Standard Deviation

Why Standard Deviation Exists

When you calculate variance, you square the deviations $(x_i - \bar{x})$. This squaring changes the units. If the original data is in centimetres, the variance is in square centimetres — a unit that has no direct physical meaning for the spread of the data. To get a measure of dispersion that is in the same units as the original observations, you take the positive square root of the variance. That square root is called the standard deviation.

The standard deviation is the most widely used measure of dispersion. It tells you, on average, how far each observation lies from the mean, but in the original units. It is denoted by the Greek letter $\sigma$ (sigma).

This is equation (1) from the textbook. Notice that the expression inside the square root is exactly the variance $\sigma^2$. So the relationship is simple:

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Worked Example: Ungrouped Data

The textbook illustrates the calculation with a concrete example. Let's walk through it step by step.

Example 8: Find the variance and standard deviation of the data:

6, 8, 10, 12, 14, 16, 18, 20, 22, 24

Step 1 — Organise the data and choose an assumed mean.

There are $n = 10$ observations. The data is evenly spaced, so the step-deviation method works well. Choose an assumed mean $A = 14$. The common class size (the step) is $h = 2$.

Step 2 — Compute the step deviations $d_i$.

For each $x_i$, calculate $d_i = \frac{x_i - A}{h} = \frac{x_i - 14}{2}$.

Step 3 — Compute the actual mean $\bar{x}$.

First find $\sum d_i$:

The mean using step deviation:

Step 4 — Compute the variance. …