?

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© Jones & Bartlett Lea NOT FOR SALE OR DIS

Self-Quiz 1. True or False: Descriptive statistics are used to summarize about © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC the sample and the measures in the data set. NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION

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References

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© Jones & Bartlett Learning, LLC © Jones & Bartlett Lea NOT isFOR SALE OR DIS NOT FOR SALE OR DISTRIBUTION 2. True or False: The variance of length of stay at a local hospital

25. The standard deviation is 5 and this is how each value differs on average from the mean. 3. True or False: There is no such a chart that allows a researcher to identifyLLC possible outliers. © Jones & Bartlett Learning, © Jones & Bartlett Learning, LLC Which of the following is not a measure of central NOT FOR SALE OR4.DISTRIBUTION NOT FORtendency? SALE OR DISTRIBUTION a. Mode b. Interquartile range c. Mean d. Median s & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC 5. Find the area under the normal distribution curve. R SALE OR DISTRIBUTIONa. To the left of z  0.59 NOT FOR SALE OR DISTRIBUTION b. To the left of z  2.41 c. To the right of z  1.32 d. To the right of z  0.27 e. Bartlett Between 0.87 and 0.87LLC © Jones & Learning, © Jones & Bartlett Lea f. Between 2.99 and 1.34 NOT FOR SALE OR DIS NOT FOR SALE OR DISTRIBUTION 6. The average time it takes for emergency nurses to respond to an emergency call is known to be 25 minutes. Assume the variable is approximately normally distributed and the standard deviation is 5 minutes. If we randomly select an emergency © Jones & Bartlett Learning, LLC © Jones & Bartlett nurses, find the probability of the selected nurse responding to Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION an emergency call in less than 20 minutes. 7. The average age of 25 local nursing home residents is known to be 72 and the standard deviation is 8. The director of the nursing home wants to compute a 95% confidence interval to of an estimate for theLearning, average ageLLC of s & Bartlett Learning, LLC understand the accuracy © Jones & Bartlett entire residents. What is the 95% confidence interval? R SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION a. (65.23, 78.77) b. (68.86, 75.14) c. (65.00, 74.00) d. (62.86, 82.14)

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REFERENCES

Kaiser Family Foundation, Health Research & Educational Trust. (2011). Employer health benefit: from http://ehbs.kff. Bartlett Learning, LLC 2011 annual survey. Retrieved © Jones & Bartlett Learning, org/pdf/8226.pdf

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LLC NOT FOR SALE OR DISTRIBUTION

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CHAPTER SIX



DESCRIPTIVE STATISTICS

© Jones & Bartlett Learning, LLC © Jones & Bartlett Lea FOR SALE OR DIS NOTKershner, FOR SALE OR DISTRIBUTION K. (n.d.). Drug effective against high blood pressure and prostate NOT problems. Retrieved from http://researchnews.osu.edu/archive/hytrin.htm Kovner, C. T., Brewer, C. S., Fairchild, S., Poornima, S., Kim, H, & Djudic, M. (2007). Newly licensed RNs’ characteristics, work ethics, and intentions to work. American Journal of Nursing, 107(9), 58–70. © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC Tzeng, H. (2011). Perspectives of patients and families about the nature of NOT FOR SALE ORand DISTRIBUTION NOT FORtime. SALE OR DISTRIBUTION reasons for call light use and staff call light response Medsurg Nursing, 20(5), 225–234.

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