Copyright © 2017, Elsevier Inc. All rights reserved. 291 Calculating Descriptive Statistics There are two major classes of statistics: descriptive statistics and
inferential statistics. Descriptive statistics are computed to reveal characteristics of the sample data set and to describe study variables. Inferential statistics
are computed to gain information about effects and associations in the population being studied. For some types of studies, descriptive statistics will be the only
approach to analysis of the data. For other studies, descriptive statistics are the fi rst step in the data analysis process, to be followed by infer-ential statistics.
For all studies that involve numerical data, descriptive statistics are crucial in understanding the fundamental properties of the variables being studied. Exer-cise
27 focuses only on descriptive statistics and will illustrate the most common descrip-tive statistics computed in nursing research and provide examples using actual
clinical data from empirical publications. MEASURES OF CENTRAL TENDENCY A measure of central tendency is a statistic that represents the center or middle of a
frequency distribution. The three measures of central tendency commonly used in nursing research are the mode, median ( MD ), and mean ( X ). The mean is the
arithmetic average of all of a variable ’ s values. The median is the exact middle value (or the average of the middle two values if there is an even number of
observations). The mode is the most commonly occurring value or values (see Exercise 8 ). The following data have been collected from veterans with rheumatoid
arthritis ( Tran, Hooker, Cipher, &Reimold, 2009 ). The values in Table 27-1 were extracted from a larger sample of veterans who had a history of biologic medication
use (e.g., infliximab [Remi-cade], etanercept [Enbrel]). Table 27-1 contains data collected from 10 veterans who had stopped taking biologic medications, and the
variable represents the number of years that each veteran had taken the medication before stopping. Because the number of study subjects represented below is 10, the
correct statistical notation to reflect that number is: n=10 Note that the n is lowercase, because we are referring to a sample of veterans. If the data being presented
represented the entire population of veterans, the correct notation is the uppercase N. Because most nursing research is conducted using samples, not popu-lations, all
formulas in the subsequent exercises will incorporate the sample notation, n. Mode The mode is the numerical value or score that occurs with the greatest frequency; it
does not necessarily indicate the center of the data set. The data in Table 27-1 contain two
EXERCISE 27 292EXERCISE 27 • Calculating Descriptive StatisticsCopyright © 2017, Elsevier Inc. All rights reserved. modes: 1.5 and 3.0. Each of these numbers occurred
twice in the data set. When two modes exist, the data set is referred to as bimodal ; a data set that contains more than two modes would be multimodal . Median The
median ( MD ) is the score at the exact center of the ungrouped frequency distribution. It is the 50th percentile. To obtain the MD , sort the values from lowest to
highest. If the number of values is an uneven number, exactly 50% of the values are above the MD and 50% are below it. If the number of values is an even number, the
MD is the average of the two middle values. Thus the MD may not be an actual value in the data set. For example, the data in Table 27-1 consist of 10 observations, and
therefore the MD is calculated as the average of the two middle values. MD=+()=15202175… Mean The most commonly reported measure of central tendency is the mean. The
mean is the sum of the scores divided by the number of scores being summed. Thus like the MD, the mean may not be a member of the data set. The formula for calculating
the mean is as follows: XXn=∑ where X = mean ∑ = sigma, the statistical symbol for summation X = a single value in the sample n = total number of values in the sample
The mean number of years that the veterans used a biologic medication is calculated as follows: X=+++++++++()=010313151520223030401019………..years TABLE 27-1
DURATION OF BIOLOGIC USE AMONG VETERANS WITH RHEUMATOID ARTHRITIS ( n = 10) Duration of Biologic Use (years) 0.10.31.31.51.52.02.23.03.04.0 Calculating Descriptive
Statistics • EXERCISE 27Copyright © 2017, Elsevier Inc. All rights reserved. The mean is an appropriate measure of central tendency for approximately normally
distributed populations with variables measured at the interval or ratio level. It is also appropriate for ordinal level data such as Likert scale values, where higher
numbers rep-resent more of the construct being measured and lower numbers represent less of the construct (such as pain levels, patient satisfaction, depression, and
health status). The mean is sensitive to extreme scores such as outliers. An outlier is a value in a sample data set that is unusually low or unusually high in the
context of the rest of the sample data. An example of an outlier in the data presented in Table 27-1 might be a value such as 11. The existing values range from 0.1 to
4.0, meaning that no veteran used a biologic beyond 4 years. If an additional veteran were added to the sample and that person used a biologic for 11 years, the mean
would be much larger: 2.7 years. Simply adding this outlier to the sample nearly doubled the mean value. The outlier would also change the frequency distribution.
