Statistics Calculator

Paste a dataset and this calculator returns the full descriptive picture: how many values you have, their sum, the three measures of center (mean, median, mode), the spread (range, quartiles, interquartile range, variance, standard deviation), the minimum and maximum, plus the distribution shape (skewness and excess kurtosis), the standard error of the mean, and the coefficient of variation. You can switch between the sample formulas (divide by n – 1, for data drawn from a larger group) and the population formulas (divide by n, for data that is the whole group).

Quartiles use the Tukey hinge method: Q1 is the median of the lower half of the sorted data, Q3 the median of the upper half, and the interquartile range (IQR) is Q3 – Q1. The IQR covers the middle 50% of your data and is not dragged around by extreme values the way the range is.

Worked example

Dataset: 2, 4, 4, 4, 5, 5, 7, 9 (the classic textbook example), using the sample formulas.

1. Count n = 8, sum = 40, mean = 40 / 8 = 5.0000
2. Sorted: 2, 4, 4, 4, 5, 5, 7, 9. Median = (4 + 5) / 2 = 4.5000
3. Mode = 4 (appears 3 times, more than any other value)
4. Range = 9 – 2 = 7.0000; Q1 = 4, Q3 = 6, IQR = 2.0000
5. Sum of squared deviations = 32. Sample variance = 32 / 7 = 4.5714; sample SD = √4.5714 = 2.1381

Paste the same numbers above with the sample mode selected and every figure matches. Switch to population mode and the variance becomes 32 / 8 = 4.0000 with SD 2.0000.

Sample vs population: which to choose

Choose sample when your data is a subset of something bigger: a survey of 200 voters, 30 product measurements from a production line, a class of students standing in for all students. Dividing by n – 1 (Bessel’s correction) compensates for the fact that a sample underestimates the true spread. Choose population only when your data is the entire group of interest: all 50 states, every employee in a 12-person company, the full set of monthly sales for a closed year. When in doubt, sample is the safer default, and it is what most textbooks and statistical software assume.

Reading the results

• Mean vs median: if the mean is clearly above the median, a few large values are pulling the average up (right skew). If it is below, low values are pulling it down. When they agree, the data is roughly symmetric.
• Standard deviation: roughly, a typical value sits within one SD of the mean. A small SD means the data clusters tightly; a large one means it is spread out.
• IQR vs range: the range uses only the two extremes, so one outlier can dominate it. The IQR covers the middle half of the data and stays honest when extremes misbehave.
• No mode: with all-unique values there is no most-frequent value, which is normal for continuous measurements like heights or times.

Frequently asked questions

Why do sample and population variance differ?

They divide the same sum of squared deviations by different denominators: n – 1 for a sample, n for a population. With small datasets the difference is noticeable (4.5714 vs 4.0000 in the example); with thousands of values it nearly vanishes.

Can I paste data from a spreadsheet column?

Yes. The calculator accepts numbers separated by commas, spaces, tabs, semicolons, or line breaks, so a copied column pastes directly. Anything that is not a number triggers a clear error instead of a silent wrong answer.

How are quartiles calculated?

With Tukey hinges: Q1 is the median of the lower half of the sorted data and Q3 the median of the upper half (the median itself is excluded from both halves when n is odd). Other software may use slightly different conventions, so small differences against Excel are expected.

What does it mean when there are two modes?

The dataset is bimodal: two values tie for most frequent. That often signals two subgroups mixed together, such as test scores from two different classes. The calculator lists all tied modes.

Why is the standard deviation the square root of the variance?

Variance is in squared units (dollars-squared, for instance), which is hard to interpret. The square root brings the spread back into the original units so it can be compared directly with the mean.

How many values can I enter?

A few thousand is comfortable. Very large pastes still compute, but the sorted-data display is capped at the first 60 values to keep the page readable.