How standard deviation is built
The calculation works outward from the mean in three steps. First it finds the average of your numbers. Then, for each value, it measures the distance to that mean and squares it — squaring removes the sign so values above and below the mean both count, and it gives extra weight to points that are far away. Finally it averages those squared distances to get the variance, and takes the square root to return to the original units. That square root is the standard deviation.
Squaring is the reason standard deviation reacts strongly to outliers: a value twice as far from the mean contributes four times as much to the variance.
Sample versus population
The only difference between the two modes is what you divide by. Population mode divides by n because you have measured every member of the group. Sample mode divides by n−1 because a sample tends to underestimate the true spread, and dividing by the smaller number corrects for that bias (this is known as Bessel’s correction).
- Choose population (n) when your numbers are the entire group you care about.
- Choose sample (n−1) when your numbers are a subset used to estimate a larger group.
- The gap between the two shrinks as the dataset grows, so for large n the choice barely matters.
Reading the result and the chart
A small standard deviation means the values huddle close to the mean; a large one means they are widely scattered. For roughly bell-shaped data, about two-thirds of values fall within one standard deviation of the mean and about 95% within two.
The bar chart shows each value in your dataset so you can see the spread directly. Tall and short bars far from the typical height are the points that inflate the standard deviation. Remember that the statistic only describes spread, not the center or the shape of the distribution, so read it together with the mean.
Formula
variance = Σ(x−mean)²/N (N = n for population, n−1 for sample); sd = √varianceFrequently asked questions
- What is variance?
- Variance is the average of the squared differences from the mean. Standard deviation is the square root of the variance, expressed in the same units as the data.