Without the outlier, the frequency distribution is approximately normal, as shown in Figure 27-1 . Including the outlier changes the shape of the distribution to
appear positively skewed. Although the use of summary statistics has been the traditional approach to describing data or describing the characteristics of the sample
before inferential statistical analysis, its ability to clarify the nature of data is limited. For example, using measures of central tendency, particularly the mean,
to describe the nature of the data obscures the impact of extreme values or deviations in the data. Thus, significant features in the data may be concealed or
misrepresented. Often, anomalous, unexpected, or problematic data and discrepant patterns are evident, but are not regarded as meaningful. Measures of disper-sion,
such as the range, difference scores, variance, and standard deviation ( SD ), provide important insight into the nature of the data. MEASURES OF DISPERSION Measures
of dispersion , or variability, are measures of individual differences of the members of the population and sample. They indicate how values in a sample are dis-persed
around the mean. These measures provide information about the data that is not available from measures of central tendency. They indicate how different the scores are
—the extent to which individual values deviate from one another. If the individual values are similar, measures of variability are small and the sample is relatively
homogeneous in terms of those values. Heterogeneity (wide variation in scores) is important in some statistical procedures, such as correlation. Heterogeneity is
determined by measures of variability. The measures most commonly used are range, difference scores, variance, and SD (see Exercise 9 ). FIGURE 27-1 ■ FREQUENCY
DISTRIBUTION OF YEARS OF BIOLOGIC USE, WITHOUT OUTLIER AND WITH OUTLIER. 0FrequencyFrequency3-3.90-0.92-2.91-1.94-4.93-3.90-.91-1.92-2.94-4.95-5.96-6.97-7.98-8.99-
9.910-10.911-11.9Years of biologic useYears of biologic use3.02.52.01.51.00.503.02.52.01.51.00.5 294EXERCISE 27 • Calculating Descriptive StatisticsCopyright © 2017,
Elsevier Inc. All rights reserved. Range The simplest measure of dispersion is the range . In published studies, range is presented in two ways: (1) the range is the
lowest and highest scores, or (2) the range is calculated by subtracting the lowest score from the highest score. The range for the scores in Table 27-1 is 0.3 and
4.0, or it can be calculated as follows: 4.0 − 0.3 = 3.7. In this form, the range is a difference score that uses only the two extreme scores for the comparison. The
range is generally reported but is not used in further analyses. Difference Scores Difference scores are obtained by subtracting the mean from each score. Sometimes a
difference score is referred to as a deviation score because it indicates the extent to which a score deviates from the mean. Of course, most variables in nursing
research are not “scores,” yet the term difference score is used to represent a value ’ s deviation from the mean. The difference score is positive when the score is
above the mean, and it is negative when the score is below the mean (see Table 27-2 ). Difference scores are the basis for many statistical analyses and can be found
within many statistical equations. The formula for difference scores is: XX− Σof absolute values95:. TABLE 27-2 DIFFERENCE SCORES OF DURATION OF BIOLOGIC USE X –X
XX– 0.1 − 1.9 − 1.80.3 − 1.9 − 1.61.3 − 1.9 − 0.61.5 − 1.9 − 0.41.5 − 1.9 − 0.42.0 − 1.90.12.2 − 1.90.33.0 − 1.91.13.0 − 1.91.14.0 − 1.92.1 The mean deviation is the
average difference score, using the absolute values. The formula for the mean deviation is: XXXndeviation=−∑ In this example, the mean deviation is 0.95. This value
was calculated by taking the sum of the absolute value of each difference score (1.8, 1.6, 0.6, 0.4, 0.4, 0.1, 0.3, 1.1, 1.1, 2.1) and dividing by 10. The result
indicates that, on average, subjects ’ duration of biologic use deviated from the mean by 0.95 years. Variance Variance is another measure commonly used in statistical
analysis. The equation for a sample variance ( s 2 ) is below. sXXn221=−()−∑ Calculating Descriptive Statistics • EXERCISE 27Copyright © 2017, Elsevier Inc. All rights
