Standard Deviation Calculator

This Standard Deviation Calculator helps you find the population standard deviation of a list of numbers and shows each step clearly. It works by finding the mean, calculating how far each value is from the mean, squaring those differences, averaging them, and then taking the square root. This makes it useful for checking answers, understanding how standard deviation is calculated, and practising statistics step by step.

Step-by-step method

1. List the numbers and count how many there are.
2. Find the mean.
3. Find each deviation from the mean (x − x̄).
4. Square each deviation and add them up.
5. Divide by n to get the population variance.
6. Take the square root to get the population standard deviation.

Formula:

Example 1: 3, 1, 4, 2

Step 1 - List the numbers and count how many there are.

In this problem: The numbers are 3, 1, 4, 2, so the count is n = 4.

Step 2 - Find the mean.

In this problem: The mean is x̄ = Σx ÷ n. Here x̄ = 2.5.

Step 3 - Find each deviation from the mean (x − x̄).

In this problem: Subtract the mean from each value: x − x̄ gives 0.5, -1.5, 1.5, -0.5.

Step 4 - Square each deviation and add them up.

In this problem: Square each deviation: (x − x̄)² gives 0.25, 2.25, 2.25, 0.25, and the sum is Σ(x − x̄)² = 5.

Step 5 - Divide by n to get the population variance.

In this problem: Population variance is σ² = Σ(x − x̄)² ÷ n, so σ² = 5 ÷ 4 = 1.25.

Step 6 - Take the square root to get the population standard deviation.

In this problem: Standard deviation is the square root of the variance: σ = √1.25 = 1.1180339887.

Final answer: Standard deviation = 1.1180339887

Example 2: 10, 12, 9, 11, 8

Step 1 - List the numbers and count how many there are.

In this problem: The numbers are 10, 12, 9, 11, 8, so the count is n = 5.

Step 2 - Find the mean.

In this problem: The mean is x̄ = Σx ÷ n. Here x̄ = 10.

Step 3 - Find each deviation from the mean (x − x̄).

In this problem: Subtract the mean from each value: x − x̄ gives 0, 2, -1, 1, -2.

Step 4 - Square each deviation and add them up.

In this problem: Square each deviation: (x − x̄)² gives 0, 4, 1, 1, 4, and the sum is Σ(x − x̄)² = 10.

Step 5 - Divide by n to get the population variance.

In this problem: Population variance is σ² = Σ(x − x̄)² ÷ n, so σ² = 10 ÷ 5 = 2.

Step 6 - Take the square root to get the population standard deviation.

In this problem: Standard deviation is the square root of the variance: σ = √2 = 1.4142135624.

Final answer: Standard deviation = 1.4142135624