Standard Deviation Calculator

What this calculator does

Standard deviation measures how far a set of numbers typically sits from its mean. A small value means the observations cluster closely around the average. A large value means they are spread across a wider range. This calculator accepts a list of numbers and reports the count, sum, mean, variance, standard deviation, and a frequency breakdown for every distinct value. You can treat the data as a complete population or as a sample drawn from a larger group.

The choice between population and sample matters. Population standard deviation divides the squared deviations by N, the total number of values. Sample standard deviation divides by N − 1, a correction that makes the estimate less biased when the data represent only part of a larger group.

How to use this calculator

1. Select Population if your numbers include every member of the group you are studying. Select Sample if they are a subset of a larger group.
2. Enter the values separated by commas, spaces, semicolons, or new lines.
3. Click Calculate.
4. Read the headline standard deviation and the summary rows beneath it.

The frequency row lists each distinct value, how often it appears, and its percentage of the data set. The summary also includes the mean and variance so you can check the intermediate quantities, not only the final answer.

Formulas

Mean: μ = (Σ x_i) / N

Population variance: σ² = Σ(x_i − μ)² / N
Population standard deviation: σ = √σ²

Sample variance: s² = Σ(x_i − x̄)² / (N − 1)
Sample standard deviation: s = √s²

Here x̄ is the sample mean and N is the number of observations. The calculator keeps full precision internally and displays up to ten decimal places.

Default example

The default data set is 10, 12, 23, 23, 16, 23, 21, 16. It contains eight values with a sum of 144 and a mean of 18. The population variance is 24, so the population standard deviation is about 4.89898. If you switch to Sample, the variance becomes about 27.42857 and the standard deviation about 5.23723.

The frequency table shows that 23 appears three times (37.5%), 16 appears twice (25%), and 10, 12, and 21 each appear once (12.5% each). Those percentages sum to 100%.

Assumptions and limitations

• The numbers are treated as individual observations, not as grouped frequencies. If your data is already grouped, expand it into one value per observation first.
• The sample calculation needs at least two values. A single value has no spread to measure.
• The population calculation accepts a single value and returns a standard deviation of zero.
• Extreme outliers can dominate the result because every squared deviation enters the calculation.
• The tool assumes the values you enter are the complete list you want to analyze. It does not infer missing observations.
• Very large lists are limited to keep the page responsive, and floating-point arithmetic can produce tiny rounding differences in the last displayed digits.

Frequently asked questions

When should I use the sample standard deviation?

Use the sample version whenever your numbers are a subset of a larger population and you want to estimate that population's spread. Dividing by N − 1 compensates for the fact that the sample mean is estimated from the same data. Use the population version only when every member of the group is included.

Why are variance and standard deviation both shown?

Variance is the average squared deviation. Squaring makes the value difficult to interpret in the original units, so the standard deviation takes the square root and returns to the original scale. The two measures always move together; one is the square of the other.

Does a single outlier change the standard deviation?

Yes, strongly. Because deviations are squared, a value far from the mean contributes disproportionately. A single extreme observation can increase the standard deviation more than several values near the mean. Check the frequency table and the sorted values for unusual observations before relying on the result.

Reference tables

Z-values and error bars

Example frequency table

The frequency table is for illustration; the calculator accepts a list of values and computes frequencies from it.

Source

Reference: Calculator.net — Standard Deviation Calculator.