Frequency
distributions

Analyzing
single
variables

Presenting univariate
data

Measures of
central
tendency

Measures of
variability

Standard
deviation and
the normal distribution

Computer
analysis
reminder

Aids

Choosing a measure of central tendency

Selection of a measure of central tendency depends on the level of measurement used. For nominal measurement, the mode is the only measure of central tendency that can be used. At a university with 3,500 men and 1,250 women, it makes sense to say that men represent the mode for enrollment by gender. It is impossible to calculate a median or mean with nominal data. Medians can be used for ordinal data, but when there are only a few categories, the mode is often a more appropriate measure of central tendency. For data measured at the interval or ratio level, the mean is usually used as the measure of central tendency.

Sometimes, however, the mean is not the most appropriate measure to use with numerical data.   When scores occur at the extreme ends of a distribution, the mean will move in the direction of the extreme scores, which can give a misleading report for central tendency.   The median is less affected by extreme scores because it does not depend on the sizes of the scores as the mean does. Some simple numbers will illustrate what we have just described.   Consider the   incomes reported by two groups of workers:

Sample A:   10, 90, 100, 120

Sample B:    80, 100, 120, 200

The mean for sample A is 320/4 = 80 and the median is 95. For sample B the mean is 125 and the median is 110. The low score of 10 in sample A pulled the mean down to 80, making the mean less than the median. Even if the lowest score had been 0, the median would have remained 95, but the mean would have declined further to 77.5. In the case of sample B, because of the high income of 200, the mean moved upward to 125 while the median (110) remained close to the center of the distribution.   Any higher score for income in sample B would further increase the mean, but would not affect the median.   Suppose that instead of 200, a person in sample B received 1,000 per month in local income. Then the mean for sample B would be 325, but the median would still remain at 110.

Distributions with a number of frequencies for scores at one end of the distribution are described as being skewed. In a negatively skewed distribution, the mean is pulled down in the direction of the lower scores; in a positively skewed distribution, the mean is located nearer the higher scores.

When there are extreme scores the median is generally the more appropriate measure for describing central tendency. If in doubt, calculate and report both the mean and median

Measures of variability

In addition to finding the central tendency of a distribution, univariate analysis also includes describing the extent of variation among the scores for the distribution. The idea of variability is quite simple, as we think you will see by examining the following two sets of grades for two classes:

Class A: 40, 50, 55, 60, 70, 80, 85, 90, 100

Class B: 66, 67, 68, 69, 70, 71, 72, 73, 74.

Which class shows the greatest amount of variation in grades?  With the scores arranged in order from the lowest to the highest, one can easily see that there is more variation in the grades of students in class A than in Class B. Grades in class A vary from 40 to 100 while whose in class B vary from 66 to 74. Statisticians use three measures of variation. These are the range, variance and the standard deviation of a distribution.

Range

The range (R) is simply the highest score minus the lowest score + 1. The "+1" is added because we want to count the lowest score in the range. To clarify this point, calculate the range for the scores of 10, 11, 12, 13, 14, and 15. The range is 15-10+1=5+1=6. If we simply subtracted 15-10, we would get 5, but there are six scores: Count them. The range gives a quick idea of how far the scores are spread out. For class A, as listed above, the range equals 100-40+1 or 61 while R for class B is 74-66+1 or 9.  Class A has almost seven times the variability of Class B.

While useful for giving a quick idea of variability, the range has limited value because it is based on only the highest and lowest scores in a distribution. Also, the range will change any time either of the end scores changes. Because of these limitations, researchers rely on the variance and its derivative, the standard deviation, for more accurate descriptions of variability.