What Is Standard Deviation?
Standard deviation is a measure of how spread out numbers are in a data set. A low standard deviation means data points tend to be close to the mean (average), while a high standard deviation means data points are spread over a wider range of values. It is one of the most commonly used measures of variability in statistics, finance, science, and engineering.
Population vs. Sample Standard Deviation
There are two slightly different formulas. The population standard deviation (σ) is used when you have data for an entire population. The sample standard deviation (s) is used when you only have a sample of the population, and it uses Bessel's correction (dividing by n − 1 instead of n) to provide an unbiased estimate.
The Formula
Population standard deviation:σ = √( Σ(xi − μ)² / N )Sample standard deviation:s = √( Σ(xi − x̄)² / (n − 1) )Where σ is the population standard deviation, s is the sample standard deviation, N is the population size, n is the sample size, xi are individual data points, μ is the population mean, and x̄ is the sample mean.
Step-by-Step Example
For the data set [2, 4, 4, 4, 5, 5, 7, 9]:
- Calculate the mean: (2+4+4+4+5+5+7+9)/8 = 40/8 = 5
- Subtract the mean from each value and square the result: 9, 1, 1, 1, 0, 0, 4, 16
- Sum the squared differences: 9+1+1+1+0+0+4+16 = 32 (this is the variance numerator)
- Population variance: 32 / 8 = 4
- Population standard deviation: √4 = 2
- Sample variance: 32 / (8-1) = 32/7 ≈ 4.57
- Sample standard deviation: √4.57 ≈ 2.14
Related Measures
- Variance: the square of the standard deviation
- Mean (average): the sum divided by the number of values
- Coefficient of variation: standard deviation divided by the mean, useful for comparing variability across different scales
- Range: the difference between maximum and minimum values
Real-World Applications
Standard deviation is used everywhere. In finance, it measures the volatility of stock prices or investment returns. In manufacturing, it quantifies the quality and consistency of a production process. In medicine, it describes the variation in patient responses to treatments. In psychology and education, it captures the spread of test scores. Essentially, whenever you need to understand how consistent or variable a set of data is, standard deviation is the go-to measure.
Frequently Asked Questions
What is the empirical rule (68-95-99.7 rule)?
For a normal distribution, approximately 68% of values lie within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.
Should I use population or sample standard deviation?
Use population standard deviation only when you have data for every member of the population you care about. If you are using a sample to estimate a larger population's variability, use the sample version (dividing by n − 1).
What is a "good" standard deviation?
It depends entirely on context. A standard deviation of 1 inch is large for human height but tiny for the distance between cities. The coefficient of variation (SD/mean) helps compare relative variability.