Quiz score consistency
Use sample standard deviation to see how tightly a class's quiz scores cluster around the average.
Data: 72, 75, 78, 80, 85
Mean = 78, sample SD ≈ 5.05
Standard Deviation Calculator
Our free standard deviation calculator computes the standard deviation, variance, mean and other statistical measures from your data. Simply input your values, choose between population or sample calculation, and get accurate results instantly.
Standard Deviation Calculator
Enter your data and select calculation type to find the standard deviation and other statistical measures.
Separate values with spaces, commas, semicolons, or line breaks
Sample (n-1) is commonly used when your data is a subset of a larger population
Results
Enter data and calculate to see results
What is Standard Deviation?
Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. Standard deviation is represented by the Greek letter σ (sigma) for a population and the letter s for a sample.
Standard Deviation Formulas
There are two main formulas for standard deviation, depending on whether you're calculating it for an entire population or a sample:
| Type | Formula | Description |
|---|---|---|
| Population SD (σ) | \( \sigma = \sqrt{\frac{\sum(x_i - \mu)^2}{N}} \) | Used when data represents the entire population |
| Sample SD (s) | \( s = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}} \) | Used when data is a sample from a larger population |
Step-by-Step Calculation Process
Calculate the mean (average) of all values
\( \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \)Find the deviation of each value from the mean
\( d_i = x_i - \bar{x} \)Square each deviation
\( d_i^2 = (x_i - \bar{x})^2 \)Calculate the variance (average of squared deviations)
\( s^2 = \frac{\sum d_i^2}{n-1},\ \sigma^2 = \frac{\sum d_i^2}{N} \)Take the square root of the variance to get the standard deviation
\( s = \sqrt{s^2},\ \sigma = \sqrt{\sigma^2} \)
How to Use the Standard Deviation Calculator
- Enter your data in the text area. You can separate values by spaces, commas, semicolons, or line breaks.
- Select whether your data represents a sample or the entire population.
- Click the "Calculate" button to compute the standard deviation and other statistics.
- View the results including standard deviation, variance, mean, minimum, maximum, count, and sum.
Use sample standard deviation when your values come from a subset of a larger group. Use population standard deviation only when the list includes every value in the group you are studying.
Applications of Standard Deviation
Standard deviation is widely used in various fields to understand variability and make informed decisions:
- Finance: Measuring investment risk and volatility in stock markets
- Quality Control: Monitoring manufacturing processes and ensuring consistency
- Research: Analyzing experiment results and determining statistical significance
- Education: Assessing test scores and student performance distributions
Standard Deviation Examples
Monthly order volume
Use population standard deviation when you have every monthly order total for a fixed period.
Data: 120, 135, 128, 140, 132, 145
Mean ≈ 133.33, population SD ≈ 8.18
Manufacturing tolerance check
A low SD means measured parts stay close to the target dimension.
Data: 10.01, 9.99, 10.02, 10.00, 9.98
Sample SD is small, so the process is stable
Investment return volatility
Compare the spread of several monthly returns before evaluating overall risk.
Data: 2.4, -1.1, 3.0, 1.8, -0.6
Higher SD indicates more volatile returns
Standard Deviation FAQ
When should I use sample standard deviation instead of population standard deviation?
Use sample standard deviation when your dataset is only part of a larger population. It divides by n-1 to reduce bias when estimating the population spread.
What does a higher standard deviation mean?
A higher standard deviation means the values are more spread out from the mean. A lower value means the dataset is more tightly clustered.
Can standard deviation be negative?
No. Standard deviation is the square root of variance, so the result is always zero or positive.
Why does this calculator also show variance and mean?
Variance and mean help you interpret standard deviation. The mean shows the center of the data, and variance shows the squared spread before taking the square root.