Consider that your school head wants to calculate the average family size of students in your school. To carry out this activity on a large scale, let us first break it down into smaller subparts.
Thus, to start with, we will ask teachers to take our family size of students of each classroom.
We need to get an answer to the following questions over here:
1. What is the intended population? (Households of students in each teachers’ class)
2. The variable to be measured. (The number of people in a household)
3. Anticipating variability. (Asking about typical household sizes)
Let us now move ahead and start getting answers to these questions step by step.
Collect/consider data
Suppose the teacher decides to work with five students at a time in the classroom and asks each student, “How many people, including yourself, are in the household that you live in?”. As an answer to this, each student represents their family size with a collection of snap cubes.
The data for “family size” is represented with snap cubes in Fig.
Snap cube stacks representing family size
Analyze the Data To examine the distribution of the household size of the collected data, students first need to arrange the stacks of snap cubes in increasing order as shown in Fig.
Order stacks representing family size
You must have realized that family sizes vary. The next question that we need to ask is, How many people would be in each family if all five families were the same size?
When we make all the family sizes the same, family size does not vary. You may use two equivalent approaches:
1. Disconnect all the snap cubes and redistribute them one at a time to the five students until all snap cubes have been allocated. In this case, there are 15 snap cubes. Redistributing them among the five students yields 5 stacks of 3 cubes each.
2. Remove one snap cube from the highest stack and place it on one of the lowest stacks, continuing until all the stacks are leveled out.
Both these approaches yield an equal family size of three, which we can consider as an equal share or a fair share.
For the second approach, you can start with removing a snap cube from the highest stack and placing it on one of the lowest stacks. This will result in a new arrangement of cubes as shown in Fig.
Moving one snap cube from the highest stack
We will continue this process until all the stacks are level, or nearly level when there is a remainder as shown in Fig.
Snap cubes representing on average family size
After the final move all five stacks are leveled with three cubes each. This represents that the family size of three is an equal share. That means, if all five family sizes were the same, the number of people in the household would have been three. This equal share is nothing but the mean of the distribution.
By now, we know how to calculate the mean by adding up all the observations and dividing it by the number of observations. However, what does mean to tell us about the distribution? How are we expected to interpret the mean? How are we expected to describe the variability in a distribution in relation to its mean?
We can investigate the following problem to get an answer to these questions:
Suppose two other groups of five students in the classroom found their equal share value to be six. What are some different snap cube representations that they could have constructed?
To answer this, we should first realize that we need to start with 30 snap cubes. We can then create two different distributions of family size where the equal share value is 6. For example, consider the following two groups, Group 1 (shown in Fig) and Group 2 (shown in Fig) of data on five family sizes from the classroom where the equal share family size for each group is 6.
Group 1 arrangement with average 6
Group 2 arrangement with average 6
Because the equal share value for each group is 6, the two groups cannot be distinguished based on the equal share value.
An analysis question in this case may be:
Which group is closer to being equal?
We can offer different answers to this question, including :
1. Group 2, as this group has the highest frequency of stacks of six snap cubes.
2. Group 1, as for this group, we need fewer snap cubes to level out all the stacks to the equal share value of six.
The second method of having fewer snap cubes to move can be thought of as counting the “number of steps to equal”, or, how many steps we need to move the snap cubes to create the equal-sized groups. Fewer steps indicate that the distribution is closer to being equal and has less variability from the mean. We can go through the process to check that for Group 1, we need to move two cubes a total of two steps. For Group 2, we need to move two cubes a total of two steps each. Thus, Group 1 and Group 2 has equal variability from the mean.
Now, it is time to interpret the results to answer the original question,
What is the average family size of students in your school?
Using the results from the last two groups, we can comment that if the families are of equal size, the number of people in a household will be six. This will be equal share or mean value.
Thus, with the help of this activity we have learnt how mean helps us to get a quick resolution to our day to day activities.
