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
Sample SD
Population SD
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.
How to Read the SD From Sums
When you know Σx, Σx² and n, you do not need the individual values. The sum of squares is SS = Σx² − (Σx)² ÷ n. With Σx = 150, Σx² = 4900 and n = 6, SS = 4900 − 3750 = 1150.
The sample variance is SS ÷ (n − 1) = 230, so s = √230 ≈ 15.17. The population SD for the same data divides by n instead. Large Σx² values can cause rounding error, so keep many digits.
Where SD Formula Is Used
Summary statistics
Get the SD from totals when the raw data are not available.
Frequency tables
Use values and frequencies to compute the SD of grouped counts.
Exam practice
Check the shortcut formula with Σx and Σx².
From variance
Take the square root of a variance to get the SD.
Common SD Formula Mistakes to Avoid
- Use n − 1 for a sample and n for a population.
- The shortcut formula can lose precision when Σx² is large and the spread is small.
- For a frequency table, multiply each value by its frequency before you sum.
- SS must not be negative. A negative value means an entry error.
SD Formula Terms and Definitions
- SS
- Sum of squared deviations from the mean.
SD Formula FAQs
What is the formula for standard deviation?
The sample SD is s = √(Σ(x − x̄)² ÷ (n − 1)). The population SD is σ = √(Σ(x − μ)² ÷ N).
How do I find SD from Σx and Σx²?
Compute SS = Σx² − (Σx)² ÷ n, divide by n − 1 for a sample, then take the square root. With Σx = 150, Σx² = 4900 and n = 6, s ≈ 15.17.
How is SD found from a frequency table?
Use Σfx and Σfx² in place of Σx and Σx², with n = Σf. Then apply the same formula.
How do I get SD from variance?
Take the square root. A variance of 230 gives an SD of about 15.17.