Calculate descriptive statistics
What is a statistics calculator for mean, variance, and standard deviation?
A statistics calculator for mean, variance, and standard deviation is a tool that takes a list of numbers and computes the average (mean), how spread out the values are (variance), and the typical distance of each value from the average (standard deviation). The mean is the sum of all values divided by how many there are. The variance is the average of the squared differences from the mean, and the standard deviation is the square root of the variance, which puts the spread back into the same units as the original data.
These three measures describe a dataset in two ways: the mean tells you its center, while variance and standard deviation tell you its dispersion. Standard deviation is the most widely used measure of spread because it is in the same units as the data, a standard deviation of 15 IQ points is directly interpretable, whereas a variance of 225 squared-points is not. This calculator is used by students, researchers, data analysts, and anyone working with test scores, measurements, or survey data. It also underpins the bell curve and IQ scoring, where IQ tests are standardized to a mean of 100 and a standard deviation of 15, so a score of 130 sits exactly 2 standard deviations above the mean (about the 98th percentile).
The formulas: mean, variance, and standard deviation
Start with the mean, the arithmetic average of the data.
- Mean (average): add up all the values and divide by the count. Mean = (sum of all values) / n
Variance measures average squared distance from the mean. There are two versions, and which you use depends on whether your numbers are the whole population or just a sample drawn from it.
- Population variance: take each value, subtract the mean, square the result, add up all those squared differences, then divide by n (the number of values).
- Sample variance: same steps, but divide by n minus 1 instead of n. This is called Bessel's correction.
Standard deviation is simply the square root of the variance.
- Population standard deviation = square root of population variance (divide by n)
- Sample standard deviation = square root of sample variance (divide by n minus 1)
The square root step matters because variance is in squared units (squared points, squared dollars), while standard deviation returns to the original units, making it directly readable.
Population vs sample: which one should you use
This is the single most common point of confusion, and choosing wrong changes your answer.
- Use the population formula (divide by n) when your data includes every member of the group you care about. Example: the exam scores of all 30 students in one specific class, and you only care about that class.
- Use the sample formula (divide by n minus 1) when your data is a subset used to estimate a larger group. Example: 30 survey responses used to estimate the spread for an entire city.
Dividing by n minus 1 for samples is Bessel's correction. Dividing by n alone would slightly underestimate the true population variance because the sample mean is, by construction, the closest center to its own data points. Subtracting 1 from the denominator corrects that bias and gives an unbiased estimate of the population variance. The smaller your sample, the bigger the difference between the two methods. With large datasets, dividing by n versus n minus 1 produces nearly identical results.
Worked example, step by step
Dataset: 4, 8, 6, 5, 3, 7 (six numbers).
Step 1, mean: 4 + 8 + 6 + 5 + 3 + 7 = 33. Divide by 6. Mean = 5.5
Step 2, squared differences from the mean:
- (4 - 5.5) squared = 2.25
- (8 - 5.5) squared = 6.25
- (6 - 5.5) squared = 0.25
- (5 - 5.5) squared = 0.25
- (3 - 5.5) squared = 6.25
- (7 - 5.5) squared = 2.25
Sum of squared differences = 17.5
Step 3, variance:
- Population variance = 17.5 / 6 = 2.9167
- Sample variance = 17.5 / 5 = 3.5
Step 4, standard deviation (square root of variance):
- Population standard deviation = square root of 2.9167 = 1.708
- Sample standard deviation = square root of 3.5 = 1.871
Notice the sample values are larger, exactly because the denominator is smaller (5 instead of 6).
Related stats: median, mode, and range
A full statistics calculator usually reports more than just mean, variance, and standard deviation. The other common measures answer different questions.
- Median: the middle value when the data is sorted from low to high. If there is an even count, average the two middle numbers. The median is resistant to outliers, while the mean is pulled toward extreme values.
- Mode: the value that appears most often. A dataset can have one mode, several modes, or none if every value is unique.
- Range: the largest value minus the smallest value, the simplest measure of spread.
- Count and sum: the number of values (n) and their total, which feed directly into the mean.
When data is skewed (for example, incomes with a few very high earners), the median often describes the typical value better than the mean. For symmetric, bell-shaped data, the mean, median, and mode all sit close together at the center.
How standard deviation connects to IQ and the bell curve
Standard deviation is the engine behind IQ scoring and the normal distribution (bell curve). Modern IQ tests set the mean to 100 and the standard deviation to 15. Once you know the mean and standard deviation, you know where any score falls.
- IQ 85 to 115 = within 1 standard deviation of the mean, covering about 68 percent of people
- IQ 70 to 130 = within 2 standard deviations, covering about 95 percent
- IQ 55 to 145 = within 3 standard deviations, covering about 99.7 percent
This 68-95-99.7 pattern is the empirical rule for any normal distribution. The same math turns a raw score into a z-score: z = (value minus mean) divided by standard deviation. An IQ of 130 has a z-score of (130 - 100) / 15 = 2.0, placing it at roughly the 98th percentile. This is why standard deviation, not variance, is the number reported on score reports: it shares the same units as the score and translates directly into percentiles.
Learn how the IQ scale uses mean 100 and SD 15 · See the full IQ score chart · IQ percentile calculator
Frequently asked questions
What is the difference between variance and standard deviation?
Variance is the average of the squared differences between each value and the mean, while standard deviation is the square root of the variance. They measure the same thing (how spread out the data is), but standard deviation is in the same units as the original data, making it easier to interpret. For example, with IQ scores the standard deviation is 15 points, whereas the variance is 225 squared-points, which has no direct real-world meaning.
Should I divide by n or n minus 1?
Divide by n when your data is the entire population you care about (the population formula). Divide by n minus 1 when your data is a sample used to estimate a larger group (the sample formula). Dividing by n minus 1 is called Bessel's correction, and it corrects a downward bias so the sample gives an unbiased estimate of the true population variance. For large datasets the two answers are nearly identical; for small samples the difference is meaningful.
How do you calculate standard deviation step by step?
First find the mean by adding all values and dividing by the count. Second, subtract the mean from each value and square the result. Third, add up all those squared differences. Fourth, divide that sum by n for a population or by n minus 1 for a sample, which gives the variance. Fifth, take the square root of the variance to get the standard deviation.
What is the mean, median, and mode?
The mean is the arithmetic average: the sum of all values divided by how many there are. The median is the middle value when the numbers are sorted in order (or the average of the two middle values if the count is even). The mode is the value that appears most frequently. The mean is sensitive to outliers, the median is resistant to them, and the mode identifies the most common value.
What does standard deviation tell you about IQ scores?
IQ tests are standardized to a mean of 100 and a standard deviation of 15. This means about 68 percent of people score between 85 and 115 (within one standard deviation), about 95 percent score between 70 and 130 (two standard deviations), and about 99.7 percent score between 55 and 145 (three standard deviations). A score of 130 is exactly 2 standard deviations above the mean, placing it near the 98th percentile.