Comparing spread

Which class is more alike?

Two classes both have a mean mark of $60$. Class A has the marks $40, 50, 60, 70, 80$. Class B has the marks $58, 59, 60, 61, 62$.

The mean says that the two classes are the same. But in class B nearly everyone got about $60$, while in class A the marks are far apart. A single average hides this. We need a number for the spread.

Three ways to measure spread

• The range is the largest value minus the smallest. For A it is $80 - 40 = 40$, and for B it is $62 - 58 = 4$. It uses only two values, so one outlier can change it a lot.
• The IQR is the width of the middle half of the data. It ignores the extreme values.
• The standard deviation is the typical distance of a value from the mean. It uses every value.

Try it in Python

Which measure of spread?

A pair that goes together: the mean with the standard deviation, and the median with the IQR. A report that says "the median is $15$" should add "and the IQR is $10$".

What if every value changes?

Look at the last two lines of the program.

• Add 10 to every value. Everything moves along by $10$. The gaps between the values do not change, so the standard deviation stays the same.
• Multiply every value by 2. Every gap doubles, so the standard deviation also doubles.

The mean and the median move with every change: they go up by $10$ when we add $10$, and they double when we multiply by $2$.

You should find that the slider for a moves the dots without changing the range or the standard deviation. The slider for k pulls the dots apart, and both measures of spread grow k times.

A real example: temperatures in degrees Celsius have a mean of $20\ ^\circ\text{C}$ and a standard deviation of $5\ ^\circ\text{C}$. In kelvin we add $273$, so the mean becomes $293\ \text{K}$, and the standard deviation is still $5\ \text{K}$. In degrees Fahrenheit, $F = 1.8C + 32$, the standard deviation becomes $1.8 \times 5 = 9$ degrees Fahrenheit, because the scale is stretched by $1.8$.

Spread and trust

A small spread means that the data is consistent, and the mean is a good guide to a single value. A large spread means that the mean is a poor guide. A bus that comes on average every $10$ minutes, with a spread of $1$ minute, is a good bus. With a spread of $8$ minutes, you may wait $20$ minutes, or see two buses together.

Practise with new numbers

Common mistakes

• Comparing only the means. Two sets with the same mean can be very different.
• Using the range with outliers. One extreme value changes it.
• Thinking that adding to every value changes the spread. It does not.
• Mixing the pairs. Use the standard deviation with the mean, and the IQR with the median.

Remember these

• The same mean does not mean the same spread.
• Use the mean with the standard deviation, and the median with the IQR.
• Adding the same number to every value leaves the spread unchanged. Multiplying every value by $k$ multiplies the spread by $k$.

Check yourself

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

1. The standard deviation of some marks is $5$. Every mark goes up by $3$. What is the new standard deviation?

2. The standard deviation of some lengths is $2\ \text{cm}$. Every length is multiplied by $4$. What is the new standard deviation?

3. Two machines fill bottles with a mean of $500\ \text{mL}$. Machine A has a standard deviation of $2\ \text{mL}$ and machine B has $15\ \text{mL}$. Which machine is better?

4. A data set is described by its median. Which measure of spread should go with it?