How the Statistics Calculator Works
Statistics describes a list of numbers with a few figures that are easy to compare: where the middle is, how spread out the numbers are, and how big and small they get. The Statistics Calculator takes a list of numbers and gives you the count, the mean, the variance, the standard deviation, the smallest and largest values, and the range. You can use it for test scores, prices, measurements or any set of data.
What you enter
- Dataset: your numbers, separated by commas or spaces. For example: 10, 12, 23, 23, 16, 23, 21, 16.
- You need at least 2 numbers. Anything that is not a number is ignored.
What you get
| Result | What it means |
|---|---|
| Count (n) | How many numbers are in the list |
| Mean | The average: the sum divided by n |
| Variance | How spread out the numbers are (squared units) |
| Standard deviation | The typical distance from the mean (same units as your data) |
| Min and Max | The smallest and the largest number |
| Range | Max minus min |
The formulas
mean = (x1 + x2 + ... + xn) / n variance = sum of (each x - mean)^2 / (n - 1) standard deviation = square root of the variance range = max - min
This calculator divides by (n - 1), not by n. That gives the sample variance and standard deviation, which is the right choice when your numbers are a sample taken from a bigger group. The results are rounded to 2 decimal places.
A worked example
The default list is 10, 12, 23, 23, 16, 23, 21, 16.
| Step | Value |
|---|---|
| Count | 8 |
| Sum (10 + 12 + 23 + 23 + 16 + 23 + 21 + 16) | 144 |
| Mean (144 / 8) | 18 |
| Sum of squared differences | 192 |
| Variance (192 / 7) | 27.43 |
| Standard deviation (square root of 27.43) | 5.24 |
| Min, max, range | 10, 23, 13 |
The squared differences from the mean of 18 are 64, 36, 25, 25, 25, 4, 4 and 9, and they add up to 192.
Chart: the data
The longer the bar, the bigger the number. The mean is 18, so most bars are around that length:
10 █████████████████ 12 █████████████████████ 23 ████████████████████████████████████████ 23 ████████████████████████████████████████ 16 ████████████████████████████ 23 ████████████████████████████████████████ 21 █████████████████████████████████████ 16 ████████████████████████████
Chart: where the variance comes from
Each bar is the squared difference of that number from the mean (18). The far-away numbers add the most:
10 ████████████████████████████████████████ 64 12 ███████████████████████ 36 23 ████████████████ 25 (three times) 16 ███ 4 (two times) 21 ██████ 9
Population or sample?
- Sample (this calculator): divide by n - 1. Use it when the numbers are only part of a bigger group, such as 8 students out of a school.
- Population: divide by n. Use it when you have every value. For the example this gives a variance of 24 and a standard deviation of 4.90.
How to read the standard deviation
- A small standard deviation means the numbers are close to the mean.
- A large one means they are spread out.
- If the data is roughly bell-shaped, about 68% of the values are within 1 standard deviation of the mean, and about 95% are within 2.
Tips
- The standard deviation is in the same unit as your data (for example points or dollars). The variance is in squared units.
- One extreme number can change the mean and the standard deviation a lot.
- This calculator does not give the median or the mode. Use the Average Calculator for those.
This tool gives an estimate for general information only. For research, grading or business decisions, check the method that applies to your data.