Mean and standard deviation from tables, totals and coded data

In the last lesson you found the mean and standard deviation of a short list of numbers:

$\bar{x} = \frac{\Sigma x}{n} \qquad \sigma = \sqrt{\frac{\Sigma x^2}{n} - \bar{x}^2}$

Look closely at those two formulas. Everything they need is three numbers: how many values there are ($n$), their total ($\Sigma x$), and the total of their squares ($\Sigma x^2$). You never need the individual values themselves.

That one observation is the key to this whole lesson. Exam questions often hide the data and give you something else instead: a frequency table, the totals, or "coded" totals. Each time, the job is the same: get hold of $n$, $\Sigma x$ and $\Sigma x^2$ (or something equivalent), then use the two formulas.

Sometimes the data come as a table of classes like "10 to 20 minutes: 12 people". You do not know exactly where each of those 12 values is, only that each is somewhere between 10 and 20.

The standard trick is to pretend every value in a class sits at the class midpoint (15 here). Some are really above the midpoint and some below, so the errors roughly cancel. Because we have guessed, the answers are estimates, and you should say so.

Now the 12 people in that class behave like the value 15 written down 12 times. Their total is $12 \times 15$, which is $f x$. Their squares total $12 \times 15^2$, which is $f x^2$. Adding these up over every class gives:

$\bar{x} = \frac{\Sigma f x}{\Sigma f} \qquad \sigma = \sqrt{\frac{\Sigma f x^2}{\Sigma f} - \bar{x}^2}$

These are exactly the old formulas. The only change is that $n$ is now $\Sigma f$ (the total frequency) and each total is built row by row. Set your working out as a table with columns $x$, $f$, $fx$ and $fx^2$.

Sometimes the question just says something like "$n = 20$, $\Sigma x = 340$, $\Sigma x^2 = 6200$". That is not a trick. As we saw at the start, those three numbers are all the formulas need, so substitute straight in.

You can also run this backwards. If you know $n$, the mean and the standard deviation, you can rebuild the totals:

$\Sigma x = n\bar{x} \qquad \Sigma x^2 = n(\sigma^2 + \bar{x}^2)$

The second one comes from rearranging $\sigma^2 = \dfrac{\Sigma x^2}{n} - \bar{x}^2$: add $\bar{x}^2$ to both sides, then multiply both sides by $n$. You will need these when combining two data sets.