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How to Solve Standard Deviation

Published in Statistics 3 mins read

Standard deviation measures the spread or dispersion of a dataset around its mean (average). A low standard deviation indicates data points are clustered closely around the mean, while a high standard deviation shows data points are more spread out. Here's how to calculate it:

Calculating Standard Deviation: A Step-by-Step Guide

The process involves several steps, applicable to both population standard deviation (σ) and sample standard deviation (s). The key difference lies in the denominator used in step 5: population uses 'N' (total number of data points), while sample uses 'n-1' (total number of data points minus 1). This adjustment in sample standard deviation accounts for the fact that a sample may not perfectly represent the entire population.

  1. Find the Mean (Average): Add all data points and divide by the total number of data points (N for population, n for sample). This is represented by μ (mu) for population and x̄ (x-bar) for sample.

  2. Calculate Deviations: Subtract the mean from each individual data point. These differences are called deviations.

  3. Square the Deviations: Square each deviation from Step 2. This eliminates negative values and emphasizes larger deviations.

  4. Sum of Squared Deviations: Add up all the squared deviations from Step 3. This is known as the sum of squares.

  5. Variance: Divide the sum of squared deviations (from Step 4) by:

    • N (the total number of data points) for population standard deviation.
    • n-1 (the total number of data points minus 1) for sample standard deviation. This result is the variance.

  6. Standard Deviation: Take the square root of the variance (from Step 5). This is the standard deviation (σ for population, s for sample).

Example: Calculating Sample Standard Deviation

Let's say we have the following sample dataset: {2, 4, 6, 8, 10}

  1. Mean (x̄): (2 + 4 + 6 + 8 + 10) / 5 = 6

  2. Deviations:

    • 2 - 6 = -4
    • 4 - 6 = -2
    • 6 - 6 = 0
    • 8 - 6 = 2
    • 10 - 6 = 4

  3. Squared Deviations:

    • (-4)² = 16
    • (-2)² = 4
    • 0² = 0
    • 2² = 4
    • 4² = 16

  4. Sum of Squared Deviations: 16 + 4 + 0 + 4 + 16 = 40

  5. Variance (s²): 40 / (5 - 1) = 10

  6. Sample Standard Deviation (s): √10 ≈ 3.16

Tools and Resources

Many calculators and software packages can compute standard deviation automatically. Online calculators like Calculatorsoup's Standard Deviation Calculator provide a quick and easy way to calculate it. Software such as Excel, SPSS, R, and Python also have built-in functions for this calculation.

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