Average (Mean)

If four friends shared all their stickers so everyone had the same amount, the equal share would be the average, also called the mean. It is a fair-share idea: the total stays the same while the uneven amounts are levelled.

Parents: let your child predict before they reveal each step. The method boxes name the formula to write before substituting numbers.

By the end you'll be able to find an average, work backwards to the total or the number of items, and solve word problems where a new value changes the average.

Average means equal share

The average is found by adding all the values and dividing by how many values there are:

average = total ÷ number of items

Imagine four pupils whose marks total 32. If their marks were levelled, each group would be the same size. Use the lab to predict that equal share before opening the calculation.

🔬

Level a total into equal groups

🤔 Predict first: If 32 marks are shared equally among 4 pupils, what is one pupil's equal share?

6812

Work backwards to the total

If you know the average and the number of items, rebuild the total by multiplying:

total = average × number of items

For 5 children with an average of 12 stickers, the total is 12 × 5 = 60 stickers. This is the useful reverse form when a new value is added or removed later.

Show the worked solution

Find the hidden total

$$\text{Total}=\text{average}\times\text{number}$$

$$5\times6=30\text{ kg}$$

Work backwards to the number of items

If you know the total and the average, divide to find the number of items:

number of items = total ÷ average

If a class collected 84 cans and the average per pupil was 12 cans, there were 84 ÷ 12 = 7 pupils. Decide which of average, total or number is missing before choosing the formula.

Equal-sized groups and group averages

It is useful to know exactly when averaging group averages works. Suppose group A has 2 values with average 10 and group B has 2 values with average 20. Their counts are equal, so decide whether a direct average is valid before calculating.

Show the worked solution

Because each group contributes two values, the combined average is (10 + 20) ÷ 2 = 15.

With unequal group sizes, directly averaging the group averages is not guaranteed to give the combined average. Suppose group A has 1 value with average 10, while group B has 3 values with average 20. Directly averaging 10 and 20 gives 15, but rebuilding the totals gives 10 + 60 = 70 across 4 values, so the combined average is 17.5. The safe P6 method is to rebuild each total and divide by the combined count.

Equal group counts are sufficient for the direct mean of the group averages. They are not necessary for the numbers to happen to agree: if all group averages are the same, the direct average can match even when the counts differ. The core total-and-count method always remains available.

When a new value changes the average

For a new value, go through the total. First find the old total, then add or remove the value, then divide by the new count. Sara's first 4 tests average 70, and her fifth mark is 80. The old total is 70 × 4 = 280. Add 80 to get 360, then divide by 5: her new average is 72.

Rebuild the total first

$$\text{Old total}=70\times4=280$$

$$\text{New total}=280+80=360$$

$$\text{New average}=360\div5=72$$

Three practice problems

1. Four boxes weigh 50 g, 60 g, 70 g and 60 g. Find the average mass.

Total divided by count

$$50+60+70+60=240\text{ g}$$

$$240\div4=60\text{ g}$$

$$\text{Average}=60\text{ g}$$

2. The average of 4 test scores is 70. A fifth test score is 80. Find the new average.

Add the new value

$$\text{Old total}=70\times4=280$$

$$\text{New total}=280+80=360$$

$$360\div5=72$$

$$\text{New average}=72$$

3. One parcel has average mass 10 kg. Three other parcels have average mass 20 kg. Find the combined average.

Rebuild both group totals

$$\text{First total}=10\times1=10\text{ kg}$$

$$\text{Second total}=20\times3=60\text{ kg}$$

$$\text{Combined average}=(10+60)\div(1+3)=17.5\text{ kg}$$

Watch out — these are easily mixed up

Quick recap

🎯 Mastery check

Answer all 8 — your progress is saved on this device.

1. The average of 4 numbers is 15. What is their total?

   196011

2. Four boxes weigh 50 g, 60 g, 70 g and 60 g. What is their average mass?

   240 g55 g60 g

3. A group raised $84 in total, with an average of $12 per person. How many people were there?

   7672

4. The average of 4 test scores is 70. A fifth score is 80. What is the new average?

   757072

5. Two groups each contain 2 values. Their averages are 10 and 20. What is the combined average?

   203015

6. One value has average 10 and three values have average 20. What is the combined average?

   17.51520

7. Five numbers have an average of 12. One number, 2, is removed. What is the average of the 4 numbers left?

   121014.5

8. Three numbers have a total of 45. What is their average?

   4215135

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