Choosing the right average

Three averages, three questions

Each average answers a different question.

• Mean: what is the fair share, if everything were equal?
• Median: what does the person in the middle have?
• Mode: what is the most common?

When a newspaper says "the average", it may mean any of the three. So ask which one it is, and whether it is the right one for the question.

An extreme value

Seven students have pocket money of $20, 25, 30, 25, 20, 30$ and $25$ rupees. Then a very rich friend joins, with $500$ rupees. What happens to the averages?

An outlier

A value that is very far from the others is called an outlier. The mean uses every value, so one outlier drags it away.

The median only cares about the middle, so it stays with the crowd.

You should find that the red line chases the orange dot, while the green line hardly moves. With the seventh student at 200 rupees, the mean is about 48 rupees, more than twice what any of the other six have.

Skewed data

Incomes, house prices and the sizes of cities have a few very large values and many small ones. In such data the mean is pulled up, above the median. Suppose a town has $9$ houses that cost $50$ lakh rupees and one that costs $5$ crore. The mean price is $95$ lakh, but nine of the ten houses cost $50$ lakh. The median, $50$ lakh, is the honest answer for a buyer.

Which average?

What if every value goes up?

If every value goes up by 10, all three averages go up by 10. If every value is doubled, all three averages double. The data only moves or stretches; its middle and its most common value move with it.

A good habit

When you see an average in a report, ask three things.

1. Which average is it: the mean, the median or the mode?
2. Does the data have outliers?
3. Is that the average that answers the question?

If the mean and the median are very different, the data is skewed or has an outlier. Look at both.

Practise with new numbers

Common mistakes

• Using the mean for data with outliers. Check the median too.
• Using the mean for words. Only the mode works for words.
• Reporting an average with no name. Always say which one.
• Removing an outlier because it is inconvenient. An outlier may be a true and important value.

Remember these

• An outlier drags the mean. It hardly moves the median.
• Use the mean for a fair share, the median for uneven data, and the mode for the most popular.
• A good report says which average it used.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. The salaries in a company are 20, 22, 24, 26 and 500 thousand rupees. Which describes a typical salary better, the mean or the median?

2. Which average works for the favourite subject of a class?

3. Every value in a data set is doubled. What happens to the median?

4. The mean of a data set is much higher than its median. What does this tell you?