Standard Deviation and Variance
Learn to measure data spread using variance and standard deviation. Understand when and how to use these statistics.
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Why Measure Variability?
Knowing the center of your data (mean, median, mode) is important, but it doesn’t tell the whole story. Two datasets can have the same mean but look completely different!
Dataset A: 50, 50, 50, 50, 50 → Mean = 50
Dataset B: 10, 30, 50, 70, 90 → Mean = 50
Both have the same mean, but Dataset B has much more variability (spread).
Measures of variability tell us how spread out or clustered our data is around the center.
Range
The simplest measure of spread is the range—the difference between the maximum and minimum values.
Range = Maximum - Minimum
Data: 12, 25, 18, 32, 15, 28
Range = 32 - 12 = 20
Variance
Variance measures the average squared deviation from the mean. It tells us how far, on average, each data point is from the center.
Population Variance (σ²)
When you have data for an entire population:
σ² = Σ(x - μ)² / N
Where:
- σ² (sigma squared) = population variance
- x = each data value
- μ (mu) = population mean
- N = population size
Sample Variance (s²)
When you have a sample from a population:
s² = Σ(x - x̄)² / (n - 1)
Where:
- s² = sample variance
- x = each data value
- x̄ = sample mean
- n = sample size
Calculating Variance Step-by-Step
Data: 4, 8, 6, 5, 3 (sample data)
Step 1: Calculate the mean x̄ = (4 + 8 + 6 + 5 + 3) / 5 = 26 / 5 = 5.2
Step 2: Find deviations from mean
| x | x - x̄ | (x - 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 |
Step 3: Sum the squared deviations Σ(x - x̄)² = 1.44 + 7.84 + 0.64 + 0.04 + 4.84 = 14.8
Step 4: Divide by (n - 1) s² = 14.8 / (5 - 1) = 14.8 / 4 = 3.7
Standard Deviation
The standard deviation is simply the square root of the variance. It’s more commonly used because it’s in the same units as the original data.
Population: σ = √(σ²)
Sample: s = √(s²)
From our previous example:
- Variance (s²) = 3.7
- Standard Deviation (s) = √3.7 = 1.92
Interpreting Standard Deviation
The standard deviation tells us the typical distance of data points from the mean.
- Small standard deviation → Data points are clustered close to the mean
- Large standard deviation → Data points are spread far from the mean
The Empirical Rule (68-95-99.7 Rule)
For normally distributed data:
- 68% of data falls within 1 standard deviation of the mean
- 95% of data falls within 2 standard deviations of the mean
- 99.7% of data falls within 3 standard deviations of the mean
Test scores have: Mean = 75, Standard Deviation = 10
- 68% of scores fall between 65 and 85 (75 ± 10)
- 95% of scores fall between 55 and 95 (75 ± 20)
- 99.7% of scores fall between 45 and 105 (75 ± 30)
Coefficient of Variation (CV)
To compare variability between datasets with different units or scales, use the coefficient of variation:
CV = (s / x̄) × 100%
Heights: Mean = 170 cm, SD = 10 cm CV = (10 / 170) × 100% = 5.9%
Weights: Mean = 70 kg, SD = 15 kg CV = (15 / 70) × 100% = 21.4%
Weights have more relative variability than heights!
Population vs. Sample: Quick Reference
| Measure | Population | Sample |
|---|---|---|
| Size | N | n |
| Mean | μ (mu) | x̄ (x-bar) |
| Variance | σ² | s² |
| Std Dev | σ | s |
| Divisor | N | n - 1 |
Summary
In this lesson, you learned:
- Range is the simplest measure of spread (max - min)
- Variance is the average squared deviation from the mean
- Standard deviation is the square root of variance, in original units
- For samples, we divide by (n - 1) to get unbiased estimates
- The empirical rule tells us what percentage of data falls within standard deviations
- Coefficient of variation allows comparing variability across different scales
Try It Yourself
Use our Standard Deviation Calculator to practice with your own data!
Next Steps
Continue your learning journey:
- Z-Scores and Standardization - Learn to compare values across distributions
- Normal Distribution - Understanding the bell curve
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