Standard deviation is a number that describes how spread out a dataset is from its mean. It matters because two datasets can have the same average but very different amounts of variation. Calculating it by hand helps you see every step instead of treating it as a calculator button.
The process turns raw data into deviations, squared deviations, variance, and finally standard deviation.
The key idea is to measure each value's distance from the mean, then combine those distances into one useful summary. Deviations are squared so that positive and negative differences do not cancel and larger differences count more. For a full population, divide by the number of data values, but for a sample, divide by one less than the number of data values.
The final square root puts the answer back into the original units, making it easier to interpret.
Key Facts
- Mean: x̄ = (sum of all data values) / n
- Deviation from the mean: deviation = x - x̄
- Population variance: σ² = Σ(x - μ)² / N
- Population standard deviation: σ = sqrt(Σ(x - μ)² / N)
- Sample standard deviation: s = sqrt(Σ(x - x̄)² / (n - 1))
- Standard deviation is measured in the same units as the original data.
Vocabulary
- Mean
- The mean is the arithmetic average found by adding all data values and dividing by the number of values.
- Deviation
- A deviation is the difference between a data value and the mean.
- Variance
- Variance is the average of squared deviations from the mean, using either a population or sample formula.
- Standard Deviation
- Standard deviation is the square root of variance and measures the typical spread of values from the mean.
- Sample
- A sample is a subset of a larger population used to estimate characteristics of the whole group.
Common Mistakes to Avoid
- Forgetting to square the deviations: this is wrong because positive and negative deviations can cancel to zero if they are added directly.
- Using n instead of n - 1 for a sample: this is wrong because sample standard deviation needs Bessel's correction to better estimate population spread.
- Stopping at variance instead of taking the square root: this is wrong because variance is in squared units, while standard deviation must return to the original units.
- Rounding too early in the calculation: this can change the final answer, so keep extra decimal places until the last step.
Practice Questions
- 1 Find the population standard deviation of the dataset 2, 4, 4, 4, 5, 5, 7, 9.
- 2 Find the sample standard deviation of the dataset 10, 12, 13, 15, 20. Round your answer to two decimal places.
- 3 Two classes both have a mean test score of 80. Class A has a standard deviation of 3, and Class B has a standard deviation of 14. Explain what this tells you about the score patterns in the two classes.