Standard Deviation

The standard deviation gives students problems at times, but its not really too bad in SL math.

### The General Idea

The simpler concepts of mean, median and mode do not always tell the whole story or at least not enough of the story. Standard deviation (and variance) add another measure in that they both provide a measure of how far the "average data point" is from the mean. Or in other words how spread out the data is.

For example the two data sets below have the same number of data points and the same mean. A simple frequency histogram shows clearly that the data sets have some important differences.

The difference is in the "spread." The second data set is spread out from the mean. Whereas the first data set is much more tightly clumped or grouped around the mean.

The standard deviation is a quantitative measure of spread. Most frequently you will use your GDC to calculator the standard deviation, but the formula is given in the data booklet…

### Formula for Standard Deviation

The standard deviation is given twice in the formula booklet. One is for the population and one is for the sample… In this formula booklet they are identical in function, but slightly different in notation.

For the population:

(1)
\begin{align} \sigma = \sqrt{\frac{\sum_{i=1}^k f_i (x_i-\mu)^2}{n}} \end{align}

For the sample:

(2)
\begin{align} s_n = \sqrt{\frac{\sum_{i=1}^k f_i (x_i-\overline{x})^2}{n}} \end{align}

In the formula $\mu$ and $\overline{x}$ stand for the mean (population and sample respectively).

### Dissecting the Formula

If we look at it piece by piece the formula actually makes some sense.

1. Calculate how far the data point is from the mean.
2. Square the difference to make all values positive.
3. Multiply the squared values by the frequency to account for how often values occur.
4. Sum up all the values…
5. Divide by the number of data point to form an average of sorts.
6. Finally square root the value to "undo" the earlier squaring.

There's a lot to this, but if you go step by step it actually makes sense.

### Using a Ti-84 to find the Standard Deviation

A 5 minute video on how to calculate the standard deviation from a frequency distribution. A bit boring but it gets to the point… It even addresses using mid-interval values for grouped data.

### Variance… Just because?

The variance doesn't come up too often, but it is just the square of the standard deviation. In fact the notation reflects that exact relationship:

Population Variance:

(3)
\begin{align} \sigma^2 = \frac{\sum_{i=1}^k f_i (x_i-\mu)^2}{n} \end{align}

Sample Variance:

(4)
\begin{align} s_n^2 = \frac{\sum_{i=1}^k f_i (x_i-\overline{x})^2}{n} \end{align}

Can't get much more straight forward than that. These equations are also given in your data booklet.