# Bar Model Approach for Averages of Numbers in Statistics

##### Ho Soo Thong

Copyright © Nov 2015 AceMath, Singapore.

**Abstract**

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

**Arithmetic Distributive Law over Addition**

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