reserved. Note that the lowercase letter s 2 is used to represent a sample variance. The lowercase Greek sigma ( σ 2 ) is used to represent a population variance, in
which the denominator is N instead of n − 1. Because most nursing research is conducted using samples, not popu-lations, formulas in the subsequent exercises that
contain a variance or standard deviation will incorporate the sample notation, using n − 1 as the denominator. Moreover, statistical software packages compute the
variance and standard deviation using the sample formu-las, not the population formulas. The variance is always a positive value and has no upper limit. In general,
the larger the variance, the larger the dispersion of scores. The variance is most often computed to derive the standard deviation because, unlike the variance, the
standard deviation reflectsimpor-tant properties about the frequency distribution of the variable it represents. Table 27-3 displays how we would compute a variance by
hand, using the biologic duration data. s213419=. s²=1.49 TABLE 27-3 VARIANCE COMPUTATION OF BIOLOGIC USE X X XX– XX–(())2 0.1 − 1.9 − 1.83.240.3 − 1.9 − 1.62.561.3
− 1.9 − 0.60.361.5 − 1.9 − 0.40.161.5 − 1.9 − 0.40.162.0 − 1.90.10.012.2 − 1.90.30.093.0 − 1.91.11.213.0 − 1.91.11.214.0 − 1.92.14.41 Σ 13.41 Standard Deviation
Standard deviation is a measure of dispersion that is the square root of the variance. The standard deviation is represented by the notation s or SD . The equation for
obtaining a standard deviation is SDX=−()−∑Xn21 Table 27-3 displays the computations for the variance. To compute the SD , simply take the square root of the variance.
We know that the variance of biologic duration is s 2 = 1.49. Therefore, the s of biologic duration is SD = 1.22. The SD is an important sta-tistic, both for
understanding dispersion within a distribution and for interpreting the relationship of a particular value to the distribution. SAMPLING ERROR A standard error
describes the extent of sampling error. For example, a standard error of the mean is calculated to determine the magnitude of the variability associated with the mean.
A small standard error is an indication that the sam 296EXERCISE 27 • Calculating Descriptive StatisticsCopyright © 2017, Elsevier Inc. All rights reserved. the
population mean, while a large standard error yields less certainty that the sample mean approximates the population mean. The formula for the standard error of the
mean ( sX ) is: ssnX= Using the biologic medication duration data, we know that the standard deviation of biologic duration is s = 1.22. Therefore, the standard error
of the mean for biologic dura-tion is computed as follows: sX=12210. sX=039. The standard error of the mean for biologic duration is 0.39. Confidence Intervals To
determine how closely the sample mean approximates the population mean, the stan-dard error of the mean is used to build a confidence interval. For that matter, a
confidence interval can be created for many statistics, such as a mean, proportion, and odds ratio. To build a confidence interval around a statistic, you must have the
standard error value and the t value to adjust the standard error. The degrees of freedom ( df ) to use to compute a confidence interval is df = n − 1. To compute the
confidence interval for a mean, the lower and upper limits of that interval are created by multiplying the sX by the t statistic, where df = n − 1. For a 95% confidence
interval, the t value should be selected at α = 0.05. For a 99% confidence inter-val, the t value should be selected at α = 0.01. Using the biologic medication duration
data, we know that the standard error of the mean duration of biologic medication use is sX=039. . The mean duration of biologic medication use is 1.89. Therefore, the
95% confidence interval for the mean duration of biologic medication use is computed as follows: XstX± 189039226…±()() 189088..± As referenced in Appendix A , the t
value required for the 95% confidence interval with df = 9 is 2.26. The computation above results in a lower limit of 1.01 and an upper limit of 2.77. This means that
our confidence interval of 1.01 to 2.77 estimates the population mean duration of biologic use with 95% confidence( Kline, 2004 ). Technically and math-ematically, it
