Standard Deviation Explained: Formula, Examples & When to Use It
Standard deviation is the most widely used measure of spread in statistics. It tells you how spread out values are around the mean. This guide explains it from first principles with worked examples.
What Standard Deviation Tells You
The mean tells you the centre of a dataset. Standard deviation tells you how far values typically stray from that centre.
Low standard deviation → values clustered tightly around the mean
High standard deviation → values spread widely from the mean
Two exam classes both average 70%, but:
- Class A: scores of 68, 69, 70, 71, 72 — SD ≈ 1.4 (very consistent)
- Class B: scores of 40, 55, 70, 85, 100 — SD ≈ 22.4 (highly variable)
Same mean, very different distributions.
The Formula
There are two versions depending on whether you have the full population or a sample.
Population Standard Deviation (σ)
Use when you have data for every member of the group.
σ = √((Σ(x_i - μ)^2) / (N))
Sample Standard Deviation (s)
Use when your data is a sample from a larger population (the most common case).
s = √( Σ(x_i − x̄)² / (n − 1) )
The denominator is n − 1 (not n) to correct for the bias that comes from estimating a population parameter from a sample. This is called Bessel's correction.
Step-by-Step Calculation
Dataset: Test scores for 6 students: 72, 85, 68, 91, 74, 80
Step 1: Find the mean
x̄ = (72 + 85 + 68 + 91 + 74 + 80) / (6) = (470) / (6) = 78.33
Step 2: Find each deviation from the mean
| Score | Deviation (x − x̄) | Squared deviation | |-------|---------------------|-------------------| | 72 | −6.33 | 40.07 | | 85 | +6.67 | 44.49 | | 68 | −10.33 | 106.71 | | 91 | +12.67 | 160.53 | | 74 | −4.33 | 18.75 | | 80 | +1.67 | 2.79 |
Step 3: Sum the squared deviations
Σ(x - x̄)^2 = 40.07 + 44.49 + 106.71 + 160.53 + 18.75 + 2.79 = 373.34
Step 4: Divide by n − 1 (sample)
(373.34) / (6 - 1) = (373.34) / (5) = 74.67
Step 5: Take the square root
s = √(74.67) = 8.64
The standard deviation is 8.64 points. A typical student score is about 8–9 points away from the class average.
The 68-95-99.7 Rule
For normally distributed data (bell curve), standard deviation has a predictable relationship with the spread:
- 68% of values fall within 1 SD of the mean
- 95% of values fall within 2 SD of the mean
- 99.7% of values fall within 3 SD of the mean
Applied to our example (mean = 78.33, SD = 8.64):
- 68% of scores: 78.33 ± 8.64 → 69.7 to 86.97
- 95% of scores: 78.33 ± 17.28 → 61.05 to 95.61
- 99.7% of scores: 78.33 ± 25.92 → 52.41 to 104.25
Variance vs Standard Deviation
Variance is the squared standard deviation: s² = 74.67 in our example.
Why use standard deviation instead of variance?
- Standard deviation is in the same units as your data (points, dollars, metres)
- Variance is in squared units — harder to interpret practically
- "The average score deviated by 8.64 points" is meaningful; "variance was 74.67 points²" is not
Real-World Uses
Finance: A stock with daily returns averaging 0.05% and SD of 1.2% is much riskier than one with the same average return and SD of 0.3%. Standard deviation is the foundation of volatility measurement.
Manufacturing: A factory producing bolts with a target diameter of 10mm and SD of 0.02mm is far more consistent than one with SD of 0.5mm. Quality control relies on SD.
Medicine: Clinical trials report SD alongside means to show how consistently a treatment worked across patients.
Weather: "Average temperature 18°C with SD 4°C" tells you far more than the average alone — you know what to pack.
Z-Scores
A z-score converts any value to standard deviation units, enabling comparison across different datasets:
z = /x - x̄s
A student scoring 91 in our example:
z = (91 - 78.33) / (8.64) = (12.67) / (8.64) = +1.47
This score is 1.47 standard deviations above the mean — better than about 93% of the class.
Calculate Standard Deviation Now
Our statistics calculator computes standard deviation, variance, mean, median, mode, and more from any dataset you enter. Paste your numbers and get full results instantly.