Measures of Central Tendency

by

Adam J. McKee, Ph.D.

January, 2004

Measures of central tendency do two jobs:

(1) Provide a summary of large amounts of data

(2) By providing common ground that lets us easily compare groups

Mode

                The mode is the least restricted measure of central tendency and is appropriate for use with all levels of measurement.   The mode is simply the value that occurs most often in the data.  In the above sample table for frequency distributions, 30 is the mode because it occurs 7 times, more than any other value.  If there are two values that occur an equal number of times, then the distribution is said to be bimodal.  For grouped data, the mode is the interval containing the greatest frequency.  The mode does not provide as much information as other central tendency statistics, and may not provide the best estimation of a "typical" value.

Median

                The median is the point in the distribution where 50% of the scores fall below it and 50% of the scores fall above it.  The median is often the best measure of central tendency in distributions that are asymmetrical (skewed).  To determine the median, arrange the scores from least to greatest.  If there are an odd number of scores, then the median is the score in the middle.  In the scores 2, 3, 4, 5, 6, the median is 4 because it is the middle score.  If an even number of scores is present, the median is halfway between the two middle values. In the scores 1, 2, 3, 4, 5, 6, the median is 3.5, the score between the middle scores of 3 and 4.  A special formula is used in calculating the median for grouped data.  This requires the use of a cumulative frequency distribution (refer back to table 2).  Calculate the median for grouped data as follows:

1.  Arrange the data in a frequency distribution.

2.  Determine the value of the frequency below which 50% of the cases fall by dividing N by two.

3.   Beginning at the bottom of the distribution, examine the cumulative frequency column until you find the interval that contains the midpoint calculated in step 2.

4.  Use the following formula to calculate the median:

Where:

 N = total number of scores

 X = lower real limit of the interval containing the median

  i = interval size

Mean

The most frequently used measure of central tendency is the arithmetic mean.  The mean is the sum of all the scores divided by the number of scores.  The formula is as follows:

Where sigma is the used to denote adding up a group of numbers, and X bar denotes the mean.  If you are given four tests with scores of 80, 90, 75, and 60 the mean is calculated as follows:

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