Standard Deviation by Hand (Step-by-Step) Cheat Sheet

A printable reference covering mean, deviations, variance, population standard deviation, sample standard deviation, and step-by-step calculations for grades 9-12.

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Standard deviation measures how spread out a dataset is from its mean. This cheat sheet gives students a clear by-hand workflow for small datasets, where every calculation can be shown in a table. It is useful for checking calculator results, understanding variability, and comparing data sets in statistics class. The main steps are to find the mean, subtract the mean from each value, square each deviation, add the squared deviations, divide by the correct count, and take the square root. For a population, use $\sigma = \sqrt{\frac{\sum (x - \mu)^2}{N}}$ . For a sample, use $s = \sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}}$ , which uses $n - 1$ because samples estimate population spread.

Key Facts

The mean of a population is $\mu = \frac{\sum x}{N}$ , and the mean of a sample is $\bar{x} = \frac{\sum x}{n}$ .
A deviation is the distance from a value to the mean, written as $x - \mu$ for a population or $x - \bar{x}$ for a sample.
The sum of deviations from the mean is always $\sum (x - \bar{x}) = 0$ , except for small rounding errors.
Population variance is $\sigma^2 = \frac{\sum (x - \mu)^2}{N}$ .
Sample variance is $s^2 = \frac{\sum (x - \bar{x})^2}{n - 1}$ .
Population standard deviation is $\sigma = \sqrt{\frac{\sum (x - \mu)^2}{N}}$ .
Sample standard deviation is $s = \sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}}$ .
Standard deviation is in the original units of the data, while variance is in squared units.

Vocabulary

Mean: The mean is the average value found by dividing the sum of the data values by the number of values.
Deviation: A deviation is the difference between a data value and the mean, such as $x - \bar{x}$ .
Variance: Variance is the average of the squared deviations from the mean, using $N$ for a population or $n - 1$ for a sample.
Standard Deviation: Standard deviation is the square root of variance and describes the typical distance of data values from the mean.
Population: A population is the entire group of data values being studied, with size $N$ .
Sample: A sample is a smaller group chosen from a population, with size $n$ , used to estimate information about the population.

Common Mistakes to Avoid

Dividing by $n$ for a sample, which is wrong because sample standard deviation uses $n - 1$ to better estimate population spread.
Forgetting to square the deviations, which is wrong because positive and negative deviations would cancel out before measuring spread.
Stopping at variance, which is wrong if the question asks for standard deviation because you must take $\sqrt{s^2}$ or $\sqrt{\sigma^2}$ .
Rounding too early, which can make the final standard deviation inaccurate because small errors build up across several steps.
Mixing population and sample notation, which is wrong because $\sigma$ and $\mu$ describe a population while $s$ and $\bar{x}$ describe a sample.

Practice Questions

1 Find the population standard deviation for the dataset $4, 6, 8, 10, 12$ .
2 Find the sample standard deviation for the dataset $3, 5, 5, 7, 10$ .
3 A dataset has $n = 6$ and $\sum (x - \bar{x})^2 = 48$ . Find the sample variance and sample standard deviation.
4 Two classes have the same mean test score, but Class A has a larger standard deviation than Class B. Explain what this means about the scores in the two classes.

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