What Is Standard Deviation?
Standard deviation is a measure of how spread out the values in a dataset are relative to the mean (average). A low standard deviation means the data points are clustered closely around the mean. A high standard deviation means the data is spread out over a wider range.
It is one of the most fundamental concepts in statistics and is used everywhere from school grades to financial risk assessment to scientific research.
Population vs. Sample Standard Deviation
There are two versions of standard deviation, and choosing the right one matters:
- Population Standard Deviation (σ): Used when your data represents the entire population you are studying. The formula divides by N (the total number of data points).
- Sample Standard Deviation (s): Used when your data is a sample taken from a larger population. The formula divides by (N − 1), known as Bessel's correction. This makes the estimate of the population's variability less biased.
In most real-world scenarios (surveys, experiments, research), you are working with a sample and should use the sample standard deviation.
The Standard Deviation Formula
Population Standard Deviation:
σ = √[ Σ(xᵢ − μ)² / N ]
Sample Standard Deviation:
s = √[ Σ(xᵢ − x̄)² / (N − 1) ]
Where:
- xᵢ = each individual data value
- μ or x̄ = the mean of all data values
- N = number of data points
- Σ = "sum of" all the values
Step-by-Step Calculation Example
Dataset: [4, 8, 6, 5, 3, 2, 8, 9, 2, 5]
We will calculate the sample standard deviation.
Step 1: Calculate the Mean
Mean (x̄) = (4 + 8 + 6 + 5 + 3 + 2 + 8 + 9 + 2 + 5) ÷ 10
= 52 ÷ 10 = 5.2
Step 2: Find Each Deviation from the Mean
Subtract the mean from each data point (xᵢ − x̄):
| xᵢ | xᵢ − x̄ | (xᵢ − x̄)² |
|---|---|---|
| 4 | −1.2 | 1.44 |
| 8 | 2.8 | 7.84 |
| 6 | 0.8 | 0.64 |
| 5 | −0.2 | 0.04 |
| 3 | −2.2 | 4.84 |
| 2 | −3.2 | 10.24 |
| 8 | 2.8 | 7.84 |
| 9 | 3.8 | 14.44 |
| 2 | −3.2 | 10.24 |
| 5 | −0.2 | 0.04 |
Step 3: Sum the Squared Deviations
Σ(xᵢ − x̄)² = 1.44 + 7.84 + 0.64 + 0.04 + 4.84 + 10.24 + 7.84 + 14.44 + 10.24 + 0.04
= 57.6
Step 4: Divide by (N − 1)
Variance (s²) = 57.6 ÷ (10 − 1) = 57.6 ÷ 9 = 6.4
Step 5: Take the Square Root
s = √6.4 ≈ 2.53
The sample standard deviation of this dataset is approximately 2.53.
Interpreting Standard Deviation
The Empirical Rule (68-95-99.7 rule) applies to normally distributed data:
- 68% of data falls within 1 standard deviation of the mean (μ ± σ)
- 95% of data falls within 2 standard deviations (μ ± 2σ)
- 99.7% of data falls within 3 standard deviations (μ ± 3σ)
Real-World Examples
Test Scores
If the average exam score is 72 with a standard deviation of 8, most students (68%) scored between 64 and 80. A score of 88 is two standard deviations above average — an excellent result.
Investment Risk
In finance, standard deviation measures the volatility of an investment's returns. A higher standard deviation means higher risk (and potentially higher reward). A conservative bond fund might have a standard deviation of 3%, while a technology stock ETF might have 20%.