Paste a list of numbers to find the sample or population standard deviation, plus the cleaned values, mean, variance, and calculation steps.
Advanced options
How to use our Standard Deviation Calculator
- Enter your list in Data values. You can separate numbers with commas, spaces, semicolons, tabs, or line breaks.
- Choose Data type: pick Sample if your list is only part of a bigger group, or Population if your list includes every value you care about.
- Open Advanced options if you want to change Decimal places or turn on Sort entered values before showing cleaned list.
- Click Calculate.
- Check Number of values and Cleaned values used first. This shows what the calculator actually used after removing blank separators.
- Read Mean, Sum of squared deviations, Variance, and Standard deviation in order to see how the final answer was built.
- Use Divisor used to confirm the mode: sample uses n-1, while population uses n.
- Sanity-check the result: if all your values are close together, Standard deviation should be small; if they are spread far apart, it should be larger.
Definitions
Data values: The list of numbers you want to analyze.
Data type: Tells the calculator whether your list is a sample or a population.
Number of values: How many valid numbers were used after cleaning the input.
Mean: The average of the cleaned values.
Sum of squared deviations: Add up each value's squared distance from the mean. This is the main middle step.
Variance: The average squared spread from the mean. It uses n for a population or n-1 for a sample.
Standard deviation: The square root of variance, so it measures spread in the same units as the original data.
Divisor used: The number under the variance formula: n in population mode or n-1 in sample mode.
Cleaned values used: The exact values the calculator included after handling separators and blanks.
Common mistakes and quick fixes
Mistake: Pasting text like 4, 8, six, 5 into Data values .
Fix: Remove words or symbols so Data values contains numbers only, such as 4, 8, 6, 5.
Mistake: Choosing the wrong Data type and then wondering why Standard deviation does not match classwork.
Fix: Use Sample for part of a larger group and Population for the whole group. Then recheck Divisor used .
Mistake: Entering only one number in Sample mode and expecting a result.
Fix: In Data type set Population for a one-value population, or add at least one more value to Data values for a sample.
Mistake: Assuming repeated commas or blank lines changed the math.
Fix: The calculator ignores empty separators, so verify the actual dataset in Cleaned values used and the total in Number of values .
Mistake: Thinking a big Variance should look similar to Standard deviation .
Fix: Remember that Variance is in squared units, while Standard deviation is in the original units, so the numbers will usually differ.
Mistake: Rounding too much with Decimal places and thinking the result is wrong.
Fix: Increase Decimal places to see more detail, especially when comparing your work to homework or another calculator.
Limitations & Key Assumptions / Boundary Conditions
- The calculator only works with numeric entries. Text, mixed symbols, or invalid number patterns cause an error instead of a result.
- Sample mode needs at least 2 valid values because the sample divisor is n-1. Population mode can use 1 value and returns 0 spread for that case.
- Repeated commas, extra spaces, tabs, semicolons, and blank lines are ignored, but they do not create extra data points.
- The result depends on choosing the correct Data type. The same list will give different variance and standard deviation in sample and population mode.
- Variance is shown in squared units, so it may feel less intuitive than Standard deviation.
- If the cleaned list is very long, the displayed Cleaned values used preview may be shortened for readability even though the full list is still used in the math.
- Results are descriptive only. They summarize spread in the entered list but do not explain why the data varies or whether the data came from a good study.
Methodology
Calculation steps
The calculator first cleans Data values by splitting on commas, spaces, line breaks, tabs, and semicolons, then removing blank pieces. If any remaining piece is not a valid number, it shows an error. Negative numbers and decimals are allowed.
After cleaning, it counts the valid values, adds them, finds the mean, computes each value's distance from the mean, squares those distances, and adds them to get the sum of squared deviations.
mean = (sum of x_i) / n
SS = sum((x_i - mean)^2)
population variance = SS / n
sample variance = SS / (n - 1)
population standard deviation = sqrt(SS / n)
sample standard deviation = sqrt(SS / (n - 1))
Why sample and population differ
Population mode uses all values in the full group, so the divisor is n. Sample mode uses only part of a larger group, so the divisor is n-1. That n-1 adjustment is commonly called Bessel's correction, and it usually makes the sample standard deviation a little larger for the same data.
Mini example
Suppose Data values are 4, 8, 6, 5, 3, 7 in Sample mode.
n = 6
sum = 33
mean = 33 / 6 = 5.5
SS = (4-5.5)^2 + (8-5.5)^2 + (6-5.5)^2 + (5-5.5)^2 + (3-5.5)^2 + (7-5.5)^2 = 17.5
sample variance = 17.5 / 5 = 3.5
sample standard deviation = sqrt(3.5) = 1.8708...
So the values are typically about 1.87 units away from the mean.
How to interpret the outputs
A small Standard deviation means the values stay close to the Mean. A larger one means the values are more spread out. If all values are identical, the sum of squared deviations is 0, so both Variance and Standard deviation are 0.
Error handling and assumptions
The calculator blocks results when no valid numbers are found, when Sample mode has fewer than 2 valid values, or when a token cannot be read as a number. It also prevents NaN and Infinity from being shown. Display rounding changes only how many digits you see, not the underlying calculation.