Bar Model Approach for Averages of Numbers in Statistics

Ho Soo Thong

   Nov 2015 

Abstract
This article illustrates

(1) the use of Bar Modelling Approach to problems involving  Statistical Situations;

 (2)  the use of the Arithmetic Distributive for Deductive Approach to solving problems involving averages (arithmetic mean) at Primary School Level.

 

1. 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.

 

2. 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 actual data situation and an ‘average’ situation for the bar modelling of the above mathematical situation, 

Figure 3 shows the actual data situation and an ‘average’ situation that displays the smallest number and the average of the remaining four numbers. 

 

Figure 4 shows the actual data situation and an ‘average’ situation  that displays the smallest number, the highest number and the average of the remaining three numbers.

 

3. 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 statical situations:
1. Bar Model for the average of eight numbers
2. Bar Model for the numbers 64 and 84, and the average of the other six numbers.

Therefore, the average of the other numbers is 42.

 

Alternatively,Figure 6 shows a deductive approach applying the Arithmetic Distributive Law.

 

Exercise 1

The average of nine whole numbers is 573.
If two of the numbers are 551 and 567, what is the average of the remaining seven numbers?                                                                            Answer 577

 

Next, we will apply the deductive approach to deal with a practical word problem.

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 statical situations for before and after review and deduce the revised equal amount of $ 400 after some simple arithmetic operations..

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.

Exercise 3

Ms Chan conducted a mathematics test for 30 pupils.
After marking their papers,  the average score was recorded as 62.1.
Later, she discovered that two test scores of  37 and 46 should have been  73 and 64 respectively.What is the revised average?                       And 63.9.

 

Remark
Pupils can apply this problem solving approach to solve harder problems (See [1] and [2]) 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