means that if we computed the mean duration of biologic medication use on an infinite number of veterans, exactly 95% of the intervals would contain the true population
mean, and 5% would not contain the population mean ( Gliner, Morgan, & Leech, 2009 ). If we were to compute a 99% confidence interval, we would require the t value that
is referenced at α = 0.01. Therefore, the 99% confidence interval for the mean duration of biologic medication use is computed as follows: 189039325…±()() 189127..±
Calculating Descriptive Statistics • EXERCISE 27Copyright © 2017, Elsevier Inc. All rights reserved. As referenced in Appendix A , the t value required for the 99%
confidence interval with df = 9 is 3.25. The computation above results in a lower limit of 0.62 and an upper limit of 3.16. This means that our confidence interval of
0.62 to 3.16 estimates the population mean duration of biologic use with 99% confidence. Degrees of Freedom The concept of degrees of freedom ( df ) was used in
reference to computing a confidence interval. For any statistical computation, degrees of freedom are the number of inde-pendent pieces of information that are free to
vary in order to estimate another piece of information ( Zar, 2010 ). In the case of the confidence interval, the degrees of freedom are n − 1. This means that there
are n − 1 independent observations in the sample that are free to vary (to be any value) to estimate the lower and upper limits of the confidence interval. SPSS
COMPUTATIONS A retrospective descriptive study examined the duration of biologic use from veterans with rheumatoid arthritis ( Tran et al., 2009 ). The values in Table
27-4 were extracted from a larger sample of veterans who had a history of biologic medication use (e.g., infliximab [Remicade], etanercept [Enbrel]). Table 27-4
contains simulated demographic data col-lected from 10 veterans who had stopped taking biologic medications. Age at study enroll-ment, duration of biologic use,
race/ethnicity, gender (F = female), tobacco use (F = former use, C = current use, N = never used), primary diagnosis (3 = irritable bowel syndrome, 4 = psoriatic
arthritis, 5 = rheumatoid arthritis, 6 = reactive arthritis), and type of biologic medication used were among the study variables examined. TABLE 27-4 DEMOGRAPHIC
VARIABLES OF VETERANS WITH RHEUMATOID ARTHRITIS Patient ID Duration (yrs) Age Race/Ethnicity Gender Tobacco Diagnosis Biologic 10.142CaucasianFF5Infl iximab20.341Black,
not of Hispanic OriginFF5Etanercept31.356CaucasianFN5Infl iximab41.578CaucasianFF3Infl iximab51.586Black, not of Hispanic
OriginFF4Etanercept62.049CaucasianFF6Etanercept72.282CaucasianFF5Infl iximab83.035CaucasianFN3Infl iximab93.059Black, not of Hispanic OriginFC3Infl
iximab104.037CaucasianFF 298EXERCISE 27 • Calculating Descriptive StatisticsCopyright © 2017, Elsevier Inc. All rights reserved. This is how our data set looks in
SPSS. Step 1: For a nominal variable, the appropriate descriptive statistics are frequencies and percentages. From the “Analyze” menu, choose “Descriptive Statistics”
and “Frequen-cies.” Move “Race/Ethnicity and Gender” over to the right. C Calculating Descriptive Statistics • EXERCISE 27Copyright © 2017, Elsevier Inc. All rights
reserved. Step 2: For a continuous variable, the appropriate descriptive statistics are means and standard deviations. From the “Analyze” menu, choose “Descriptive
Statistics” and “Explore.” Move “Duration” over to the right. Click “OK.” INTERPRETATION OF SPSS OUTPUT The following tables are generated from SPSS. The fi rst set of
tables (from the fi rst set of SPSS commands in Step 1) contains the frequencies of race/ethnicity and gender. Most (70%) were Caucasian, and 100% were female.
Frequencies Frequency Table RaceEthnicityFrequencyPercentValidPercentCumulativePercentValidBlack, not of Hispanic
Origin330.030.030.0Caucasian770.070.0100.0Total10100.0100.0GenderFrequencyPercentValid PercentCumulative PercentValidF10100.0100.0 300EXERCISE 27 • Calculating
Descriptive StatisticsCopyright © 2017, Elsevier Inc. All rights reserved. DescriptivesStatisticStd. ErrorDuration of Biologic Use1.890.3860Lower Bound1.017Upper
Bound2.7631.8721.7501.4901.2206.14.03.92.0.159.687-.4371.334Mean95% Confidence Interval for Mean 5% Trimmed MeanMedianVarianceStd.
