Calculate population and sample standard deviation with full step-by-step working. Shows mean, variance, sum of squares, coefficient of variation, and a visual deviation chart. Accepts comma, space, or line-separated data.
Type or paste your numbers separated by commas, spaces, or new lines. The tool accepts integers, decimals, and negative numbers. As you type, the count, sum, and mean update in real time.
Population standard deviation (σ) divides by N — use when you have data for the entire group. Sample standard deviation (s) divides by N-1 — use when your data is a sample from a larger population. Both are calculated and shown simultaneously.
The step-by-step panel shows: the mean, each value's deviation from the mean, each squared deviation, the sum of squared deviations, the variance, and finally the standard deviation. A visual deviation chart plots each data point relative to the mean and marks ±1σ, ±2σ, and ±3σ boundaries.
Standard deviation measures how spread out data points are from the mean (average). A low standard deviation means data points are clustered close to the mean. A high standard deviation means they are spread widely. It is the most widely used measure of statistical dispersion. For a normal distribution, approximately 68% of data falls within ±1 standard deviation of the mean, 95% within ±2σ, and 99.7% within ±3σ (the empirical rule).
Population standard deviation (σ) is used when you have data for the entire population you are studying. It divides by N. Sample standard deviation (s) is used when your dataset is a subset (sample) of a larger population, and you want to estimate the population standard deviation. It divides by N-1 (Bessel's correction), which corrects for the bias introduced by using a sample instead of the full population. In practice, sample SD is more commonly needed because we rarely have access to the complete population.
Variance is the average of the squared deviations from the mean. Standard deviation is simply the square root of variance. Variance is expressed in squared units (e.g. metres²), which makes it less intuitive. Taking the square root gives standard deviation in the same units as the original data, making it more interpretable. The standard deviation is the more commonly reported statistic, but variance is used extensively in mathematical statistics because it has useful additive properties.
The coefficient of variation (CV) is the ratio of the standard deviation to the mean, expressed as a percentage: CV = (σ/μ) × 100%. It measures relative variability independent of the unit of measurement, making it useful for comparing the spread of datasets with different units or magnitudes. For example, a dataset of heights and a dataset of weights both measured in different units can be compared using CV.
When you calculate standard deviation from a sample rather than the full population, using N in the denominator underestimates the true population standard deviation. Bessel's correction uses N-1 instead of N, which inflates the estimate slightly to correct for this bias. The intuition is that a sample of N values has only N-1 independent pieces of information about spread (because knowing N-1 values and the mean determines the Nth value exactly). N-1 is called the degrees of freedom.
Step 1: Calculate the mean (μ = sum of all values ÷ N). Step 2: For each value, subtract the mean and square the result: (xᵢ - μ)². Step 3: Sum all squared differences: Σ(xᵢ - μ)². Step 4: Divide by N (population) or N-1 (sample) to get variance. Step 5: Take the square root to get standard deviation. This tool shows every one of these steps with your actual data values.
