Standard Deviation & Variance
Statistics, Descriptive
Intro: We compute mean, deviations, squared deviations, and then variance/standard deviation using your chosen denominator: sample uses $n-1$; population uses $n$.
Worked example
- Data: 2,2,3,5,9; Type: Sample
- Mean: $$\bar{x}=\frac{2+2+3+5+9}{5}=\frac{21}{5}=4.2.$$
- Deviations: $$(x_i-\bar{x})=(-2.2,\,-2.2,\,-1.2,\,0.8,\,4.8).$$
- Squared deviations sum: $$(-2.2)^2+(-2.2)^2+(-1.2)^2+0.8^2+4.8^2=4.84+4.84+1.44+0.64+23.04=34.80.$$
- Sample variance and standard deviation: $$s^2=\frac{34.80}{5-1}=8.70,\qquad s=\sqrt{8.70}\approx2.953.$$
- Answer: $$\boxed{s^2=8.70,\; s\approx2.953}.$$
- Data: 2,2,3,5,9; Type: Population
- Population mean equals sample mean here: $$\mu=4.2.$$
- Use $n=5$ in the denominator: $$\sigma^2=\frac{34.80}{5}=6.96,\qquad \sigma=\sqrt{6.96}\approx2.640.$$
- Answer: $$\boxed{\sigma^2=6.96,\; \sigma\approx2.640}.$$
FAQs
When should I use n−1?
Use the sample formula (n−1) when your data are a sample from a larger population and you are estimating the population variance/SD. Use the population formula (n) only when your dataset is the entire population.
Do you handle decimals and fractions?
Yes—values can be integers, decimals, or fractions; we keep exact arithmetic where possible and also provide decimals.
What about missing values or non-numeric entries?
We ignore blank entries and flag non-numeric tokens so you can correct them.
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