Range and Outliers

What you will learn

• Find the range as a measure of spread.
• Spot an outlier.
• Explain how an outlier changes the mean, median and range.

Before you start: you should know how to find the mean and median. See Mean, Median and Mode.

The idea

An average tells you what is typical. The range tells you how spread out the data is. Subtract the smallest value from the largest.

An outlier is a value that sits far away from the rest of the data. It might be a mistake, or it might be a real but unusual result.

Spread and consistency

A small range means the values are close together, so the data is consistent. A large range means they are spread out.

What an outlier does

An outlier stretches the range and pulls the mean towards it. The median hardly moves.

Worked examples

Example 1: Finding the range

Find the range of $12, 5, 9, 20$.

1. Largest $20$, smallest $5$.
2. $20 - 5 = 15$.

Answer: $15$

Example 2: The effect of an outlier

Scores: $4, 7, 7, 8, 9, 10, 25$. Compare the mean with and without $25$.

1. With $25$: $70 \div 7 = 10$.
2. Without $25$: $45 \div 6 = 7.5$.
3. The outlier raised the mean by $2.5$.

Answer: the mean drops from $10$ to $7.5$ without the outlier.

Example 3: Deciding what to do

Heights in cm: $140, 143, 145, 146, 148, 41$. What should you do about $41$?

1. $41$ is far below the others and is not a believable height for this group.
2. It is probably a typing mistake, such as $141$.

Answer: check the original record before deciding. Never delete a value just because it is awkward.

Tip: report the median with the range if there are outliers, and say how you dealt with them.

Why it matters

Outliers matter in medicine, finance and sport. They can show errors, fraud or something genuinely special. Knowing the range and checking for outliers keeps averages honest. For a better measure of spread, see standard deviation.