TABLE 27-1

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

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.

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

X
0.1	−1.9	−1.8
0.3	−1.9	−1.6
1.3	−1.9	−0.6
1.5	−1.9	−0.4
1.5	−1.9	−0.4
2.0	−1.9	0.1
2.2	−1.9	0.3
3.0	−1.9	1.1
3.0	−1.9	1.1
4.0	−1.9	2.1

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

X
0.1	−1.9	−1.8	3.24
0.3	−1.9	−1.6	2.56
1.3	−1.9	−0.6	0.36
1.5	−1.9	−0.4	0.16
1.5	−1.9	−0.4	0.16
2.0	−1.9	0.1	0.01
2.2	−1.9	0.3	0.09
3.0	−1.9	1.1	1.21
3.0	−1.9	1.1	1.21
4.0	−1.9	2.1	4.41
		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

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.

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:

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.

TABLE 27-4

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.”

 

299

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.”

 

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?

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?

1. The mean is defined as the arithmetic average of a set of numbers.

2. s² 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 1.22²or 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 outliers, the median is relatively unaffected.

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?

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 diagnosis?