Descriptive statistics are computed to reveal characteristics of the sample data set and to describe study variables.
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 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. iniximab [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 reect 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 signicant 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 of 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 ( 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…
