Measures of Central Tendency

A mass of data is hard to grasp, so we summarise it with a single value that represents the whole — a measure of central tendency (an average). It is the central or typical value around which the data cluster. The three main averages are the mean, the median and the mode.

The arithmetic mean is the most common average. It is found by adding all the values and dividing by the number of values:

Mean (X̄) = ΣX ÷ N

where ΣX is the sum of all the values and N is the number of values. For example, the mean of 10, 20, 30, 40, 50 is (10 + 20 + 30 + 40 + 50) ÷ 5 = 150 ÷ 5 = 30.

Sometimes the items are not equally important. Then we use the weighted mean, where each value X is multiplied by its weight W (its importance):

Weighted Mean = ΣWX ÷ ΣW

For instance, if a student scores 80 in a 3-credit subject and 60 in a 1-credit subject, the simple mean (70) misjudges the result. The weighted mean = [(80×3) + (60×1)] ÷ (3 + 1) = (240 + 60) ÷ 4 = 300 ÷ 4 = 75, which gives the more important subject its proper weight. The mean is easy to calculate and uses every value, but its big weakness is that it is badly affected by extreme values (one very large or very small figure can distort it).

The median is the middle value of the data when the values are arranged in order (ascending or descending). It divides the data into two equal halves — half the values lie below it and half above. Because it depends on position, not size, the median is not affected by extreme values, which is its great advantage.

To find the median of ungrouped data:

• First arrange the values in order.
• If the number of values N is odd, the median is the value at position (N + 1) ÷ 2.
• If N is even, there are two middle values, and the median is their average.

For example, for 7, 12, 3, 9, 5 — first arrange: 3, 5, 7, 9, 12. N = 5 (odd), so the median is the (5 + 1)÷2 = 3rd value = 7. For 4, 8, 6, 2 — arrange: 2, 4, 6, 8. N = 4 (even), so the median is the average of the 2nd and 3rd values = (4 + 6) ÷ 2 = 5.

The median is a positional average. It is especially useful when the data have extreme values or open-ended classes (like "income above ₹1,00,000"), where the mean cannot be calculated reliably. A related idea: the median is the value of the middle item, and the data can also be split into four quartiles or hundred percentiles in the same positional way.

The mode is the value that occurs most frequently in the data — the most common or most fashionable value. For example, in 2, 3, 3, 5, 7, 3, 8 the value 3 appears most often, so the mode is 3. Data may have one mode (unimodal), two modes (bimodal) or no mode at all (if every value occurs once). Like the median, the mode is not affected by extreme values, and it can even be found for qualitative data (e.g. the most popular shoe size or the best-selling colour) — which the mean cannot.

So which average should we use? Each has its merits and demerits:

• Mean — uses all values and is best for further calculation, but is distorted by extreme values and cannot be found for open-ended classes.
• Median — not affected by extremes and good for open-ended data, but does not use all the values.
• Mode — simple, works for qualitative data and shows the most typical value, but may not exist or may not be unique.

For a perfectly symmetrical distribution, the three are equal: Mean = Median = Mode. For a moderately skewed distribution they are linked by the empirical relation: Mode = 3 Median − 2 Mean. The practical rule: use the mean for general purposes and further analysis; use the median when there are extreme values or open-ended classes (e.g. average income); and use the mode when you want the most typical/popular item (e.g. the most demanded size of shirt).