Standard Deviation
Spread of values.
What does Standard Deviation mean?
Standard deviation is a measure of how spread out values are from the mean (average) of a data set. A low standard deviation means data points cluster close to the mean, while a high standard deviation indicates they are spread over a wider range. It is one of the most fundamental concepts in statistics, used in fields ranging from finance and science to quality control and social research.
How to calculate Standard Deviation
To calculate the population standard deviation (σ), find the mean of all values, subtract the mean from each value and square the result, compute the average of those squared differences (this is the variance), then take the square root. The formula is: σ = √(Σ(xᵢ − μ)² / N). For the sample standard deviation (s), divide by (N − 1) instead of N to correct for bias: s = √(Σ(xᵢ − x̄)² / (N − 1)). For example, for the values 10, 20, 30, 40, 50, the mean is 30, and the population standard deviation is approximately 14.1421.
FAQ
Population standard deviation (σ) is used when you have data for the entire population. Sample standard deviation (s) is used when your data is a subset (sample) of a larger population. The sample version divides by (N − 1) instead of N, which corrects for the bias that occurs when estimating population variability from a sample. This correction is known as Bessel's correction.
Standard deviation is the square root of variance. While variance measures the average squared deviation from the mean, standard deviation converts that back to the original units of measurement, making it easier to interpret. For example, if your data is in dollars, variance is in dollars squared, but standard deviation is back in dollars.
There is no universal threshold. A standard deviation is high or low relative to the mean and the context of your data. The coefficient of variation (CV = standard deviation / mean × 100%) helps compare spread across different scales. A CV below 15% is often considered low variability, while above 30% is high, though this depends on the field.
In finance, standard deviation measures the volatility of an investment's returns. A higher standard deviation means greater price fluctuation and therefore higher risk. Investors use it to assess risk-adjusted returns, construct portfolios, and compare the stability of different assets.
Standard deviation can be zero when all values in the data set are identical (no spread at all). It can never be negative because it is calculated as the square root of variance, which is a sum of squared differences — both squaring and square root operations always produce non-negative results.