# Calculating Descriptive Statistics

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## Calculating Descriptive Statistics essay assignment

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 first step in the data analysis process, to be followed by inferential statistics. For all studies that involve numerical data, descriptive statistics are crucial in understanding the fundamental properties of the variables being studied. Exercise 27 focuses only on descriptive statistics and will illustrate the most common descriptive statistics computed in nursing research and provide examples using actual clinical data from empirical publications.

Calculating Descriptive Statistics essay assignment

## 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 (). 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 [Remicade], 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.

TABLE 27-1

DURATION OF BIOLOGIC USE AMONG VETERANS WITH RHEUMATOID ARTHRITIS (n = 10)

Duration of Biologic Use (years)

0.1

0.3

1.3

1.5

1.5

2.0

2.2

3.0

3.0

4.0

Because the number of study subjects represented below is 10, the correct statistical notation to reflect that number is:

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 populations, 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 292modes: 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.

### 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:

where

= 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:

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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 represent 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.

FIGURE 27-1  Frequency distribution of years of biologic use, without outlier and with outlier.

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 dispersion, 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 dispersed 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).

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### 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:

TABLE 27-2

DIFFERENCE SCORES OF DURATION OF BIOLOGIC USE

The mean deviation is the average difference score, using the absolute values. The formula for the mean deviation is:

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 (s2) is below.

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Note that the lowercase letter s2 is used to represent a sample variance. The lowercase Greek sigma (σ2) 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 populations, 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 formulas, 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 reflects important 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.

TABLE 27-3

VARIANCE COMPUTATION OF BIOLOGIC USE

### 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

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 s2 = 1.49. Therefore, the s of biologic duration is SD = 1.22. The SD is an important statistic, both for understanding dispersion within a distribution and for interpreting the relationship of a particular value to the distribution.

Calculating Descriptive Statistics

Calculating Descriptive Statistics

## 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 sample mean is close to 296the 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 () is:

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 duration is computed as follows:

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 standard 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  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 interval, 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 . 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:

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 mathematically, 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:

Calculating Descriptive Statistics

297

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 independent 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 collected from 10 veterans who had stopped taking biologic medications. Age at study enrollment, 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

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This is how our data set looks in SPSS..

Calculating Descriptive Statistics

Step 1: For a nominal variable, the appropriate descriptive statistics are frequencies and percentages. From the “Analyze” menu, choose “Descriptive Statistics” and “Frequencies.” Move “Race/Ethnicity and Gender” over to the right. Click “OK.”

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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 first set of tables (from the first set of SPSS commands in Step 1) contains the frequencies of race/ethnicity and gender. Most (70%) were Caucasian, and 100% were female.

## Calculating Descriptive Statistics

300

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

301

## Study Questions

1. Define mean.

2. What does this symbol, s2, represent?

3. Define 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?

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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 a variable?

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