Standard Deviation Calculator
Calculate mean, median, variance, and standard deviation for any data set.
The Formula
σ = √[Σ(xᵢ − μ)² / N]
Sample SD:
s = √[Σ(xᵢ − x̄)² / (N−1)]
μ or x̄ = mean · N = count
Mean = 40/8 = 5
Variance = [(9+1+1+1+0+0+4+16)/8] = 4
SD = √4 = 2
What Standard Deviation Tells You
Standard deviation measures how spread out data points are around the mean (average). A low standard deviation means data clusters tightly around the mean. A high standard deviation means data is spread widely. For example, a class where all students scored between 78–82 has a much lower standard deviation than one where scores ranged from 40 to 100, even if both classes had the same average of 80. Standard deviation is the most widely used measure of variability in statistics.
Population vs. Sample Standard Deviation
Population standard deviation (σ) is used when you have data for every member of the group you care about. Sample standard deviation (s) is used when your data is a sample drawn from a larger population — it divides by (n-1) rather than n in its formula. This adjustment, called Bessel's correction, compensates for the tendency of a sample to underestimate the true population spread. When in doubt in most practical situations (surveys, experiments, quality control), use sample standard deviation.
The 68-95-99.7 Rule
In a normal (bell-curve) distribution, approximately 68% of data falls within 1 standard deviation of the mean, about 95% falls within 2 standard deviations, and about 99.7% falls within 3 standard deviations. This rule (also called the empirical rule) is remarkably useful for quickly interpreting data. A test score 2 standard deviations above the mean is in roughly the top 2.5% of all scores — an unusually high result by any measure.
Frequently Asked Questions
What does a standard deviation of 0 mean?
All data points are identical — every value equals the mean exactly. There is no spread whatsoever. In practice, a standard deviation near zero means very low variability in the data, such as a highly precise manufacturing process producing nearly identical parts.
What is variance?
Variance is the square of the standard deviation. It is the average of the squared differences from the mean. Variance is used extensively in statistical theory and calculations, but standard deviation (the square root of variance) is more interpretable because it is in the same units as the original data. A height measurement in inches has a standard deviation in inches, but variance in inches squared.
When should I use median instead of mean?
Use median instead of mean when your data has significant outliers or is heavily skewed. The mean is sensitive to extreme values — a few billionaires in a salary dataset pull the mean much higher than most people earn, while the median is unaffected. For income, home prices, and any right-skewed distribution, median is usually more representative of the typical value.
What is a z-score?
A z-score tells you how many standard deviations a data point is from the mean: z = (x - mean) / standard deviation. A z-score of 2 means the value is 2 standard deviations above the mean. Z-scores allow you to compare values from different datasets on a common scale — for example, comparing a score on one standardized test to a score on a different test.
What is the difference between descriptive and inferential statistics?
Descriptive statistics summarize and describe a dataset you have — mean, median, standard deviation, charts. Inferential statistics use a sample to make conclusions about a larger population — hypothesis tests, confidence intervals, regression analysis. Standard deviation plays a role in both: it describes spread in descriptive stats, and it underpins the calculation of confidence intervals and p-values in inferential stats.