Standard Deviation vs Variance: What's the Difference?
Standard deviation and variance both measure how spread out a set of data is from its mean. Variance is the average of the squared differences from the mean. Standard deviation is the square root of variance, which brings the measurement back into the same units as the original data. Both are foundational tools in statistics, data science, and any field that involves analyzing numerical data.
Quick Answer
- 1. Variance = average of squared deviations from the mean. Units are squared (e.g., cm2).
- 2. Standard deviation = square root of variance. Same units as the data (e.g., cm).
- 3. In a normal distribution, 68% of data falls within 1 SD, 95% within 2 SDs, 99.7% within 3 SDs.
- 4. For samples, divide by n-1 (Bessel's correction); for populations, divide by n.
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Calculate Standard Deviation FreeWhat Is Variance?
Variance measures the average degree to which each data point differs from the mean. It quantifies the "spread" or "dispersion" of a data set. A high variance means the data points are far from the mean and from each other; a low variance means they are clustered closely around the mean.
The Variance Formula
For a population (every member of the group):
Variance (sigma squared) = Sum of (each value - mean) squared / N
For a sample (a subset of the population):
Variance (s squared) = Sum of (each value - mean) squared / (n - 1)
The only difference is the denominator: N for population, n-1 for sample. The n-1 adjustment is called Bessel's correction and prevents underestimation of the true population variance when working with samples.
What Is Standard Deviation?
Standard deviation is simply the square root of variance. While variance tells you the average squared deviation from the mean, standard deviation converts that back to the original units, making it far more intuitive to interpret.
For example, if you measure heights in centimeters, variance is in "centimeters squared" (a unit that does not exist in the real world), while standard deviation is in centimeters, the same unit as your original measurements. This is the primary practical advantage of standard deviation over variance.
Side-by-Side Comparison
| Feature | Variance | Standard Deviation |
|---|---|---|
| Formula | Average of squared deviations | Square root of variance |
| Units | Squared units (e.g., cm2) | Same as data (e.g., cm) |
| Interpretation | Less intuitive | Directly interpretable |
| Mathematical use | Additive (variances can be added) | Not additive |
| Sensitivity to outliers | Very high (squares amplify outliers) | High (inherits from variance) |
| Common use | Statistical theory, ANOVA, regression | Reporting, z-scores, confidence intervals |
Worked Example: Calculating Both
Consider a data set of five exam scores: 72, 85, 90, 78, 95.
Step 1: Find the Mean
Mean = (72 + 85 + 90 + 78 + 95) / 5 = 420 / 5 = 84
Step 2: Find the Squared Deviations
| Score | Deviation (x - mean) | Squared Deviation |
|---|---|---|
| 72 | -12 | 144 |
| 85 | 1 | 1 |
| 90 | 6 | 36 |
| 78 | -6 | 36 |
| 95 | 11 | 121 |
| Sum | 338 |
Step 3: Calculate Variance
Population variance: 338 / 5 = 67.6
Sample variance: 338 / 4 = 84.5
Step 4: Calculate Standard Deviation
Population SD: Square root of 67.6 = 8.22
Sample SD: Square root of 84.5 = 9.19
Interpretation: the scores deviate from the mean by roughly 8 to 9 points on average. This is immediately meaningful because it is in the same units (points) as the original data.
The 68-95-99.7 Rule (Empirical Rule)
For data that follows a normal distribution (bell curve), standard deviation has a powerful interpretation known as the empirical rule:
- 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
Applied to our exam example (mean = 84, SD = 8.22): about 68% of scores would fall between 75.8 and 92.2, and about 95% would fall between 67.6 and 100.4. This makes standard deviation an incredibly practical tool for understanding the distribution of any data set.
When to Use Variance vs Standard Deviation
Use Variance When...
- Doing advanced statistical analysis. Variance is additive: if two independent variables each have known variances, the variance of their sum equals the sum of their variances. Standard deviation does not have this property.
- Performing ANOVA or regression. Analysis of variance (ANOVA) literally decomposes total variance into components. Regression models minimize the variance of residuals.
- Working in machine learning. Many algorithms use variance for feature selection, dimensionality reduction (PCA), and model evaluation.
Use Standard Deviation When...
- Reporting results. Standard deviation is in interpretable units. Saying "the standard deviation of heights is 3.2 inches" is meaningful; "the variance is 10.24 square inches" is not.
