Medians
The three measures of central tendency used to describe distributions of scores are the mean, median, and mode. Each has its own particular advantages and disadvantages depending on the shape of the score distribution. The mean is the most familiar and is the arithmetic average, calculated by adding up all the scores and dividing by the total number of scores. The median is the point on the scale that divides the distribution of scores in half (half of the scores fall above the median and half fall below). The mode is simply the score that occurs most frequently. Note that both the mean and the median are points on a scale and are found by computation; they aren't necessarily whole numbers.
If the score distribution is bell-shaped, the mean, median, and mode are identical and fall in the middle of the scale. Because of the way they're computed, means are influenced by extreme scores whereas medians are not. If a distribution is skewed, the mean is pulled out toward the tail of the distribution, while the median remains in the middle. Course ratings tend to be left-skewed, and for this reason IAS average ratings are reported in the form of medians.
The computation of IAS medians is based on the method described by Guildford (1965)^{1} and illustrated below. You may recognize this method as that used most commonly for calculating the median of grouped data. This method represents the actual ratings more precisely than does the "ordinal" median computed using un-grouped data.
Computation Example
In our example, 160 students rated a single item. The scale is 0-5 (very poor to excellent), and the mean is 3.76.
Rating | f | cf | |
---|---|---|---|
5 | 32 | 160 | 32 cases above |
4 | 74 | 138 | 74 cases within the interval containing the median |
3 2 1 0 |
40 12 1 1 |
54 14 2 1 |
54 cases below |
The median is the point on the scale that divides the distribution into halves, with 80 scores above and 80 scores below. As shown in the table, the scores don't divide themselves evenly into two groups, and the median would fall somewhere in the interval 4. The upper and lower limits of this interval are 3.5 and 4.5, respectively, and the exact value of the median is determined by the process of interpolation. In this process, the 74 scores are 'spread evenly' along the interval, and the median is located proportionately above the lower limit of 3.5 or below the upper limit of 4.5.
Interpolating up from the lower limit
The formula to compute the median by interpolating up from the lower limit is:
Lm + Im ( ( N / 2 – cf ) / fm )
Where:
Lm = lower limit of the interval containing the median
Im = the width of the interval containing the median
N = total number of scores
cf = cumulative frequency
f m = number of scores within the interval containing the median
This is illustrated in the following steps.
Step | Result |
---|---|
Identify the lower limit of the interval containing the median | 3.5 |
Find the width of the interval | 4.5 - 3.5 = 1.0 |
Determine the number of scores needed above the lower limit | 80 – 54 = 26 |
Determine the proportion of the interval above the lower limit | 26 / 74 = .35 |
Convert the proportion to scale units | 35 * 1.0 = .35 |
Find the scale value of the median | 3.5 + .35 = 3.85 |
Interpolating down from the upper limit
Interpolating down from the upper limit will give the same median value as interpolating up from the lower limit, as shown below.
Step | Result |
---|---|
Identify the upper limit of the interval containing the median | 4.5 |
Find the width of the interval | 4.5 - 3.5 = 1.0 |
Determine the number of scores needed below the upper limit | 80 – 32 = 48 |
Determine the proportion of the interval below the upper limit | 48 / 74 = .64 |
Convert the proportion to scale units | .64 * 1.0 = .64 |
Find the scale value of the median | 4.5 - .64 = 3.85 |
Note: Although we have reported medians to two decimals in this example to illustrate the method of computation, they are reported to only a single decimal on summary reports.
^{1} Guilford, J.P., Fundamental Statistics in Psychology and Education, United States of America, McGraw-Hill, 1965, pp. 49-55.