Question

How do you find the mean of `{25,31,31,20,30,26,32,31,35}`?

Answer 1

Add the values up, then divide this sum by the number of values in the set. The mean is 29.

Explanation:

The mean of a dataset is a statistical term for the set's average. It is calculated by adding up all the values in the set, and then dividing this sum by the number of elements in the set.

For a set of `n` data points `x_1, x_2, x_3, ..., x_n`, the mean `barx` is found using this formula:

`barx=(x_1+x_2+x_3+...+x_n)/n`

For example, the given dataset is `{25,31,31,20,30,26,32,31,35}`. This set has 9 elements, so `n=9`. The final value of `barx` is then

`barx=(25+31+31+20+30+26+32+31+35)/9`
`color(white)(barx)=261/9`

`color(white)(barx)=29`

This value of 29 is like the centre of gravity of the data. If we placed marbles on a number line at the values in the set, one for each data point, the mean `barx` is where we'd be able to balance the line, with the same amount of total distance to all the marbles on its left as to those on its right.

Using this analogy, it's easy to see what happens to the average if we move one of the data points. If we move one marble to the right by 5, for example, the centre of gravity for all 9 marbles would shift to the right as well. But by how much? The one marble moved by 5, but its weight is only `1/9`th of the total weight of all marbles. So, the centre of gravity will move `5 xx 1/9`, which is `5/9`, to the right.

So you see, the mean of a dataset is like a weighted average of the set, but with all the weights the same. This is easy to see, because the mean formula above can be written with the denominator distributed out, as

`barx=1/n x_1+1/nx_2+1/nx_3+...+1/nx_n`.

(Notice how all `n` weights are `1/n`, and so they all add to `1`.)