Variance and standard deviation

Japanese school year: Math I

What you learn

You will learn to measure data dispersion by calculating variance, the mean of squared deviations, and standard deviation. These statistics are essential for analyzing fluctuations in test scores and measurement errors. Prior knowledge of calculating the mean and basic operations with square roots will be very helpful.

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Key points

Use this formula to quantify how widely data is spread around the average (variance $s^2$ ). It computes the average of the squared deviations between each data point $x_i$ and the mean $\bar{x}$ across $n$ values.

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

This is a convenient shortcut formula for calculating variance $s^2$ . You can compute it by subtracting the square of the mean $(\bar{x})^2$ from the average of the squared data points $x_i^2$ .

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

Use this formula to express data spread in the same measurement units as the original data (standard deviation $s$ ). Taking the square root of the variance restores original units for intuitive comparison.

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

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Practice 1Coming soon