Comparing mean, median and mode

You now know three ways to summarise a list with a single number: the mean, the median and the mode. They often give different answers, and each is the best in different situations. Picking the wrong one can mislead readers , sometimes badly.

Idea

Think of the three measures as three different cameras pointed at the same data.

• The mean is the fair-share value: $\dfrac{\text{sum}}{\text{count}}$. It uses every value, so it is sensitive to extremes.
• The median is the middle of the sorted list. It only depends on order, so a few wild values barely move it.
• The mode is the most popular value. It can be used even when the data are categories, not numbers.

Why they differ. Consider the salaries (in $\rupee$ thousands) at a small office: $15, 18, 20, 22, 25, 28, 30, 350.$ Mean $= \tfrac{508}{8} = 63.5$. But seven of the eight people earn less than $30$! The mean is misleading because the boss's $\rupee 3{,}50{,}000$ pulls it up. The median is $\tfrac{22+25}{2} = 23.5$ , a much better picture of the typical salary.

When to use which.

A useful rule of thumb. If mean and median are close, the data are fairly symmetric. If they differ a lot, look for outliers or a skew.

The mode is special. It is the only one of the three that always exists for non-numerical data. You cannot find the mean shirt colour, but you can find the modal (most worn) colour.

Worked examples

Example 1. Data: $4, 5, 5, 6, 7, 8, 30$. Compare mean, median, mode.

Mean $= \tfrac{4+5+5+6+7+8+30}{7} = \tfrac{65}{7} \approx 9.3$. Median (4th of 7 sorted) $= 6$. Mode $= 5$. The outlier $30$ pulled the mean above all but one value, but the median and mode show the typical centre clearly.

Example 2. Daily temperature (°C) for a week: $28, 29, 30, 30, 31, 31, 32$. Find all three.

Mean $= \tfrac{211}{7} \approx 30.1$. Median (4th) $= 30$. Mode: $30$ and $31$ both occur twice , bimodal. The data are symmetric, so mean and median nearly agree.

Example 3. Shoe sizes ordered by a shop: $6, 7, 7, 7, 8, 8, 9, 10$. Which measure should the manager use to decide which size to stock most?

The mode , $7$ , because that is the most demanded size. Mean (about $7.75$) is irrelevant; you cannot sell three-quarters of a shoe.

Example 4. Five friends' pocket money (in $\rupee$): $40, 50, 60, 70, 280$. Which is more "typical": mean or median?

Mean $= \tfrac{500}{5} = 100$. Median $= 60$. Four of the five values are below $100$. The median $60$ is more typical.

Try it yourself

1. Marks $4, 5, 5, 6, 7$. Find mean, median, mode.
2. Salaries (in $\rupee$ thousand): $10, 12, 12, 14, 200$. Which measure best describes "typical salary"?
3. Number of goals scored by a team in $7$ matches: $0, 1, 1, 2, 2, 2, 4$. Find all three measures.
4. Heights (cm): $140, 142, 144, 146, 148$. Are mean and median close? What does that suggest?
5. A class collects favourite ice-cream flavours. Which measure makes sense , mean, median or mode? Why?
6. Data $3, 4, 4, 5, 6, 100$. Mean and median differ a lot. Explain.
7. List a small data set where mean $=$ median $=$ mode.
8. Why is the median preferred for reporting "typical income" in a country?

Activity

Collect the daily pocket money of $10$ classmates. Compute mean, median and mode. Discuss in groups: which one best describes "what a typical student gets"? Why?