Calculating Standard Deviation | CodeSignal Learn

Calculating Standard Deviation | CodeSignal Learn

Introduction

Welcome back to Understanding Standard Deviation! You have reached lesson four of five — the moment where everything comes together. In the previous three lessons, you computed deviations from the mean, developed a conceptual picture of standard deviation as typical distance, and learned why squaring those deviations solves the cancellation problem. Today, we connect all of those pieces into a single, complete calculation that produces the population standard deviation for any small dataset.

From Building Blocks to a Complete Formula

Think of the work so far as gathering ingredients for a recipe. You know how to find the mean, measure each value's distance from it, and square those distances to prevent cancellation. What we have not done yet is combine those ingredients in sequence and finish the dish. The population standard deviation formula does exactly that. It chains together the building blocks from our previous lessons and adds two final operations — dividing by the number of values and taking a square root — to produce a single number that tells us how spread out the data is. Let us start by looking at the formula, and then we will walk through a full example.

The Population Standard Deviation Formula

\sigma

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Suppose we record how many hours of sleep a person gets over five nights: 6, 8, 5, 9, 7. Let us compute the population standard deviation one step at a time.

Step 1 — Find the mean.

\bar{x} = \frac{6 + 8 + 5 + 9 + 7}{5} = \frac{35}{5} = 7

The mean is $7$ hours per night. This is the reference point every deviation will be measured from.

Step 2 — Compute each deviation.

Subtract the mean from every data value:

Night	Hours $(x)$	Deviation $(x - \bar{x})$
1	6	$-1$
2	8	$+1$
3	5	$-2$
4	9	$+2$
5	7	$0$

As expected, these deviations sum to zero: $(-1) + 1 + (-2) + 2 + 0 = 0$ . That is exactly why we cannot stop here.

Step 3 — Square each deviation.

Night	Deviation	Squared Deviation $(x - \bar{x})^2$
1	$-1$	$1$
2	$+1$	$1$
3	$-2$	$4$
4	$+2$	$4$
5	$0$	$0$

Squaring eliminates the signs and gives extra weight to larger gaps, just as we explored in lesson three.

Step 4 — Sum the squared deviations.

\sum (x - \bar{x})^2 = 1 + 1 + 4 + 4 + 0 = 10

This total captures all of the spread information in a single number.

Step 5 — Divide by $n$ .

There are $n = 5$ values, so we compute:

\frac{\sum (x - \bar{x})^2}{n} = \frac{10}{5} = 2

This result, $2$ , is the variance — the average squared deviation from the mean. On its own, the variance is hard to interpret because its units are squared hours rather than hours. That is why we need one final step.

Step 6 — Take the square root.

\sigma = \sqrt{2} \approx 1.41

The population standard deviation is approximately 1.41 hours. In plain language, this person's nightly sleep typically differs from the 7-hour average by about 1.41 hours.