Free standard deviation calculator: paste numbers separated by commas or spaces and choose sample or population to get mean, variance and standard deviation.
Your results will appear here.
Enter your values and press Calculate.
Paste numbers separated by commas, spaces or new lines, then choose sample or population. The calculator finds the mean, subtracts it from each value, squares the deviations, and averages them: dividing by n - 1 for a sample (Bessel's correction) or by n for a population. That average is the variance; its square root is the standard deviation, back in the original units. It also reports the count, minimum and maximum.
Standard deviation measures how spread out numbers are around their mean. The calculation has five steps: find the mean, subtract the mean from each value, square each deviation, average the squared deviations (this average is the variance), and take the square root.
Sample standard deviation: s = sqrt( sum(x - mean)² / (n - 1) ). Population standard deviation: sigma = sqrt( sum(x - mean)² / n ). The only difference is the denominator. The formula looks intimidating in symbols, but it is just the five steps listed below. Sample mode divides by n - 1, known as Bessel's correction, which corrects the downward bias that comes from estimating a whole population's spread from a smaller sample. Because the deviations are squared before averaging, standard deviation is always zero or positive, and it comes out in the same units as your data.
Paste or type your numbers into the data box. Commas, spaces, tabs and new lines all work, so you can copy straight from a spreadsheet, for example: 2, 4, 4, 4, 5, 5, 7, 9. Decimals and negative numbers are accepted.
Choose the method: Sample if your data is a subset used to estimate a larger population, Population if your data is the entire group you care about. Sample mode needs at least 2 values, since dividing by n - 1 is undefined for a single data point. The calculator returns the mean, variance, standard deviation, count, minimum and maximum.
Take the dataset 2, 4, 4, 4, 5, 5, 7, 9 in sample mode.
Step 1, mean: (2 + 4 + 4 + 4 + 5 + 5 + 7 + 9) / 8 = 40 / 8 = 5. Step 2 and 3, squared deviations: (2-5)² = 9, (4-5)² = 1 three times, (5-5)² = 0 twice, (7-5)² = 4, (9-5)² = 16. Step 4, variance: sum = 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32; divide by n - 1 = 7, giving 4.57. Step 5, standard deviation: sqrt(4.57) = 2.14.
The same data in population mode divides by 8 instead of 7: variance = 4.00, standard deviation = 2.00. The sample figure is always a little larger, and the gap matters most for small datasets.
The choice changes only the denominator, but it is the most common source of wrong answers:
| Question | Sample | Population | |---|---|---| | When to use it | Your data is a sample used to estimate a larger population | Your data is the entire group you want to describe | | Symbol | s | sigma | | Mean symbol | x-bar | mu | | Variance denominator | n - 1 | n |
Examples: 50 people surveyed from a city is a sample; every student in one specific class, when the question is about exactly that class, is a population. Weekly quality checks from ongoing production are a sample; every product from one finished batch is a population. When in doubt, most statistics courses and tools like Excel's STDEV.S default to sample mode.
Variance and standard deviation describe the same spread on different scales. Variance is the average of the squared deviations, so it lives in squared units: dollars squared, kg squared, seconds squared. That makes it awkward to interpret directly.
Standard deviation is the square root of variance, which brings the measure back into the original units: dollars, kg, seconds. A standard deviation of 2.14 on test scores means scores typically vary by about 2.14 points around the mean, an immediately meaningful sentence. Variance stays useful inside statistical formulas (variances of independent sources add directly), while standard deviation is the number you report and interpret.
Choosing sample or population by habit. Define the population from the question you are answering, not from how the data arrived.
Mixing units or conditions. Do not combine inches with millimeters, or measurements taken under different conditions, without converting first. Standard deviation assumes the observations are comparable.
Ignoring outliers. Because deviations are squared, one extreme value inflates standard deviation far more than it inflates the mean. Check for data entry errors before trusting the result.
Deleting a flagged outlier automatically. A large deviation only proves distance from the mean, not that the value is wrong. Investigate, do not auto-delete.
Reading meaning into zero. A standard deviation of zero only means every value is identical; it says nothing about whether the values are correct.
Sample standard deviation divides the sum of squared deviations by n - 1 and is used when your data is a subset estimating a larger population. Population standard deviation divides by n and is used when your data is the entire group you care about. The sample figure is always slightly larger. Choose the method in the calculator above to get the right one.
Use sample mode when your observations are a subset used to characterize or estimate a larger population: survey respondents, weekly quality checks, experimental trials. Use population mode only when the data is every member of the group your question is about. When in doubt, sample mode is the safer default and matches Excel's STDEV.S.
Bessel's correction is the n - 1 denominator used in sample variance and sample standard deviation. Dividing by one less than the sample size corrects the downward bias that appears when estimating a population's spread from a smaller sample. For large datasets the difference is tiny; for small ones it matters a lot.
First find the mean by adding all values and dividing by the count. Subtract the mean from each value and square each result. Add those squared deviations, divide by n - 1 for a sample (or n for a population) to get variance, then take the square root. The worked example section above walks through 2, 4, 4, 4, 5, 5, 7, 9 in full.
Variance is the average of the squared deviations, so it is expressed in squared units (dollars squared). Standard deviation is the square root of variance, which returns the measure to the original units (dollars). Report and interpret the standard deviation; use variance inside statistical formulas, where variances of independent sources add directly.
It tells you the typical distance of values from the mean, in the original units. A small standard deviation means values cluster near the mean; a large one means they are scattered widely. It is meaningful only in context: compare it with the scale of the measurements and the mean. For roughly normal data, about 68% of values fall within one standard deviation of the mean.
Sample mode requires at least 2 values, because dividing by n - 1 is undefined for a single data point. Population mode works with a single value and returns zero. Enter at least two numbers separated by commas, spaces or new lines; decimals and negative numbers are fine.
For a sample, Excel uses =STDEV.S(range); for a full population, use =STDEV.P(range). STDEV.S applies the n - 1 method and STDEV.P divides by n, so they should match this calculator's two modes exactly. This makes a spreadsheet a handy independent check of your result.