Standard Deviation Calculator – Find Mean, Variance & Std Dev
The Standard Deviation Calculator helps you quickly analyze any data set by computing the mean, variance, standard deviation, and more. Whether you're working with a sample or an entire population, this free tool delivers accurate statistical results in seconds. Simply enter your numbers and choose between sample or population mode to get started.
Enter numbers separated by commas, spaces, or new lines.
Use 'Sample' for a subset of a population; use 'Population' if you have the entire data set.
Your results will appear here
How to Use This Calculator
1. Enter your data values into the 'Data Set' field, separated by commas, spaces, or new lines. 2. Select 'Sample' if your data represents a subset of a larger population, or 'Population' if you have every value in the data set. 3. Click 'Calculate' to instantly see the count, mean, median, mode, range, variance, standard deviation, standard error, and sum of squares. 4. Review the results — the standard deviation is highlighted as the primary output. 5. To analyze a different data set, clear the field and enter your new numbers.
What Is Standard Deviation?
Standard deviation is a measure of how spread out the values in a data set are around the mean. A low standard deviation means values are clustered close to the mean, while a high standard deviation indicates they are spread over a wider range.
Sample vs. Population Standard Deviation
There are two types of standard deviation depending on your data:
- Population Standard Deviation (σ): Used when you have data for every member of the group. The variance is divided by N (the total count).
- Sample Standard Deviation (s): Used when your data is a subset of a larger population. The variance is divided by N − 1 (Bessel's correction), which corrects for bias in the estimation.
Step-by-Step Formulas
1. Calculate the Mean
Mean (x̄) = (Σx) / N
Add all values together and divide by the count of values.
2. Calculate the Sum of Squares
SS = Σ(xᵢ − x̄)²
Subtract the mean from each value, square the result, then sum all squared differences.
3. Calculate Variance
Population Variance: σ² = SS / N
Sample Variance: s² = SS / (N − 1)
4. Calculate Standard Deviation
Take the square root of the variance: σ = √(σ²) or s = √(s²)
5. Standard Error
Standard Error (SE) = s / √N
The standard error estimates how much the sample mean is likely to vary from the true population mean.
Other Statistics Explained
- Median: The middle value when the data is sorted in order. For an even count, it is the average of the two middle values.
- Mode: The value(s) that appear most frequently in the data set.
- Range: The difference between the maximum and minimum values.