How Do You Calculate Standard Deviation and Variance?

Variance is the mean of the squared deviations from the mean, while standard deviation is the positive square root of the variance. In IB Mathematics AA SL 4.3, these values are normally calculated using a GDC on Paper 2.

The Calculation Process

For data values $x_1,x_2,\ldots,x_n$, first calculate the mean:

$$\bar{x}=\frac{\sum_{i=1}^{n}x_i}{n}$$

Then subtract the mean from each value, square each deviation, and add the results. Squaring prevents positive and negative deviations from cancelling.

Consider the population data $2,4,4,6$. Its mean is

$$\bar{x}=\frac{2+4+4+6}{4}=4.$$

The squared deviations are $4,0,0,4$, so their sum is $8$. Therefore,

$$\sigma^2=\frac{8}{4}=2$$

and

$$\sigma=\sqrt{2}\approx1.41.$$

The variance is therefore $2$, while the standard deviation is approximately $1.41$. Standard deviation has the same units as the original data; variance has squared units.

To interpret the result, compare the standard deviation with the scale of the data. A small standard deviation means values are clustered near the mean; a larger value means they are more spread out. Because variance uses squared units, it is less directly interpretable, but it is the quantity used in many later statistical formulas.

If the data are supplied in a frequency table, enter each value in one GDC list and its frequency in a second list. Set the frequency-list option correctly before running one-variable statistics; otherwise, the calculator treats every listed value as occurring once.

Population or Sample?

In IB Mathematics AA SL, the data set is treated as the population unless the question states otherwise. On a GDC, select $\sigma_x$ for population standard deviation and $S_x$ for sample standard deviation.

Exam Technique

This is primarily an SL Paper 2 technology skill. Enter the data into a statistics list, run one-variable statistics, and report the correct GDC value, usually to three significant figures. A common misconception is that variance is the square root of standard deviation; the correct relationship is $\text{variance}=(\text{standard deviation})^2$.