Standard deviation

How far from the mean?

The range and the IQR look at a few values. The standard deviation uses every value. It asks: how far is a typical value from the mean?

We could find the distance of each value from the mean and take the average. But the distances below the mean are negative and those above are positive, so they cancel. In Course 3.1 you saw that they always add up to $0$.

The fix is to square each distance. A square is never negative.

The steps

1. Find the mean.
2. Find the distance of each value from the mean.
3. Square each distance.
4. Find the mean of the squares. This is the variance.
5. Take the square root of the variance. This is the standard deviation.

The Greek letter $\sigma$, sigma, is the standard deviation, and $\bar{x}$ is the mean. The square root in step 5 puts the answer back in the same units as the data.

You should find that a value far from the mean gives a very tall bar, because squaring makes large distances count much more. Put all five values together, and every bar, and the variance, drop to zero.

Try it in Python

What the number means

• A small standard deviation means that the values are close to the mean.
• A large one means that they are spread out.
• A standard deviation of $0$ means that all the values are equal.

A second version, for later

Here we divide by $n$, the number of values, because we have all the data. When the data is only a sample of a bigger group, a second version divides by $n - 1$. You will meet it in the next stage. The idea is the same.

Practise with new numbers

Common mistakes

• Adding the distances without squaring. They always add up to zero.
• Stopping at the variance. The standard deviation is its square root.
• Forgetting a zero distance. A value equal to the mean adds $0$ to the squares, but it still counts in $n$.
• Thinking a standard deviation can be negative. It is a square root, so it is zero or more.

Remember these

• Standard deviation: distances from the mean, squared, averaged, then square-rooted.
• The mean of the squared distances is the variance.
• It has the same units as the data. Zero means all the values are equal.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. The values are $3, 5, 7, 9$, with a mean of $6$. What are the squared distances, and what is the variance?

2. Can a standard deviation be negative?

3. The variance of some times is $9\ \text{s}^2$. What is the standard deviation?

4. Every value of a list is $12$. What is its standard deviation?