Paste or type a list of numbers to get the full set of descriptive statistics — mean, median, mode, variance and standard deviation, quartiles, IQR and outliers — with a histogram.
Mean
5
Std deviation
2.1381
Calculator
Standard deviation (Sample (s))
2.1381
8 values, mean 5.
Variance
4.5714
Median
4.5
Range
7
Distribution
Count (n)
8
Sum
40
Mean
5
Median
4.5
Mode
4
Range
7
Minimum
2
Maximum
9
Q1 (25%)
4
Q3 (75%)
6
IQR
2
Outliers
none
Population vs sample standard deviation
The mean is the sum divided by the count. Variance is the average squared distance from the mean: the population version divides by n, the sample version divides by n − 1 (Bessel’s correction) to correct the bias when your data is only a sample. The standard deviation is the square root of the variance, back in the original units.
The median is the middle value; the quartiles Q1 and Q3 are the medians of the lower and upper halves, and the interquartile range IQR = Q3 − Q1 measures the spread of the middle 50%. A value more than 1.5·IQR below Q1 or above Q3 is flagged as an outlier.
Should I use population or sample standard deviation?
Use the population formula (÷ n) when your data is the entire group you care about. Use the sample formula (÷ n − 1) when your data is a sample drawn from a larger population — the most common case in statistics.
How are the quartiles calculated?
This calculator uses the median-of-halves method: Q1 is the median of the values below the overall median and Q3 is the median of the values above it. For an odd number of values the middle value is excluded from both halves.
What makes a value an outlier?
By the standard 1.5·IQR rule, any value below Q1 − 1.5·IQR or above Q3 + 1.5·IQR is treated as an outlier. It is a heuristic, not proof that the value is an error.
Results are estimates. Verify with a professional for important decisions.
About this calculator
This calculator analyses any list of numbers and returns a full summary: mean, median, mode, range, population and sample standard deviation, variance, quartiles, IQR, and outliers. Use it whenever you need to understand how spread out a data set is, whether for a school assignment, a scientific study, or a quality-control check.
How to read your results
The headline figure is the standard deviation — either population (σ, divides by n) or sample (s, divides by n−1), switchable with the toggle. Below it you see variance, median, and range at a glance. The full statistics table lists every measure including Q1, Q3, IQR, and any flagged outliers. The histogram plots the frequency of each bin and shades the ±1σ band around the mean, with a dashed line marking the mean itself.
Worked example
Enter the classic dataset 2, 4, 4, 4, 5, 5, 7, 9 (eight values used in most statistics textbooks) and select the population mode.
The mean is 5, the population standard deviation is exactly 2, and the variance is 4. Switching to sample mode gives s ≈ 2.1381 and variance ≈ 4.5714, because the denominator becomes n−1 = 7.
Frequently asked questions
What is the difference between population and sample standard deviation?
Population standard deviation (σ) divides the sum of squared deviations by n, the total count. Sample standard deviation (s) divides by n−1 — a correction called Bessel's correction — which removes a small downward bias when you are estimating from a subset of a larger group. Use population mode when your list is the entire group; use sample mode when it is a subset drawn from a larger population.
How are outliers detected?
The calculator uses the interquartile range (IQR) fence rule: any value below Q1 − 1.5 · IQR or above Q3 + 1.5 · IQR is flagged as a potential outlier. This method works well for roughly symmetric distributions; very skewed data may need a different approach.
What does a higher standard deviation tell me?
A high standard deviation means the values are spread widely around the mean; a low one means they cluster closely. Two datasets with the same mean can behave very differently in practice — the one with the larger standard deviation carries more variability and, in many fields, more risk.
How it's calculated
The mean is the arithmetic average (sum ÷ count). Variance is the average of squared deviations from the mean: divide by n for population, by n−1 for sample. Standard deviation is the square root of variance. Median is the middle value of the sorted list; for even counts it is the average of the two central values. Quartiles use the median-of-halves method, excluding the median itself for odd-length lists. Outliers are flagged using the 1.5 · IQR rule applied to the resulting fences. Sources: Wolfram MathWorld and Khan Academy (see references).
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