How to Calculate Standard Deviation

Standard deviation is a statistical measurement that shows how much variation or dispersion exists from the mean (average) of a data set. It's widely used in fields such as finance, science, engineering, and more, to measure risk, uncertainty, and variability in datasets.

In this guide, we'll walk through how to calculate standard deviation, step by step, using the two main formulas: population standard deviation and sample standard deviation. We'll also provide calculators to make the process easier.

What is Standard Deviation?

Standard deviation measures how spread out numbers are in a dataset. If the numbers are close to the mean, the standard deviation will be low. If the numbers are spread out far from the mean, the standard deviation will be higher.

The two main types of standard deviation calculations are:

• Population Standard Deviation: Used when you have the entire dataset (population) for the study.
• Sample Standard Deviation: Used when you have a sample of the population and want to estimate the population's standard deviation.

Standard Deviation Equation

The standard deviation formula differs depending on whether you're calculating the population or sample standard deviation. Below are both formulas.

Population Standard Deviation Formula

When you have data for an entire population, use this formula:

σ = √( Σ (x - μ)² / N )

Where:

• σ is the population standard deviation.
• Σ denotes the sum of all the data points.
• x is each individual data point.
• μ is the population mean (average).
• N is the total number of data points in the population.

Sample Standard Deviation Formula

If you're working with a sample of the population, use the sample standard deviation formula:

s = √( Σ (x - x̄)² / (n - 1) )

Where:

• s is the sample standard deviation.
• Σ denotes the sum of all the data points.
• x is each individual data point.
• x̄ is the sample mean (average).
• n is the number of data points in the sample.
• n - 1 is used to correct bias in the estimation of the population variance from a sample.

Step-by-Step: How to Calculate Standard Deviation

Step 1: Calculate the Mean

Start by calculating the mean (average) of the dataset. To find the mean, sum all the data points and divide by the total number of data points:

Mean (μ or x̄) = Σx / N for population mean, or x̄ = Σx / n for sample mean.

For example, in the dataset [5, 10, 15, 20, 25], the mean is:

Mean = (5 + 10 + 15 + 20 + 25) / 5 = 15

Step 2: Subtract the Mean from Each Data Point

Next, subtract the mean from each data point. For example, in the dataset [5, 10, 15, 20, 25], with a mean of 15, we get the following deviations:

[5 - 15, 10 - 15, 15 - 15, 20 - 15, 25 - 15] = [-10, -5, 0, 5, 10]

Step 3: Square Each Deviation

Square each of the resulting deviations to eliminate negative values:

[-10², -5², 0², 5², 10²] = [100, 25, 0, 25, 100]

Step 4: Find the Mean of These Squared Deviations

For population standard deviation, divide the sum of squared deviations by the number of data points (N). For sample standard deviation, divide by (n - 1) to account for sample size bias.

Population Variance = (100 + 25 + 0 + 25 + 100) / 5 = 50

Step 5: Take the Square Root

The last step is to take the square root of the variance. This gives you the standard deviation:

Population Standard Deviation = √50 ≈ 7.07

The sample standard deviation would be calculated similarly, but you would divide by n - 1 instead of N in Step 4.

Conclusion

Standard deviation is a powerful tool in statistics that helps to measure how much variation exists in a dataset. Whether you are calculating the population or sample standard deviation, both formulas follow similar steps. You can also use our calculators below to quickly find the standard deviation for your dataset.