Enter a list of numbers to calculate the mean, variance, and standard deviation of your data set using the standard deviation formula, for both population and sample statistics.
A larger standard deviation means your values are more spread out from the mean (5.833); a smaller one means they cluster closer together.
Standard deviation quantifies how spread out a set of numbers is from its mean (average). A low standard deviation means values cluster tightly around the mean; a high standard deviation means values are more spread out. It's one of the most widely used measures of variability in statistics.
The sample formula divides by (n−1) instead of n — known as Bessel's correction — which corrects for the tendency of sample variance to underestimate the true population variance when working from a subset of data.
| Step | Value |
|---|---|
| Mean | 5 |
| Sum of squared differences | 32 |
| Population variance (÷8) | 4 |
| Population standard deviation | 2 |
| Population (÷N) | Sample (÷n−1) | |
|---|---|---|
| Divisor for {2,4,4,4,5,5,7,9} | 8 | 7 |
| Variance | 4.00 | 4.57 |
| Standard deviation | 2.00 | 2.14 |
Note the sample standard deviation is always slightly larger than the population figure for the same data — Bessel's correction intentionally inflates the estimate to account for the extra uncertainty of working from a sample.
A z-score tells you how many standard deviations a value sits from the mean. Once you have a z-score, a z-table (standard normal table) gives the proportion of data falling below it.
| Z-score | Area to the left (percentile) |
|---|---|
| -2.00 | 0.0228 (2.28%) |
| -1.00 | 0.1587 (15.87%) |
| -0.50 | 0.3085 (30.85%) |
| 0.00 | 0.5000 (50.00%) |
| 0.50 | 0.6915 (69.15%) |
| 1.00 | 0.8413 (84.13%) |
| 1.96 | 0.9750 (97.50%) |
| 2.00 | 0.9772 (97.72%) |
Example: a value of 68 in a dataset with mean 60 and standard deviation 8 has a z-score of (68−60)/8 = 1.00, meaning it's higher than about 84% of the distribution.
When should I use sample vs. population standard deviation? Use the population formula when you have data for an entire group; use the sample formula when your data is a subset used to estimate a larger population's variability.
Why do we square the differences instead of just taking absolute values? Squaring avoids values canceling out, gives larger weight to bigger deviations, and produces a formula with convenient mathematical properties used throughout statistics.
What's the difference between variance and standard deviation? Standard deviation is simply the square root of variance, expressed in the same units as the original data, which makes it easier to interpret.
What does a standard deviation of zero mean? It means every value in the dataset is identical to the mean — there's no variability at all.
How is standard deviation used in the empirical (68-95-99.7) rule? For roughly normally distributed data, it defines the width of the intervals around the mean that capture 68%, 95%, and 99.7% of values respectively — a quick way to gauge how typical or unusual a given value is.
Can standard deviation be negative? No — since it's derived from a square root of squared (non-negative) values, standard deviation is always zero or positive.
How does an outlier affect standard deviation? A single extreme value can noticeably inflate standard deviation, since squaring the differences gives disproportionate weight to values far from the mean.
What is a z-table used for? A z-table converts a z-score into the percentage of data falling below that point on a standard normal distribution, which is useful for finding percentiles and probabilities.
For the data set 4, 8, 6, 5, 3, the mean is 5.2. The sample standard deviation (dividing by n−1 = 4) comes out to approximately 1.92, indicating the values are moderately clustered around the mean.
Mean: The arithmetic average of a data set.
Variance: The average of the squared differences between each value and the mean.
Sample vs. population: Sample statistics estimate a larger population from a subset; population statistics describe the entire group directly.