Statistics
Variance, std dev, Z-score
About This Calculator
Descriptive statistics summarize a data set's central tendency (mean) and spread (variance, standard deviation). The standard deviation measures how spread out values are from the mean, while the Z-score tells you how many standard deviations a specific value is from the mean. These are foundational tools for data analysis, quality control, and scientific research.
Formula
Mean: μ = Σx / n
Variance: σ² = Σ(x − μ)² / n (population) or Σ(x − μ)² / (n−1) (sample)
Standard deviation: σ = √variance
Z-score: Z = (x − μ) / σ
Example Calculation
Data: [4, 7, 13, 2, 1, 3]
- Mean = (4+7+13+2+1+3)/6 = 30/6 = 5
- Deviations²: (4-5)²=1, (7-5)²=4, (13-5)²=64, (2-5)²=9, (1-5)²=16, (3-5)²=4
- Variance = (1+4+64+9+16+4)/6 = 98/6 = 16.33
- Std Dev = √16.33 = 4.04
Mean=5, Variance=16.33, Std Dev≈4.04
Standard Deviation Interpretation (Empirical Rule)
| Range | % of Data | Real-world example |
|---|---|---|
| μ ± 1σ | ~68% | 68% of adults within 1 SD of mean height |
| μ ± 2σ | ~95% | 95% of test scores within 2 SDs of mean |
| μ ± 3σ | ~99.7% | 99.7% of quality parts in spec (6-sigma is ±3σ per side) |
Frequently Asked Questions
When do I use population vs sample variance?
Use population variance (divide by n) when you have data for the entire group. Use sample variance (divide by n−1) when your data is a sample, because dividing by n−1 gives an unbiased estimate of the true population variance.
What does a high standard deviation mean?
A high standard deviation means data points are spread far from the mean — high variability. A low standard deviation means data clusters tightly around the mean. It's always interpreted relative to the mean: a SD of 10 is large if the mean is 5, but small if the mean is 1000.
What is a Z-score used for?
A Z-score tells you how unusual a value is relative to its distribution. Z=0 is average, Z=2 means the value is 2 standard deviations above average (rarer than 97.7% of values). It also lets you compare values from different scales.
What is the difference between variance and standard deviation?
Variance is the average squared deviation from the mean — it's useful mathematically but in squared units. Standard deviation is the square root of variance, bringing it back to the original units and making it directly interpretable.