How to Calculate Standard Deviation
Quick Answer
Population SD (σ): σ = √[ Σ(xi - μ)² ÷ N ]
Sample SD (s): s = √[ Σ(xi - x̄)² ÷ (n - 1) ]
Dataset: {4, 7, 13, 2, 9}
Mean = (4+7+13+2+9) ÷ 5 = 7
Variance = [(4-7)²+(7-7)²+(13-7)²+(2-7)²+(9-7)²] ÷ 5 = [9+0+36+25+4] ÷ 5 = 14.8
SD = √14.8 = 3.85
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Step-by-Step Worked Example
Dataset: {10, 14, 8, 6, 12}
- Mean = (10+14+8+6+12) ÷ 5 = 50 ÷ 5 = 10
- Subtract mean and square: (10-10)²=0, (14-10)²=16, (8-10)²=4, (6-10)²=16, (12-10)²=4
- Population variance = (0+16+4+16+4) ÷ 5 = 40 ÷ 5 = 8. Sample variance = 40 ÷ 4 = 10.
- Population SD (σ) = √8 = 2.83. Sample SD (s) = √10 = 3.16.
Population SD vs Sample SD: When to Use Which
- Population SD (σ): Use when you have data for the entire population. Example: exam scores for all students in one class. Divides by N.
- Sample SD (s): Use when your data is a sample from a larger population. Example: a survey of 500 people representing a city. Divides by n - 1 (Bessel's correction).
- Why n - 1? Dividing by n - 1 corrects for bias. A sample tends to underestimate the true population spread.
What Standard Deviation Tells You
| SD Value | Meaning |
|---|---|
| SD = 0 | All values are identical. No variation. |
| Small SD relative to mean | Values are tightly clustered. High consistency. |
| Large SD relative to mean | Values are widely spread. High variability. |
| Within ±1 SD of mean | 68% of values in a normal distribution |
| Within ±2 SD of mean | 95% of values in a normal distribution |
| Within ±3 SD of mean | 99.7% of values in a normal distribution |
Standard Deviation in Real-World Applications
- Finance and investing: Measures the volatility or risk of an investment. High SD = high risk, high potential return.
- Education and exam grading: Used to normalise exam scores. If an exam was too easy or too hard, grading on a curve uses SD.
- Quality control: Products more than 3 SD from the mean are considered defective. Basis of Six Sigma quality methodology.
- Cricket statistics (India, Pakistan, UK, Australia): Batting average and SD of scores quantify both a batsman's ability and consistency.
Frequently Asked Questions
What is a high or low standard deviation?
Whether a SD is high or low depends entirely on context and units. The most useful comparison is the coefficient of variation (CV = SD ÷ mean × 100%), which expresses SD as a percentage of the mean.
What is variance and how does it relate to standard deviation?
Variance is the average of the squared deviations from the mean. Standard deviation is the square root of variance. Standard deviation is in the same units as the original data, making it more intuitive.
Can I calculate standard deviation in Excel or Google Sheets?
Yes. Use STDEV.S() for sample standard deviation or STDEV.P() for population SD. For example: =STDEV.S(A1:A10) calculates the sample SD of values in cells A1 to A10.
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