# Bar Model Approach for Averages of Numbers in Statistics

##### Ho Soo Thong

Abstract
This article illustrates the use of mathematical implications of the Arithmetic Distributive Law for solving problems involving averages at Primary School Level.

In algebraic expressions, we write

a = b + c       and       na = nb + nc

or

a − b = c       and      na − nb = nc

Figure 1 shows the bar modelling of the mathematical implications of the Distributive Law over Addition.

Bar Modelling of the Average of a list of numbers.
At Primary school level, the average for a list of numbers is the arithmetic mean given by

For example, the average of the five numbers 42, 43, 44, 51 and 70 is

(42 + 43 + 44 + 51 + 70)÷ 5 = 50

This gives the mathematical situation

42 + 43 + 44 + 51 + 70 = 5×50

Figure 2 shows the bar modelling of this mathematical situation.

Figure 3 shows the bar modelling of another mathematical situation that displays the smallest number and the average of the remaining four numbers.

Figure 4 shows a different mathematical situation that displays the smallest number, the highest number and the average of the remaining three numbers.

## Problems at Primary School Level

we begin with a simple example.

### Example 1

The average of eight whole numbers is 50.
If two of the numbers are 64 and 84, what is the average of the remaining six numbers?

### Solution

Figure 5 shows two bar models for the two mathematical situations:
1. Bar Model with the average of eight numbers
2. Bar Model with 40 and 84, and the average of the other six numbers.

Figure 5 shows two bar models for the two mathematical situations:
1. Bar Model with the average of eight numbers
2. Bar Model with 64 and 84, and the average of the other six numbers.

Therefore, the average of the other numbers is 42.

Figure 6 shows an alternative approach using the mathematical implications of the arithmetic distributive law.

Next, we will apply the mathematical implications in Distributive Law to deal with word problems involving changes in averages.

### Example 2

In a school financial assistance scheme, a certain amount of donated cash is equally distributed among 8 pupils.
After further review, the school decided to include three more pupils. As a result, each pupil will receive \$ 150 less.
How much will each pupil get after the review?

### Solution

In Figure 7, we construct a comparison bar model of the mathematical situations for before and after review and deduce the revised equal amount of \$ 400.

After the review, each pupil will receive \$ 400.

### Example 3

Ms Chan gave her pupils a mathematics test.
After marking their papers, she noted that the average score was 62.1.
Later, she discovered that a test score 37 should have been a score of 73. The revised average was 63.6.
How many pupils took the test?

### Solution

In Figure 8, the average score increases by 63.3 − 62.1 = 1.5 after correction and the total score increases by 73 − 37 = 36 as shown.

Therefore, there are 24 ( = 36 ÷1.5) pupils.

Remark
Pupils can apply this problem solving approach to solve harder problems at Primary Olympiad level.

References
[1] Ho Soo Thong, Distributive Law for Excess and Shortage Problems in PSLE Math, barmodelhost.com, Feb 2015
[2] Ho Soo Thong, Distributive Law for Indeterminate Problems, barmodelhost.com, June 2015