Range, IQR and Standard Deviation: Quick Revision

A brief note on Range, IQR and Standard Deviation. An average says where the data is; the spread says how far apart the values are. Try each question at the end before you open its answer.

Three measures of spread

• Quartiles. Sort the data. $Q_1$ is the median of the lower half, $Q_3$ of the upper half. With an odd count, leave the median out of both halves.
• Box plot. The smallest value, $Q_1$, the median, $Q_3$ and the largest value. The box is the middle half.
• Pairs. The mean goes with the standard deviation; the median with the IQR.

When every value changes

• Add a number to every value: the averages move, the spread does not.
• Multiply every value by $k$: the averages and the spread are all multiplied by $k$.

A worked example

Nine students need $11, 13, 14, 14, 16, 17, 19, 22$ and $41$ minutes to reach school.

1. Range. $41 - 11 = 30$ minutes.
2. Quartiles. The median, $16$, is left out. The lower half $11, 13, 14, 14$ gives $Q_1 = 13.5$; the upper half $17, 19, 22, 41$ gives $Q_3 = 20.5$.
3. IQR. $20.5 - 13.5 = 7$ minutes, much smaller than the range, because one student needs 41 minutes.
4. Which to report. With that outlier, give the median, 16 minutes, with the IQR, 7 minutes.

Practice with new numbers

A program to try

The program works out the worked example, and the standard deviation too.

Mistakes to avoid

• Forgetting to sort before finding the quartiles.
• Adding the distances from the mean. They always make 0; square them first.
• Stopping at the variance. The standard deviation is its square root.
• Thinking that adding to every value changes the spread. It only moves the data.

Check yourself

1. What is the range of $8, 3, 15, 6$ and $11$?

2. A box plot has the five numbers $4, 9, 13, 20, 31$. What is the IQR?

3. Some marks have a standard deviation of $7$. Every mark is multiplied by $3$, and then $5$ is added. What is the new standard deviation?

4. Two shops sell the same mean number of loaves a day. Shop A's daily sales have a standard deviation of 4 loaves and shop B's 15 loaves. Which shop can plan its baking more easily?