Statistics Calculator

What the Statistics Calculator Does

This Statistics Calculator takes a list of numbers and instantly returns the core descriptive statistics for that data set: the mean, median, mode, range, variance, and standard deviation. You enter your values (separated by commas, spaces, or new lines) and it summarizes both the center and the spread of your data.

It is useful for students checking homework, teachers grading, and anyone working with measurements, test scores, prices, or survey results. Because it reports both sample and population versions of variance and standard deviation, it works whether your numbers are a complete population or a sample drawn from a larger group.

How It Works: The Formulas

Each statistic comes from a clear, standard formula. With n numbers in your list:

• Mean (average): add all values and divide by the count. Mean = (x1 + x2 + ... + xn) / n
• Median: sort the values; the median is the middle value. With an even count, average the two middle values.
• Mode: the value (or values) that appears most often. A set can have one mode, several, or none.
• Range: largest value minus smallest value. Range = max - min
• Population variance: average squared distance from the mean. Var = Σ(xi - mean)² / n
• Sample variance: same numerator, but divide by (n - 1) instead of n: Σ(xi - mean)² / (n - 1)
• Standard deviation: the square root of the variance (population or sample, matching the divisor used).

Worked Example

Suppose your data set is: 4, 8, 6, 5, 3, 8.

Count n = 6. Sum = 4 + 8 + 6 + 5 + 3 + 8 = 34, so the mean = 34 / 6 = 5.6667. Sorted, the values are 3, 4, 5, 6, 8, 8; the two middle values are 5 and 6, so the median = (5 + 6) / 2 = 5.5. The value 8 appears twice while every other value appears once, so the mode = 8. The range = 8 - 3 = 5.

For variance, find each squared deviation from the mean (5.6667): (-1.6667)², (2.3333)², (0.3333)², (-0.6667)², (-2.6667)², (2.3333)². These sum to about 20.8333. Population variance = 20.8333 / 6 = 3.4722, so population standard deviation = √3.4722 = 1.8634. Sample variance = 20.8333 / 5 = 4.1667, so sample standard deviation = √4.1667 = 2.0412.

Sample vs. Population: Which to Use

The only difference between the two is the divisor: n for a population, n - 1 for a sample. Use the population version when your numbers represent every member of the group you care about (for example, the test scores of all students in one class). Use the sample version when your numbers are a subset used to estimate a larger group.

Dividing by n - 1 (Bessel's correction) makes the sample variance a less biased estimate of the true population variance, which is why it is the default in most statistics courses and software.

Tips and Common Mistakes

Small errors usually come from data entry or from confusing similar statistics. Keep these points in mind:

• Sort before finding the median by hand; using the unsorted middle value is a frequent mistake.
• Mean vs. median: with skewed data or outliers, the median often describes the typical value better than the mean.
• A single extreme outlier pulls the mean and inflates the range, variance, and standard deviation; check whether such values are genuine.
• Confirm your count n. A stray comma or duplicated value changes every result.
• Match the divisor to your intent: choose sample (n - 1) or population (n) deliberately, since the two standard deviations differ.
• Standard deviation is in the same units as your data; variance is in those units squared, which makes it harder to interpret directly.