Measures of central tendency are measures of a set in the units of the measures of the individuals in the set. Central tendency identifies a measure of the set that attempts to identify with a single measure something about the magnitude of the individual measures.

1. Mode--Technically the score value around which there is the greatest density of scores. For a meaningful mode, the distribution of scores should be smoothed out. Often selected from a frequency distribution with 10 to 20 intervals. The middle of the most frequently represented interval, i.e., the score value around which the scores are most densely distributed is the mode. If the data are not grouped and there are multiple occurrences of different scores, the score value with the highest frequency is the mode, but this measure may be quite uninformative.

2. Median--The median is the score value at the 50^th percentile. Half the scores are larger and half the scores are smaller than the median. It is approximately the value of the middle score when individuals are ordered from largest to smallest. Count up or down (N+1)/2 scores for the median.

3. Mean--Average of all of the scores in the set. Based on adding all of the scores and dividing by the number of scores. Means of populations are represented by m, means of samples by  . The formula for computing the mean is  . The mean represents something like the center of gravity of the scores. If every score had an equal weight and was placed on a teeter-totter (see-saw) at a distance equal to its deviation from the mean, the teeter-totter would balance.

Finding the Mode, Median, and Mean from actual scores.

Note that the mode is at the center of the densest location of scores.

The median is given to the nearest score value by this method, which is almost always precise enough. For the men the median is as exact as can be computed. For the women there is rounding error using this method. For a more accurate score, one needs a linear interpolation for the median for women.

Linear Interpolation for the median of the women's heights

We see that 12 of the 25 scores are below the interval of the median. Thus we need only 1 1/2 of the 4 scores in the interval 63 1/2--64 1/2 for the midpoint. To find out how much to add to 63 1/2 we need to compute: 1 1/2, equivalently 3/2 or 1.5, is to 4 as x is to 1.  ,  ,  . Thus the 

The arithmetic means can be found by the formula  , for men  , for women,  .

The Grand mean or Weighted mean height for the entire set of scores can be computed by  . The grand mean treats the individuals in the different subsets (groups) as if there were no subgroups, but only individual measures in the set. The grand mean is thus simply the mean of all of the scores.