Standard Deviation Calculator
Enter your numbers to calculate the standard deviation and variance of a data set.
What Is a Standard Deviation Calculator?
A standard deviation calculator measures how spread out the numbers in a data set are from their mean (average). A low standard deviation means the values are clustered close to the average, while a high standard deviation means they are widely scattered. Along with the standard deviation, this tool also reports the variance, mean, and count of values, giving you a complete picture of your data’s variability.
How to Use This Standard Deviation Calculator
Using the tool is straightforward:
- Enter your numbers in the fields. Use the + Add button to include more values.
- Choose whether your data is a sample or a population.
- Click Calculate to see the standard deviation, variance, mean, and value count.
You need at least two numbers to get a meaningful result.
The Formula Behind the Standard Deviation Calculator
The standard deviation is the square root of the variance. The variance is the average of the squared differences from the mean:
Variance (σ² or s²) = Σ(xᵢ − x̄)² / n
where x̄ is the mean and n is the number of values. For a sample, the variance uses (n − 1) as the divisor to give an unbiased estimate of the population variance:
- Population variance: divide by n
- Sample variance: divide by (n − 1)
The standard deviation is simply the square root of the variance.
Example Calculation
Let’s calculate the standard deviation of the data set {4, 8, 6, 5, 7} as a sample:
- Mean: (4 + 8 + 6 + 5 + 7) / 5 = 6
- Squared differences: (4−6)² + (8−6)² + (6−6)² + (5−6)² + (7−6)² = 4 + 4 + 0 + 1 + 1 = 10
- Sample variance: 10 / (5−1) = 2.5
- Standard deviation: √2.5 = 1.5811
| Metric | Value |
|---|---|
| Mean | 6 |
| Variance (sample) | 2.5 |
| Standard deviation | 1.5811 |
Why Standard Deviation Matters
Standard deviation is one of the most important concepts in statistics. It tells you how reliable the mean is as a summary of your data and how much variation exists. In finance, it measures the volatility or risk of an investment. In quality control, it tracks consistency in manufacturing. In education and medicine, it identifies how scores or measurements vary across a population. Understanding spread is essential to interpreting any data set correctly.
How This Standard Deviation Calculator Works
When you click Calculate, the tool collects all non-empty numbers and computes their mean. It then sums the squared difference of each value from the mean. Depending on whether you selected sample or population, it divides this sum by (n − 1) or n to get the variance, and finally takes the square root to produce the standard deviation. All results are rounded to four decimal places, and the entire calculation runs locally in your browser.
Frequently Asked Questions
What is the difference between sample and population standard deviation?
A population includes every member of the group, while a sample is a subset. The population standard deviation divides by n; the sample standard deviation divides by (n − 1) to correct for bias when estimating from a sample.
What does a low vs. high standard deviation mean?
A low standard deviation means values cluster tightly around the mean; a high one means they are spread over a wider range.
Why is variance the square of standard deviation?
Variance is the average of squared deviations, so its units are squared. Taking the square root gives the standard deviation, which is in the same units as the original data and easier to interpret.
Can standard deviation be negative?
No. Standard deviation is always zero or positive because it is the square root of a variance, which cannot be negative.
Which should I use for my data?
Use the population formula if you have data for the entire group you’re studying. Use the sample formula when you have only a sample and want to estimate the population’s standard deviation.