## Averages

You hear the word ‘average’ a lot nowadays, average height, weight, football etc. But what exactly do people mean when they say this, and how is this ‘average’ calculated exactly?

The word average is usually taken to mean the “middle” value, the most common or most likely to occur. There are several different averages we can take, each calculated in a different way. The averages we’ll be looking at in this article are the Mean, Median, Mode, Range and Moving Averages.

### Mean

The Mean – or Arithmetic Mean – is one of the most commonly used averages today. It’s calculated by adding up your values, and dividing by the “number of numbers”.

Raw Data – Raw data simply means that you have the actual data, it’s not been grouped in anyway. For example, if you have the numbers, 3, 5, 6, 7, 8, the average of these numbers would be $\frac{3 + 5 + 6 + 7 +8}{5} = \frac{29}{5} = 5.8$.

Grouped Data – When you are presented with grouped data, you cannot use the formula given above because you don’t have access to the individual pieces of data. The technique we use in this situation is to calculate an estimate of the mean using the midpoint of the groups. The formula we use is the sum of the midpoints multiplied by the frequency of the group, divided by the sum of the frequencies, written as $\frac{\sum fx}{\sum f}$. Don’t worry I assure you this isn’t as complicated as it seems.

Note – In Mathematics the Greek letter Sigma, $\sum$ is shorthand for ‘The Sum of’

Example – calculate the mean value for the following data.

Height (cm) Frequency (f) Midpoint (x) fx
130 – 140 2 135 270
140 – 150 5 145 725
150 – 160 8 155 1240
160 – 170 9 165 1485
170 – 180 8 175 1400
180 – 190 2 185 370
190 – 200 0 195 0

You won’t necessary get the data presented like this in the exam, I’ve just laid it out in what I believe is the easiest way to make calculations.

We then work out the sum of the frequencies, $\sum f = 34$, and the sum of the frequencies multiplied by the midpoint, $\sum fx = 5490$.

We can now work out the estimate for the mean height, $\frac{\sum fx}{\sum f} = \frac{5490}{34} = 161.5$ to 1 decimal place.

### Median

As the name suggests, the median of an ordered set of data is the middle number. Finding the median value of a set of data is quite simple, first arrange the data in size order, then the $\frac{n + 1}{2}$th value is the median.

For example, if we have a set of 9 numbers, the $\frac{9+1}{2} = 5$th item is the median.

If we have an even number of items, for example 6, then we take the mean of the middle two. For example, the numbers 1, 2, 3 and 4, we take the average of 2 and 3, which is 2.5.

### Mode

The Mode value of a set of numbers is simply the most common number in that set.

For example, the modal value of 2, 3, 5, 5, 6, 7, 7, 7, 8, 8, 9, 10, 10 would be 7.

### Range

The Range of a set of data is just the largest number minus the smallest number. Honestly, it really is that simple!

### Moving Averages

The Moving Average of a set of data is used to compare how the data changes over time.

Let’s suppose that you’ve measured the weight (kg) of one of the 34 individuals in the example for the Mean over a year, 12 months, and the figures are, 70.2, 70.4, 70.5, 70.6, 70.6, 70.4, 70.2, 70, 69.9, 69.9, 69.8, 69.9.

If we wanted a 4 year moving average, we simply take the (Mean)Average of 4 year intervals. For example, the first 3 four year average of our data would be: $\frac{1}{4} (70.2 + 70.4 + 70.5 + 70.6) = 70.425$ $\frac{1}{4} (70.4 + 70.5 + 70.6 + 70.6 = 70.525)$ $\frac{1}{4} (70.5 + 70.6 + 70.6 + 70.4 = 70.525)$

To calculate moving averages for different length intervals, just take the average of the different values. For example, for a 3 year moving average, take 3 year intervals etc.