Averages that hide the spread

By the end of this lesson you can see how two groups with the same average can be very different.

Why it matters

An average squeezes a whole group into one number. That is handy, but it throws information away. A school that plans one bus time from an average could leave some pupils waiting for an hour.

The idea in plain words

Imagine two rivers, each with an average depth of 1 metre. In the first, the water is about 1 metre deep everywhere. In the second, it is very shallow in most places but 3 metres deep in the middle. Would you walk across both? The average is the same. The rivers are not.

Average (mean): add up all the values, then divide by how many there are. In Google Sheets: =AVERAGE(range).

Spread: how far apart the values are. A simple way to see it: the smallest value (=MIN(range)) and the largest (=MAX(range)).

Here is Lucía's survey, average minutes to school, by class.

Two identical bars. It looks as if both classes have the same journey.

See it done

Data: Lucía's tab, class-survey.csv (synthetic). Column G is travel_minutes. Class 5A is rows 2 to 16; 5B is rows 17 to 31.

Question: are the two classes' journeys to school really alike?

1. Work out each average. =AVERAGE(G2:G16) gives 20. =AVERAGE(G17:G31) gives 20. The same.
2. Check one by hand. Class 5A's 15 values add up to 300 (=SUM(G2:G16)), and 300 ÷ 15 = 20. The formula agrees.
3. Find the smallest and largest. 5A: =MIN(G2:G16) gives 17 and =MAX(G2:G16) gives 23. 5B: 5 and 40.
4. Work out the spread. 5A: 23 − 17 = 6 minutes from shortest to longest. 5B: 40 − 5 = 35 minutes.
5. Look at every pupil. In the dot chart below, each dot is one pupil.

Same average. Very different smallest and largest values.

Class 5A's pupils all take about 20 minutes. In 5B, some walk 5 minutes and some travel 40.

6. Say what it means. "Both classes average 20 minutes, but in 5A everyone takes between 17 and 23 minutes, while in 5B it ranges from 5 to 40 minutes. A 5B pupil might arrive 35 minutes after a classmate who left at the same time."

Your turn, with help

Amara sells pawpaws and pineapples. Each has an average of 2.3 fruits per sale. Compare the spread of quantity for each fruit: find the smallest and largest quantity.

Hint: filter fruit to Pawpaw, then read the smallest and largest quantity that show (or sort them). Do not use MIN and MAX here: they look at the whole column, even rows a filter hides. Then do the same for Pineapple.

Your turn, alone

1. In Lucía's survey, what is the average of siblings for all pupils who answered? (=AVERAGE(H2:H31); give two decimal places.) Why does AVERAGE ignore the 4 blanks, and is that right?
2. What are the smallest and largest siblings values?
3. Two clinics both have an average waiting time of 30 minutes. Clinic A: 25 to 35 minutes. Clinic B: 5 to 120 minutes. Which would you choose if you had an important appointment after, and why?
4. Write one sentence about Lucía's classes that uses the average and the spread.

Watch out

• "Average" alone. When you see an average, ask for the smallest and largest values, or a picture of every value.
• One extreme value. A single huge value can drag an average up. The average salary in a room jumps if a millionaire walks in, though nobody else got richer.
• Blanks treated as zero. That pulls an average down. Leave blanks blank.

Say it back

1. How do you work out an average?
2. Which two functions show a simple spread?
3. What were the spreads (largest minus smallest) for 5A and 5B?

What is next

On Amara's hottest days, she sells more watermelons and more coconuts. So does selling watermelons make people buy coconuts? Next lesson: one of the best-known traps of all.