--- title: "Standard Deviation and Variance" description: "Measure spread by how far the values sit from the mean on average, using squared deviations so that distances on both sides count. Compare data sets by mean and standard deviation together." canonical: https://lightmysky.com/learn/mathematics/standard-deviation-and-variance-mt_eNpow5tIX2 source: https://lightmysky.com/learn/mathematics/standard-deviation-and-variance-mt_eNpow5tIX2.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Standard Deviation and Variance Measure spread by how far the values sit from the mean on average, using squared deviations so that distances on both sides count. Compare data sets by mean and standard deviation together. Subject: Mathematics · Area: Data & Statistics · Ages 16 to 17 Page: https://lightmysky.com/learn/mathematics/standard-deviation-and-variance-mt_eNpow5tIX2 ## Ready when they can - Calculate the standard deviation of a small data set by hand - Compute it from Σx and Σx² for a frequency table - Compare two sets with the same mean but different spreads ## Lesson: Measuring spread with standard deviation Variance and standard deviation measure spread around the mean. For 2, 4, 4, 4, 5, 5, 7, 9, the values add to 40 and there are 8 of them, so the mean is 5. The gaps from 5 are negative 3, negative 1, negative 1, negative 1, 0, 0, 2 and 4. Square each gap to get 9, 1, 1, 1, 0, 0, 4 and 16, which add to 32. Divide by 8 for the population variance 4, and take the root for the standard deviation 2, back in the original units. With big tables, totals do the same job faster. For those 8 values the total is 40 and the total of squares is 232. The mean of squares is 232 over 8, which is 29, and the mean squared is 25. Variance is the mean of squares minus the mean squared: 29 minus 25 is 4. **Example.** Compare sets by mean and spread together. Sets A with 4, 5, 6 and B with 1, 5, 9 share the mean 5. A strays at most 1 from the mean while B strays up to 4, so B has the larger standard deviation. Stretching values out pushes the number up. **Tip.** Standard deviation is the root of the variance, so it always carries the original units. Check the means match before you compare spreads, or the wider number may just follow a bigger mean. **Recap.** Square the gaps from the mean, average them for variance, and root it for standard deviation. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Estimating the Mean from Grouped Data](https://lightmysky.com/learn/mathematics/estimating-the-mean-from-grouped-data-mt_3PHP7NLZAw) - [Quartiles and the Interquartile Range](https://lightmysky.com/learn/mathematics/quartiles-and-the-interquartile-range-mt_vN10d1I9dA) ## Opens up - [The Normal Distribution and Standardised Scores](https://lightmysky.com/learn/mathematics/the-normal-distribution-and-standardised-scores-mt_Cs32DT9KnA) - [Operators, Eigenvalues and Measurement](https://lightmysky.com/learn/science/operators-eigenvalues-and-measurement-mt_E9C_9DlwD2) - [Point Estimation: Bias, Variance and Consistency](https://lightmysky.com/learn/mathematics/point-estimation-bias-variance-and-consistency-mt_I374r2UU_i) - [Uncertainty in a Measurement and Where It Comes From](https://lightmysky.com/learn/science/uncertainty-in-a-measurement-and-where-it-comes-from-mt_JXw4D9DReI) - [Signal, Noise and the Limit of Detection](https://lightmysky.com/learn/science/signal-noise-and-the-limit-of-detection-mt_NLtCWo_qlB)