Find Spread, Variance, and Mean with a Full Step-by-Step Breakdown
Paste a list of numbers to instantly calculate sample or population standard deviation, variance, mean, median, mode, and range. Toggle step-by-step view to see every deviation and squared deviation.
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Standard Deviation Calculator
Paste or type numbers to get standard deviation, variance, mean, median, mode, and range with a full step-by-step breakdown.
Enter numbers
Separate with spaces, commas, semicolons, or line breaks.
Calculation Type
Use when your data is a subset drawn from a larger group.
Enter numbers above to see results
Supports spaces, commas, semicolons, and line breaks. Copy from Excel or Sheets.
Sample and Population Modes
Switch between sample standard deviation (uses n-1, Bessel's correction) and population standard deviation (uses n) with a single click. The choice determines whether your data is treated as a subset or the complete group, affecting variance and standard deviation output.
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Nine Statistical Measures at Once
A single calculation returns standard deviation, variance, mean, count, sum, median, min, max, and mode. All values update live as you type or paste numbers, separated by commas, spaces, semicolons, or line breaks.
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Full Step-by-Step Calculation
Click "Show Calculation Steps" to expand a complete walkthrough: data values, mean calculation, a table of each value's deviation and squared deviation, variance formula with substituted numbers, and the final square root step.
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Enter your numbers
Type or paste numbers into the input field. Separate them with spaces, commas, semicolons, or line breaks. You can also copy a column directly from Excel or Google Sheets.
Choose sample or population
Select "Sample" if your data is a subset drawn from a larger group. Select "Population" if you have every member of the group. Results update instantly when you switch.
Read results and expand steps
The main result cards show standard deviation, variance, mean, and count. Use "Show Calculation Steps" to see a full derivation including deviations and the step-by-step formula.
Deep Dive
What Standard Deviation Measures and How It Is Calculated
Standard deviation quantifies how spread out values are relative to their mean. The formulas and their applications are explained below.
What This Calculator Provides
Sample and Population
Switch between n-1 and n formulas with one click.
Paste from Spreadsheets
Copy a column from Excel or Sheets and paste directly.
9 Stats at Once
Std dev, variance, mean, median, mode, range, min, max, count.
Step-by-Step
Full derivation table showing every deviation and squared deviation.
Who Uses It
Who Uses This Calculator
Students, researchers, analysts, and anyone working with numerical data.
Frequently Asked Questions
What is standard deviation and what does it tell me?
Standard deviation measures how far individual values in a dataset typically stray from the mean. A low standard deviation means the data is clustered tightly around the mean (consistent). A high value means data is spread widely (variable). For example, exam scores with a low standard deviation indicate a uniformly performing class; high standard deviation indicates a wide range of performance levels.
What is the difference between sample and population standard deviation?
Sample standard deviation (s) uses n-1 in the denominator. This Bessel's correction makes it an unbiased estimator when your data is a sample from a larger group. Population standard deviation (sigma) uses n and is used when you have data for every member of the group. When in doubt, choose sample -- it is the right choice for most surveys, experiments, and measurement sets.
How do I enter data into the calculator?
Type or paste numbers separated by spaces, commas, semicolons, or line breaks. You can copy a column from Excel or Google Sheets and paste it directly -- the calculator handles the spacing and delimiters automatically. Results update live as you type.
What additional statistics does the calculator show?
Along with standard deviation and variance, the calculator shows: mean (average), count (n), sum, median (middle value), mode (most frequent value or "None" if all values appear once), and range expressed as min - max.
What does the step-by-step view show?
Clicking "Show Calculation Steps" expands a full derivation: (1) all data values with count, (2) mean calculation (sum / n), (3) a table of each value, its deviation from the mean, and its squared deviation, (4) variance formula substituted with actual numbers, and (5) standard deviation as the square root of variance.
What does a high or low standard deviation mean in practice?
For normally distributed data, about 68% of values fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3 (the 68-95-99.7 rule). In practical terms: investment returns with low standard deviation are less volatile; test scores with low standard deviation reflect consistent performance; manufacturing measurements with low standard deviation indicate a stable process.
Ready to Analyze Your Data?
Paste your numbers, choose sample or population mode, and get a complete statistical summary instantly.