Standard Deviation Calculator

How it works

For n values with mean x̄, population variance is Σ(x−x̄)²/n. Sample variance divides the same centered sum by n−1; taking a square root gives the corresponding standard deviation, in the original units. Use the population result when describing the complete population of interest, or the sample result when the data are a sample and that estimator fits your purpose. The tool cannot decide whether the data were sampled appropriately. A single value has population standard deviation zero, but sample variance and sample standard deviation are undefined.

The implementation uses compensated summation and a centered second pass, with a correction for rounding of the mean. This avoids the particularly unstable shortcut E[x²]−E[x]² for a large offset and small spread. It still uses IEEE 754 binary64 browser numbers: decimal inputs are approximate, and digits lost during parsing cannot be recovered by a later formula. A decimal whose normalized value does not survive conversion to a browser number and back to decimal text is rejected with a precision error, preventing distinct entered values from silently collapsing together. This round-trip guard is not a promise of exact decimal arithmetic: 0.1 still has a binary approximation. Large magnitudes and very small relative spreads trigger a visible precision warning. Rescale at the original source when necessary; subtracting from already rounded values cannot restore missing information.

Median, Q1 and Q3 use type-7 linear interpolation at index (n−1)p in the sorted values, matching R's default quantile convention. Quartiles are not uniquely defined across software: a method based on separate lower and upper halves can give different answers. Interquartile range is Q3−Q1. Values below Q1−1.5×IQR or above Q3+1.5×IQR are flagged, never removed. A flag is a reason to review a value, not evidence that it is an error. The box plot's whiskers deliberately show the full minimum and maximum, including flagged values, and are labeled accordingly.

Number-list mode treats commas and whitespace as separators. Thus 1,234 means values 1 and 234, not a grouped integer; use 1234 for one thousand two hundred thirty-four. Decimal points and e notation are accepted, but currency symbols, units, arithmetic expressions, Infinity, NaN and localized decimal commas are not. CSV mode requires a header row with unique, nonblank names and a consistent record shape. Choose comma, semicolon or tab, read the columns, select one numeric column and set the blank-cell policy. Invalid values in that selected column are never converted to zero. Unselected text columns are not analyzed.

Limits are 1 MiB of UTF-8 input, 10,000 numbers in list mode, or 10,000 CSV records including the header, with 200 columns and 200,000 cells under the shared CSV guard. Each number may contain at most 200 characters; nonzero magnitudes must lie between 1e−100 and 1e100. Inputs outside the range, including values that would underflow to zero, are rejected. A local file must be nonempty valid UTF-8. Blank lines in list mode are ignored; an empty comma-separated field is rejected.

The displayed summary rounds to 12 significant digits. CSV preserves every computed numeric value and original record number, deviation from the mean, and a 0/1 outlier flag. JSON also includes the full summary, method, blank policy, selected column index and omitted count. Original values are represented as parsed numbers, not guaranteed exact original decimal strings. Changing data or a setting removes the old answer and exports. The dataset remains in tab memory with no upload, persistent history or URL input. Optional site analytics records fixed action labels and input mode, never the values, filenames or column names.

Common use cases

• Compare consistency in a small manufacturing or timing dataset using the appropriate sample or population measure.
• Review a CSV column without treating blanks or text as zero.
• Check how quartile conventions affect an IQR outlier review.
• Keep observations and calculation settings together for a spreadsheet review.

Frequently asked questions

Standards and limitations

• NIST: measures of scale, including standard deviation
• R documentation: nine quantile algorithms and default type 7
• NIST: box plots and interquartile range