Standard Deviation Calculator
Calculate sample or population standard deviation from a list or frequency table. Explore real data points, their distances and contributions to spread.
Separate observations with commas, spaces or new lines. Use a decimal point; 1,234 means two observations, not a thousands-grouped number.
Mean 5 · 8 observations · divided by 7
Standard deviation is in the same units as your observations. It measures spread, not the uncertainty of the mean. Nonzero SD is rounded to 6 significant digits.
See the spread
Dots show actual values; their size indicates frequency. The shaded band is one standard deviation either side of the mean. Select a row below to inspect its contribution.
6 of 8 observations lie within mean ± 1 SD. This is the observed count, not a probability prediction.
Mean, variance and calculation steps
- Add the observations and divide by their count. Mean = 5.
- Subtract the mean from each observation, square each difference and add them. Sum of squared deviations = 32.
- Sample variance = 32 ÷ (8 − 1) = 4.57143.
- Standard deviation = √4.57143 ≈ 2.13809. The square root is calculated before display rounding.
Frequency rows count as repeated observations, not reliability weights. Mean and signed distances are rounded to at most 6 decimal places, using scientific notation for very small values; variance and squared distances use 6 significant digits.
Sample or population?
Choose sample when the observations are a subset used to estimate the spread of a larger group. Choose population when they are the complete group you want to describe. The sample formula divides the sum of squared deviations by n − 1; the population formula divides by N. Sample standard deviation needs at least two observations. A one-value population has standard deviation zero.
What the graph can and cannot tell you
The mean is the center line; the band extends one calculated standard deviation in each direction. Values are placed using their exact differences from the mean, so a large common offset does not collapse small differences. Repeated values share a dot. Dots are spread vertically only to keep close values visible; vertical position has no numerical meaning.
The band count describes your entered data. It does not guarantee 68% coverage or a normal distribution. A large distance contributes more to variance because it is squared; this does not by itself make an observation erroneous or justify deleting it. The row inspector lets you make an explicit, reversible edit to explore that effect.
For over 80 distinct values, the plot shows 80 evenly selected groups including both ends; the full dataset still determines the answer and count. Every original row remains available in the inspector.
Input rules and exact arithmetic
Enter finite decimal numbers, optionally scientific notation such as 1.2e3. Use a decimal point; commas are separators, not decimal marks or thousands grouping. In frequency mode use one value, positive integer count per line: 10, 3 means three observations of value 10. No headers, missing cells, fractional counts, arbitrary weights or grouped class intervals are accepted.
Limits: 2,000 observations or frequency rows, total frequency 1,000,000, 100,000 characters, 24 coefficient digits per number, 12 fractional coefficient digits and an exponent from −24 to 24. All decimal sums and variance are calculated with scaled integers; the square root is rounded directly from the exact rational variance, avoiding loss of small differences between large values. Display rounding never changes your input. This is a descriptive calculation, not a confidence interval or a significance test.
For expressions, use the scientific calculator. Method reference: NIST: measures of scale.
