Grades Geometry & Functions StatisticsStandard Deviation & Variance

Standard Deviation & Variance

📊 Statistics

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Standard Deviation & Variance

Measure the spread of data using variance and standard deviation.

While the mean measures center, variance and standard deviation measure spread. Variance is the average squared deviation from the mean; standard deviation is its square root and shares the same units as the data.

\bar{x} = \frac{1}{n}\sum_{i=1}^n x_i \qquad \sigma^2 = \frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})^2 \qquad \sigma = \sqrt{\sigma^2}

Computing Standard Deviation

1Calculate the mean x̄.

2Subtract the mean from each data point to find deviations.

3Square each deviation.

4Average the squared deviations to get the variance σ².

5Take the square root of the variance to get σ.

🔢Computing σ for {2, 4, 4, 4, 5, 5, 7, 9}

Mean = 40/8 = 5. Deviations²: (2−5)²=9, (4−5)²=1, (4−5)²=1, (4−5)²=1, (5−5)²=0, (5−5)²=0, (7−5)²=4, (9−5)²=16. Variance = 32/8 = 4. σ = 2.

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Remember This!

Sample standard deviation uses n − 1 in the denominator (Bessel's correction) instead of n. Use σ (population) when you have all data; use s (sample) when estimating from a sample.

✏️ Try It!

A dataset has values {10, 20, 30}. What is the mean and population standard deviation?

Take Quiz 📝 — 24 Questions