Standard Deviation Calculator

What Is Standard Deviation?

Standard deviation is a statistical measure that describes how spread out a set of values is around the mean. A low standard deviation indicates that values are clustered close to the average, while a high standard deviation shows greater variability.

Standard Deviation Formula

Population: σ = √( Σ(x − μ)² / N )
Sample: s = √( Σ(x − x̄)² / (n − 1) )

Use the population formula when your data includes every member of the group being studied. Use the sample formula when your data represents only a subset of a larger population.

Example

For the values 10, 12, 15, 18, and 20, the mean is 15. The standard deviation shows how far each value typically deviates from this average, providing insight into data consistency and variability.

How to Calculate Standard Deviation Step by Step

Standard deviation is a statistical measure that shows how spread out a set of values is around the average (mean). It helps you understand whether most values are close to the mean or widely scattered. This makes standard deviation especially useful in statistics, finance, science, education, and data analysis.

A small standard deviation means the values are tightly grouped, while a larger standard deviation indicates greater variability. Simply knowing the average is often not enough — standard deviation tells you how reliable that average is.

Step 1: List Your Data Values

Start by writing down all the values in your dataset. These could be test scores, prices, measurements, or any numerical observations. For example:

10, 12, 15, 18, 20

Step 2: Calculate the Mean (Average)

Add all the values together and divide by the total number of values. This gives you the mean, which represents the central value of the dataset.

Mean = (10 + 12 + 15 + 18 + 20) ÷ 5 = 15

Step 3: Subtract the Mean from Each Value

Next, subtract the mean from each value. This shows how far each number is from the average.

10 − 15 = −5
12 − 15 = −3
15 − 15 = 0
18 − 15 = 3
20 − 15 = 5

Step 4: Square Each Difference

Square each result to remove negative values and emphasize larger differences. This step ensures that all deviations contribute positively to the calculation.

(−5)² = 25
(−3)² = 9
0² = 0
3² = 9
5² = 25

Step 5: Find the Variance

Add all the squared differences together, then divide by the appropriate number depending on whether you are working with a population or a sample.

Sum of squares = 25 + 9 + 0 + 9 + 25 = 68

For a population, divide by the total number of values (N). For a sample, divide by one less than the number of values (n − 1).

Population variance = 68 ÷ 5 = 13.6
Sample variance = 68 ÷ 4 = 17

Step 6: Take the Square Root

Finally, take the square root of the variance. This gives you the standard deviation, expressed in the same units as the original data.

Population standard deviation = √13.6 ≈ 3.69
Sample standard deviation = √17 ≈ 4.12

Standard Deviation Formulas

Population: σ = √( Σ(x − μ)² ÷ N )
Sample: s = √( Σ(x − x̄)² ÷ (n − 1) )

Use the population formula when your data includes every member of the group being studied. Use the sample formula when the data represents only a portion of a larger population.

Why Standard Deviation Matters

Standard deviation is used to measure consistency, risk, and reliability. In finance, it indicates volatility. In education, it shows how spread out test scores are. In science and research, it helps assess the accuracy and stability of experimental results.

Understanding how standard deviation is calculated helps you interpret results more confidently and recognize whether differences in data are meaningful or expected.