Standard Deviation Examples in Statistics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Standard Deviation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Statistics.

Concept Recap

Standard deviation is a measure of how spread out data values are from the mean, representing the typical distance of data points from the average. A small standard deviation means data clusters tightly around the mean; a large one means data is widely spread.

If the mean is 'home base,' standard deviation tells you how far data points typically wander from home. Small SD = data clusters close to the mean (like a tight group of friends). Large SD = data is scattered (friends spread all over town).

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Standard deviation is the typical distance of data points from the mean. A small SD means data is tightly clustered; a large SD means it is widely spread.

Common stuck point: Students confuse standard deviation with variance. Variance is the average squared distance; SD is the square root of variance and has the same units as the data.

Sense of Study hint: First, find the mean of the data. Then subtract the mean from each value and square the result. Next, find the average of those squared differences (that is the variance). Finally, take the square root of the variance to get the standard deviation.

Worked Examples

Example 1

medium
Calculate the population standard deviation of: 4, 8, 6, 2, 10.

Solution

  1. 1
    Step 1: Mean = \frac{4+8+6+2+10}{5} = 6.
  2. 2
    Step 2: Squared deviations: (4-6)^2=4, (8-6)^2=4, (6-6)^2=0, (2-6)^2=16, (10-6)^2=16.
  3. 3
    Step 3: Variance = \frac{4+4+0+16+16}{5} = \frac{40}{5} = 8. Standard deviation = \sqrt{8} \approx 2.83.

Answer

\sigma \approx 2.83
Standard deviation measures the typical distance of data points from the mean. It is the square root of the average of squared deviations, giving a measure of spread in the same units as the data.

Example 2

hard
Dataset A: {5, 5, 5, 5}. Dataset B: {2, 4, 6, 8}. Without full calculation, which has a larger standard deviation and why?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
Calculate the population standard deviation of: 10, 12, 14.

Example 2

medium
Calculate the population standard deviation of the data set: 3, 3, 7, 7.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

mean fair sharevariability intro