Thanks Emilia for our newest Homework Headache. Emilia’s son is in 7^{th }Form (~6^{th} grade in the American system) and is having some difficulties solving the problem below. Emilia, we’re going to walk you thorough it step by step so you can explain it to your son.

This problem involves some terms that you may not use on a daily basic so let’s review.

**Mean:** Mean is simply the average value of a series of numbers. You can find the mean of a group of numbers by adding up all the numbers in that group and then dividing that sum by the amount of numbers in that group.

For example, if we want to calculate the mean of 3, 4, 5, 6 and 13

We have to add all the number: 3 + 4 + 5 + 6 + 13 = 30, then divided 30 by the number of numbers in this group, in this case 5.

But in the problem above we aren’t’ simply given a list of numbers. We are asked to solve for the mean given only the total number of students—30—and the frequencies of the shoe sizes.

**Frequency:** Frequency is the number of occurrences of a particular number in a series.

For example, in the following group, 4, 4, 4, 7, 7, 9

The frequency of 4 is 3, since it shows up 3 times in the series. The frequency of 7 is 2 and 9 is 1. If we were only given the frequency we could also figure out how many of each number occurred. For example

Using the frequency information we are given, we can reconstruct what the series looked like, 2 6’s, 3 5’s and 1 4.

6, 6, 5, 5, 5, 4

Now that we have reviewed ‘mean’ and ‘frequency’ let’s return to our original problem.

The first row tells us all of the shoe sizes for our 30 pupils and the second row shows us how frequently each of those shoe size occur in the group. To calculate mean we need to add up all of the shoe sizes and then divide by the total number of students. We already know that the total number is 30. And we can add everything up since what we know from frequency.

Mean =

= 164

Alternatively, a faster way of approaching this problem is just to:

- Multiply each shoe size by the frequency
- Add the products, and
- Divide by the total number of students

Mean =

= 164

Emilia, we hope this helps you explain this problem to your son.

Sincerely,

Homework Unlocked

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