DeviationMinimumMaximumRangeInterquartileRangeSkewnessKurtosis The second set of output (from the second set of SPSS commands in Step 2) contains the descriptive
statistics for “Duration,” including the mean, s (standard deviation), SE , 95% confidence interval for the mean, median, variance, minimum value, maximum value, range,
and skewness and kurtosis statistics. As shown in the output, mean number of years for duration is 1.89, and the SD is 1.22. The 95% CI is 1.02–2.76. Explore
Calculating Descriptive Statistics • EXERCISE 27Copyright © 2017, Elsevier Inc. All rights reserved. STUDY QUESTIONS 1. Defi ne mean. 2. What does this symbol, s 2 ,
represent? 3. Defi ne outlier. 4. Are there any outliers among the values representing duration of biologic use? 5. How would you interpret the 95% confidence interval
for the mean of duration of biologic use? 6. What percentage of patients were Black, not of Hispanic origin? 7. Can you compute the variance for duration of biologic
use by using the information presented in the SPSS output above? 302EXERCISE 27 • Calculating Descriptive StatisticsCopyright © 2017, Elsevier Inc. All rights
reserved. 8. Plot the frequency distribution of duration of biologic use. 9. Where is the median in relation to the mean in the frequency distribution of duration of
biologic use? 10. When would a median be more informative than a mean in describing Copyright © 2017, Elsevier Inc. All rights reserved. 303 Answers to Study Questions
Duration of biologic useMean = 1.89Std. Dev. = 1.221N = 10321001.02.03.04.05.0Frequency 1. The mean is defined as the arithmetic average of a set of numbers. 2. s 2
represents the sample variance of a given variable. 3. An outlier is a value in a sample data set that is unusually low or unusually high in the context of the rest of
the sample data. 4. There are no outliers among the values representing duration of biologic use. 5. The 95% CI is 1.02–2.76, meaning that our confidence interval of
1.02–2.76 estimates the population mean duration of biologic use with 95% confidence. 6. 30% of patients were Black, not of Hispanic origin. 7. Yes, the variance for
duration of biologic use can be computed by squaring the SD presented in the SPSS table. The SD is listed as 1.22, and, therefore, the variance is 1.22 2 or 1.49. 8.
The frequency distribution approximates the following plot: 9. The median is 1.75 and the mean is 1.89. Therefore, the median is lower in relation to the mean in the
frequency distribution of duration of biologic use. 10. A median can be more informative than a mean in describing a variable when the variable ’ s frequency
distribution is positively or negatively skewed. While the mean is sensitive to outli-ers, the median is relatively unaffected. Copyright © 2017, Elsevier Inc. All
rights reserved. 305 Questions to Be Graded EXERCISE 27 Follow your instructor ’ s directions to submit your answers to the following questions for grading. Your
instructor may ask you to write your answers below and submit them as a hard copy for grading. Alternatively, your instructor may ask you to use the space below for
notes and submit your answers online at http://evolve.elsevier.com/Grove/statistics/ under “Questions to Be Graded.” 1. What is the mean age of the sample data? 2.
What percentage of patients never used tobacco? 3. What is the standard deviation for age? 4. Are there outliers among the values of age? Provide a rationale for your
answer. 5. What is the range of age values? Name: _______________________________________________________ Class: _____________________ Date:
________________________________________________________________ 306EXERCISE 27 • Calculating Descriptive StatisticsCopyright © 2017, Elsevier Inc. All rights
reserved. 6. What percentage of patients were taking infliximab? 7. What percentage of patients had rheumatoid arthritis as their primary diagnosis? 8. What percentage
of patients had irritable bowel syndrome as their primary diagnosis? 9. What is the 95% CI for age? 10. What percentage of patients had psoriatic arthritis as their
primary di Calculating Pearson Product-Moment Correlation Coefficient Correlational analyses identify associations between two variables. There are many differ-ent
kinds of statistics that yield a measure of correlation. All of these statistics address a research question or hypothesis that involves an association or
relationship. Examples of research questions that are answered with correlation statistics are, “Is there an associa-tion between weight loss and depression?” and “Is
there a relationship between patient satisfaction and health status?” A hypothesis is developed to identify the nature (positive or negative) of the relationship
between the variables being studied. The Pearson product-moment correlation was the fi rst of the correlation measures developed and is the most commonly used. As is
explained in Exercise 13 , this coefficient (statistic) is represented by the letter r , and the value of r is always between − 1.00 and + 1.00. A value of zero
indicates no relationship between the two variables. A positive cor-relation indicates that higher values of x are associated with higher values of y . A negative or
inverse correlation indicates that higher values of x are associated with lower values of y . The r value is indicative of the slope of the line (called a regression
line) that can be drawn through a standard scatterplot of the two variables (see Exercise 11 ). The strengths of different relationships are identified in Table 28-1 (
Cohen,