--- 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: 2026-11-12 --- > **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. Take 3, 4, 7, 10 and 11. They add to 35 and there are 5 of them, so the mean is 7. The gaps from 7 are negative 4, negative 3, 0, 3 and 4. Square each gap, and a negative times a negative is positive: 16, 9, 0, 9 and 16, which add to 50. Divide by 5 for the population variance, 10. The standard deviation is its square root, about 3.16 on a calculator, back in the original units. Totals give the same answer faster. For 3, 4, 7, 10 and 11, the total is 35 and the total of squares is 9 + 16 + 49 + 100 + 121 = 295. The mean of squares is 295 over 5, which is 59, and the mean squared is 7 × 7 = 49. Variance is the mean of squares minus the mean squared: 59 minus 49 is 10. *(drawing: Values furthest from 7 give the biggest squares. The 7 itself adds nothing, and the five squares add to 50.)* As when finding a mean from a frequency table, count each value f times. Σ means add them all up. Values 1, 2 and 3 with frequencies 2, 5 and 3 make n = 10. Σfx = 2 + 10 + 9 = 21, so the mean is 2.1. Σfx² = 2 × 1 + 5 × 4 + 3 × 9 = 49, so the mean of squares is 4.9. Variance is 4.9 minus 2.1 squared, which is 4.9 minus 4.41 = 0.49. The standard deviation is 0.7. **Example.** Compare sets by mean and spread together. C with 6, 7, 8 and D with 2, 7, 12 share the mean 7. C strays at most 1 from the mean while D strays up to 5, so D has the larger standard deviation. Stretching values away from the mean pushes the number up. **Tip.** Standard deviation is the root of the variance, so it carries the original units. Adding the same amount to every value moves the mean but leaves the standard deviation unchanged. **Recap.** Square the gaps from the mean, average them for variance, and take the square root 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) ## Optional connections - [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) - [Principal Components and Dimensionality Reduction](https://lightmysky.com/learn/computing/principal-components-and-dimensionality-reduction-mt_l-9XHg_ChZ) - [Stochastic Gene Expression: Bursting, Noise and Cell-to-Cell Variation](https://lightmysky.com/learn/science/stochastic-gene-expression-bursting-noise-and-cell-to-cell-variation-mt_YD6DOmrvng)