- Calculating z-scores. A z-score is (value - mean) / standard deviation, telling you how many standard deviations a value is from the mean.
- Building confidence intervals. Confidence intervals use standard deviation (or standard error, which is derived from it) to define the range of likely population values.
- Applying quality control. Six Sigma and other quality methodologies define process limits in terms of standard deviations from the target.
Common Mistakes to Avoid
Using Population Formulas for Sample Data
If your data is a sample (which it almost always is), use n-1 in the denominator. Using n underestimates the true population variance. For large samples (n greater than 30), the difference is small, but for small samples it can be significant.
Confusing Standard Deviation with Standard Error
Standard deviation measures the spread of individual data points. Standard error measures the precision of the sample mean as an estimate of the population mean. Standard error = standard deviation / square root of n. They answer different questions.
Interpreting Variance in the Wrong Units
A variance of 100 does not mean data points deviate by 100 from the mean. It means the average squared deviation is 100, and the actual average deviation (standard deviation) is 10. Always take the square root to get the interpretable measure.
Real-World Applications
Finance: Portfolio Risk
In finance, variance and standard deviation of returns measure investment risk. A stock with a standard deviation of 20% is more volatile than one with 8%. Portfolio theory uses the additive property of variance to calculate the risk of combined assets.
Manufacturing: Quality Control
Standard deviation is the backbone of Six Sigma methodology. A process is considered "six sigma" when the nearest specification limit is 6 standard deviations from the process mean, meaning only 3.4 defects per million opportunities.
Data Science: Feature Scaling
Standardization (z-score normalization) uses standard deviation to scale features so they have a mean of 0 and a standard deviation of 1. This is critical for machine learning algorithms that are sensitive to feature magnitudes, like support vector machines and k-nearest neighbors.
The Bottom Line
Variance and standard deviation measure the same thing: the spread of data around the mean. Variance is the raw measure (in squared units), and standard deviation is its interpretable sibling (in original units). Use variance for mathematical operations and advanced analysis; use standard deviation for reporting, z-scores, and practical interpretation. In most everyday situations, standard deviation is the number you want.
Our free standard deviation calculator computes both measures instantly from any data set, and our statistics calculator provides a full suite of descriptive statistics including mean, median, mode, range, and quartiles.
Frequently Asked Questions
When should I use standard deviation instead of variance?
Use standard deviation when you need to communicate results to a non-technical audience or when you want a measure of spread in the same units as your data. For example, if you are reporting the variability of test scores measured in points, standard deviation gives you a result in points (e.g., 12.5 points), while variance gives you points squared (e.g., 156.25 points squared). Standard deviation is also the basis for z-scores, confidence intervals, and the empirical rule (68-95-99.7 rule).
Why do we divide by n-1 instead of n for sample data?
Dividing by n-1 instead of n is called Bessel's correction. When you calculate variance from a sample (a subset of a population), using n in the denominator produces a biased estimate that systematically underestimates the true population variance. This happens because sample data points tend to be closer to the sample mean than to the true population mean. Dividing by n-1 inflates the result slightly to compensate, giving an unbiased estimate. For large samples (n > 30), the difference between n and n-1 is minimal, but for small samples it matters significantly.
Can variance or standard deviation be negative?
No. Variance is calculated by squaring differences from the mean, so it is always zero or positive. Standard deviation is the square root of variance, so it is also always zero or positive. A variance of zero means every data point is identical to the mean, indicating no variability at all. If your calculation produces a negative value, there is an error in your math.
What is the relationship between standard deviation and the normal distribution?
In a normal (bell-shaped) distribution, standard deviation determines the width of the curve. The empirical rule states that approximately 68% of data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and about 99.7% falls within three standard deviations. This is also called the 68-95-99.7 rule. For example, if exam scores have a mean of 75 and a standard deviation of 10, then about 68% of students scored between 65 and 85.
How do I calculate standard deviation in Excel or Google Sheets?
For sample standard deviation (divides by n-1), use the STDEV.S function (or just STDEV): =STDEV.S(A1:A20). For population standard deviation (divides by n), use STDEV.P: =STDEV.P(A1:A20). For variance, use VAR.S for sample variance or VAR.P for population variance. Most of the time, you are working with sample data and should use the .S versions. Google Sheets uses the same function names.
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