Choosing the right measure: mean, median or mode
Use the mean for balanced data, the median when a few extreme values pull the mean, and the mode for the most common value.
Which measure, when?
| Measure | What it tells you | Best used for | Example |
|---|---|---|---|
| Mean | The average of all values | Data without extreme values; comparing two groups | Average marks of Class 10A and Class 10B |
| Median | The middle value | Data with a few very large or very small values | Typical pay in a firm where the owner earns far more than the staff |
| Mode | The most common value | Finding the most popular item or size | The kurta size a shop should stock the most |
Why extreme values matter
Suppose five friends get ₹ 20, ₹ 30, ₹ 30, ₹ 40 and ₹ 380 as pocket money. The mean is 500 ÷ 5 = ₹ 100, more than four of them get. The median is ₹ 30, which describes the group much better. One very large value pulled the mean up. So for data with extreme values, prefer the median.
Worked example 1: using the empirical relation
For a distribution, the mean is 35.4 and the mode is 34.2. Find the median using the empirical relation.
- Formula: 3 Median = Mode + 2 Mean.
- 3 Median = 34.2 + 2 × 35.4 = 34.2 + 70.8 = 105.
- Median = 105 ÷ 3 = 35.
Median = 35.
Worked example 2: all three for one table
The weights of 50 students are: 30-35: 4, 35-40: 10, 40-45: 18, 45-50: 12, 50-55: 6. For this table, the mean is 43.1 kg, the median 43.06 kg and the mode 42.86 kg. Check the empirical relation.
- Formula: 3 Median = Mode + 2 Mean.
- Left side: 3 × Median = 3 × 43.06 = 129.18.
- Right side: Mode + 2 × Mean = 42.86 + 2 × 43.1 = 42.86 + 86.2 = 129.06.
- The two sides are almost equal: 129.18 and 129.06.
The relation holds approximately. It is an empirical rule, found by observation, so it is not exact.
Quick rules
- The mean changes with every value, so one extreme value can pull it a lot.
- The median depends only on the middle position.
- The mode can be found even for data that are not numbers, like favourite colours.
- Rearranged: Mode = 3 Median − 2 Mean, and Mean = (3 Median − Mode) ÷ 2.
News reports often give the median age or the average rainfall of a city. A shopkeeper in a Jaipur market restocks the mode, the size bought most. The right measure gives an honest picture.
1 marks: “The mode of a distribution is 24 and the mean is 27. Find the median using the empirical relation.” 3 Median = 24 + 2 × 27 = 78, so Median = 26.
2 marks: “The median and mode of a distribution are 22 and 20. Find the mean.” 3 × 22 = 20 + 2 Mean, so 2 Mean = 66 − 20 = 46 and Mean = 23.
2 marks: “Which measure of central tendency would you use to find the most popular shoe size? Give a reason.” The mode, because it is the value that occurs most often. It shows the size most people buy.
Mean for fair, median for extreme, mode for most. And 3 Median = Mode + 2 Mean.