Enter your numbers below, separated by commas, spaces, or new lines. The calculator finds the standard deviation, variance, mean, sum, and count, and shows the step-by-step working.
The standard deviation calculator on this page computes the standard deviation, variance, mean, sum, and count of any data set, and shows every step of the calculation. Standard deviation is one of the most important and widely used measures in all of statistics because it answers a simple but powerful question: how spread out are the numbers in a data set? A small standard deviation means the values cluster tightly around the average; a large standard deviation means they are scattered widely. From test scores and stock returns to manufacturing tolerances and scientific measurements, standard deviation is the standard yardstick for variability.
This calculator handles both the population standard deviation and the sample standard deviation, plus it estimates confidence intervals for the mean. Just type your numbers into the box, choose whether your data represents an entire population or a sample, and click Calculate.
Standard deviation is a number that describes the typical distance between each data point and the mean (average) of the data set. It is expressed in the same units as the original data, which makes it easy to interpret. If a set of exam scores has a mean of 75 and a standard deviation of 5, you know most students scored roughly within 5 points of 75. If another class has the same mean but a standard deviation of 15, that class had far more variation in performance even though the averages are identical.
The standard deviation is the square root of the variance. Variance is the average of the squared differences from the mean. We square the differences so that values above and below the mean do not cancel each other out, and so that larger deviations are weighted more heavily. Taking the square root at the end returns the measure to the original units, which is why standard deviation is usually reported instead of variance.
The population standard deviation, denoted by the Greek letter sigma (σ), is used when you have data for every member of the group you are studying - the entire population. Because nothing is left out, no correction is needed. The formula is:
σ = √ [ Σ(xi − μ)2 ÷ N ]
Here xi is each individual value, μ (mu) is the population mean, and N is the total number of values. In words: subtract the mean from each value, square the result, add up all those squares, divide by the number of values, and take the square root.
For example, take the data set 1, 3, 4, 7, 8. The mean is (1 + 3 + 4 + 7 + 8) ÷ 5 = 4.6. The squared deviations are (1 − 4.6)² = 12.96, (3 − 4.6)² = 2.56, (4 − 4.6)² = 0.36, (7 − 4.6)² = 5.76, and (8 − 4.6)² = 11.56. Their sum is 33.2, divided by 5 gives a variance of 6.64, and the square root gives a population standard deviation of about 2.577.
In most real research it is impossible to measure every member of a population, so we work with a random sample and use it to estimate the population's standard deviation. The sample standard deviation, denoted by the letter s, uses a small but crucial adjustment: instead of dividing by N, it divides by N − 1. The formula is:
s = √ [ Σ(xi − x̄)2 ÷ (N − 1) ]
Here x̄ (x-bar) is the sample mean. Dividing by N − 1 instead of N is known as Bessel's correction. It compensates for the fact that a sample tends to underestimate the true variability of the full population, because the sample values are, on average, slightly closer to the sample mean than to the unknown population mean. Subtracting one from the denominator makes the estimate larger and, on balance, more accurate. This corrected version is the most commonly used estimator and is what people usually mean when they say "sample standard deviation."
Because the denominator is smaller, the sample standard deviation is always at least as large as the population standard deviation for the same data. The difference matters most for small data sets and shrinks as the sample grows. For the data set 10, 12, 23, 23, 16, 23, 21, 16, the population standard deviation is about 4.899 while the sample standard deviation is about 5.237.
Choosing the right version is essential for correct results. Ask yourself one question: does my data include every value I care about, or is it a subset used to draw conclusions about a larger group?
When in doubt in scientific and business contexts, the sample standard deviation is the safer choice, because most real data sets are samples rather than complete populations.
Whether you use the population or sample formula, the procedure is the same except for the final divisor. Here is the full process:
The calculator above performs all six steps instantly and displays the intermediate sum of squares, the variance, and the final standard deviation so you can follow along and check your own work.
Variance and standard deviation measure the same thing - spread - but in different units. Variance is reported in squared units (squared dollars, squared centimeters), which are hard to interpret intuitively, so standard deviation is usually preferred for communication. Variance, however, is mathematically convenient and appears throughout statistics, especially when combining the variability of independent quantities. The relationship is simple: standard deviation is the square root of variance, and variance is standard deviation squared.
When data follows a normal distribution (the familiar bell curve), standard deviation has a beautifully predictable meaning known as the empirical rule:
This rule is what makes standard deviation so useful for spotting outliers and understanding probability. A value more than two or three standard deviations from the mean is unusual, which is the basis for quality control limits, grading curves, and many statistical tests.
The calculator also reports the standard error of the mean (SEM) and a table of confidence intervals. The standard error measures how precisely your data's mean estimates the true mean of the population, and it equals the standard deviation divided by the square root of the sample size:
SEM = σ ÷ √N
Because the standard error shrinks as the sample size grows, larger samples produce more precise estimates of the mean. A confidence interval expresses this precision as a range: a 95% confidence interval, for instance, is the mean plus or minus roughly 1.96 standard errors, and it means that if you repeated the study many times, about 95% of such intervals would contain the true population mean. The confidence-interval table on this page shows the margin of error at common confidence levels from 68.3% up to 99.999%.
Standard deviation appears in nearly every quantitative field:
Standard deviation is not the only way to measure variability, but it is usually the best. The range (maximum minus minimum) is simple but depends entirely on the two most extreme values, making it sensitive to outliers and ignorant of everything in between. The mean absolute deviation averages the absolute differences from the mean and is easier to explain, but it lacks the convenient mathematical properties that make standard deviation central to statistical theory. The interquartile range describes the middle 50% of the data and is robust to outliers, which makes it a good companion to standard deviation for skewed data. Standard deviation remains the default because it uses every data point and underpins the normal distribution and most inferential statistics.
Population standard deviation (σ) divides the sum of squared deviations by N and is used when your data includes every member of the group. Sample standard deviation (s) divides by N − 1 and is used when your data is a sample of a larger population. Dividing by N − 1 corrects for the slight bias that samples have, making s a better estimate of the true population spread.
If you simply added the raw deviations from the mean, the positives and negatives would cancel out and always sum to zero. Squaring makes every term positive and gives extra weight to larger deviations. Taking the square root at the end returns the measure to the original units.
No. Standard deviation is the square root of an average of squared numbers, so it is always zero or positive. A standard deviation of zero means every value in the data set is identical.
The same units as the original data. If your data is in dollars, the standard deviation is in dollars. Variance, by contrast, is in squared units, which is one reason standard deviation is usually reported instead.
For a population standard deviation you need at least one value. For a sample standard deviation you need at least two, because the formula divides by N − 1 and dividing by zero is undefined. In practice, larger data sets give more reliable measures of spread.
No. All calculations run entirely in your browser. Nothing you type is uploaded or saved to any server, so your data stays private.
This Standard Deviation Calculator is provided for educational and general informational purposes. It uses the standard population and sample formulas and is suitable for coursework, research, and everyday analysis. For high-stakes statistical decisions, confirm your methodology and results with appropriate statistical software or a qualified statistician.