SD Formula Calculator: Standard Deviation From Sums

Sample√(SS / (n − 1))

Population√(SS / n)

SSΣx² − (Σx)²/n

From variance√variance

What Is SD Formula?

The standard deviation formula measures how far values typically are from the mean. When you only have summary sums, the computational form SS = Σx² − (Σx)² ÷ n gives the same answer without the raw list.

Choose whether you know the sums, a frequency table or the variance, and pick sample or population. The calculator shows the mean, sum of squares, variance and SD with each step, and the other SD for comparison.

Below are the formulas, a worked example and answers on frequency tables and precision.

How to Use the SD Formula Calculator

Choose what you know.
Choose sample or population.
Enter the values.
Read the SD and the working.

SD Formula Formula

Sum of squares

SS=\sum x^2-\frac{(\sum x)^2}{n}

Sample SD

s=\sqrt{\frac{SS}{n-1}}

Population SD

\sigma=\sqrt{\frac{SS}{n}}

SD Formula Variables Explained

Symbol	Meaning	Notes
Σx	Sum of values
Σx²	Sum of squared values
n	Count
SS	Sum of squared deviations

SD Formula Worked Example

With Σx = 150, Σx² = 4900 and n = 6: SS = 4900 − 150² ÷ 6 = 4900 − 3750 = 1150. The sample variance is 1150 ÷ 5 = 230 and s = √230 = 15.166.

With the example values loaded in the calculator above, the steps are:

Mean = Σx ÷ n = 150 ÷ 6 = 25.
Sum of squares SS = Σx² − (Σx)² ÷ n = 4900 − 3750 = 1150.
Variance = SS ÷ n − 1 = 1150 ÷ 5 = 230.
SD = √variance = 15.165751.

Result: Sample standard deviation s = 15.165751; Sample variance s² = 230.

Where SD Formula Is Used

Summary data

Get SD when only sums are given.

Frequency tables

Use Σf·x and Σf·x².

Checking work

Confirm a hand calculation.

From variance

Take the square root.

Common SD Formula Mistakes to Avoid

Check that Σx² ≥ (Σx)²/n.
For a frequency table use Σf·x and Σf·x².
Do not round the sums.

SD Formula Terms and Definitions

SS: Sum of squared deviations from the mean.

SD Formula FAQs

What is the formula for standard deviation?

The square root of the variance, where variance is SS divided by n − 1 for a sample or n for a population.

How do I find SD from Σx and Σx²?

Compute SS = Σx² − (Σx)² ÷ n, divide by n − 1 or n, then take the square root.

How is SD found from a frequency table?

Use n = Σf, Σx = Σf·x and Σx² = Σf·x² in the same formula.

How do I get SD from variance?