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CalcMyWay

Statistics Calculator

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Result

Mean (average)

14.6
Median
15
Mode
15
Standard deviation
3.3066
Count (n)
10
Sum
146
Variance
10.9333
Range
11
Minimum
9
Maximum
20
First quartile (Q1)
12.5
Third quartile (Q3)
16.5
Interquartile range
4
  • Calculated as a sample, so variance divides by n − 1 = 9. This is Bessel's correction, and it is the right choice unless your data is the entire population.

Deviation from the mean

Each value's distance from the mean, and that distance squared — the terms the variance sums.

Valuex − x̄(x − x̄)²
9-5.631.36
11-3.612.96
12-2.66.76
14-0.60.36
150.40.16
150.40.16
150.40.16
172.45.76
183.411.56
205.429.16

How do you calculate mean, median, mode and standard deviation?

The mean is the sum of all values divided by how many there are. The median is the middle value once the data is sorted. The mode is the value appearing most often. The standard deviation is the square root of the average squared distance of each value from the mean.

x̄ = Σx / n s² = Σ(x − x̄)² / (n − 1) s = √s²
x̄
The mean, pronounced "x bar"
Σx
The sum of every value
n
The number of values
s²
Sample variance
s
Sample standard deviation

The mean and median are both averages and they answer different questions. The mean uses every value, so a single extreme number drags it; the median only cares about position, so it is unaffected. When the two disagree substantially the distribution is skewed, and the median is usually the more honest summary. Standard deviation measures spread. Squaring each deviation removes the sign — otherwise the deviations cancel to exactly zero — and the square root at the end returns the figure to the original units, which is why standard deviation is quoted rather than variance. The divisor is the part people get wrong. A sample divides by n − 1 rather than n, known as Bessel's correction: a sample's own mean sits closer to its own values than the true population mean does, so dividing by n underestimates the real spread. Use n only when the data is the entire population.

  1. Add every value together to get the sum.
  2. Divide the sum by the number of values to get the mean.
  3. Subtract the mean from each value and square each result.
  4. Add all of those squared deviations together into a single total.
  5. Divide by n − 1 for a sample, or n for a population, to get variance.
  6. Take the square root of the variance to get standard deviation.

Worked example: The data set 12, 15, 11, 18, 14, 15, 20, 9, 15, 17 treated as a sample

The data set 12, 15, 11, 18, 14, 15, 20, 9, 15, 17 treated as a sample

There are 10 values and they sum to 146, so the mean is 14.6. Sorted, the data reads 9, 11, 12, 14, 15, 15, 15, 17, 18, 20. With an even count the median is the average of the fifth and sixth values, both 15, so the median is 15. The value 15 appears three times and nothing else repeats, so the mode is 15. For the spread, subtract 14.6 from each value and square it: (9 − 14.6)² = 31.36, (11 − 14.6)² = 12.96, and so on. Those squares total 98.4. Because this is a sample we divide by n − 1 = 9, giving a variance of 10.9333, and the square root is a standard deviation of 3.3066. Had we treated it as a population, dividing by 10 instead, the variance would be 9.84 and the standard deviation 3.1369 — lower, which is exactly the understatement Bessel's correction exists to prevent.

What each statistics input means

Your data set
The numbers you are analysing, separated by commas, spaces or line breaks. Paste a column straight from a spreadsheet and it will parse.
Is this a sample or the whole population?
A sample is a subset you are using to infer something about a larger group; a population is every member of the group. This changes the divisor in the variance calculation.
Decimal places
How many decimal places to display. The calculation itself is carried at full precision.

What this statistics does not account for

  • A data set where no value repeats has no mode. That is different from a mode of zero, and this calculator reports "None" rather than inventing one.
  • When several values tie for the highest frequency the set is multimodal and every tied value is reported. Many calculators silently return only the first.
  • Sample variance is undefined for a single value, because it divides by n − 1 = 0. Switch to Population if you genuinely have one observation.
  • Quartiles have no single agreed definition. This calculator uses linear interpolation, matching Excel's PERCENTILE.INC and NumPy's default; textbooks using Tukey's hinges or the nearest-rank method will differ slightly on small data sets.
  • Outlier flagging uses Tukey's 1.5 × IQR rule, which is a convention and not a test of significance. A flagged value may be entirely genuine.
  • Non-numeric tokens are ignored rather than treated as zero, so a stray header row pasted from a spreadsheet will not distort the mean.

Frequently asked questions

What is the difference between sample and population standard deviation?

Sample standard deviation divides the squared deviations by n − 1; population divides by n. The sample version is larger and corrects for the fact that a sample underestimates true spread. Use the sample version unless your data covers the whole group.

What is the difference between mean, median and mode?

The mean is the arithmetic average of all values. The median is the middle value when sorted. The mode is the most frequent value. For skewed data the median is usually the most representative, because extreme values pull the mean.

How do you find the median of an even number of values?

Sort the data and take the two middle values, then average them. For ten values that is the fifth and sixth. With an odd count there is a single middle value and no averaging is needed.

Why is variance squared?

Squaring each deviation makes all of them positive, which stops them cancelling to zero, and it weights large deviations more heavily. The square root taken at the end returns the result to the original units as standard deviation.

What does standard deviation tell you?

It measures how spread out the data is around the mean. A small standard deviation means values cluster tightly; a large one means they are dispersed. For roughly normal data, about 68% of values fall within one standard deviation.

How do you identify an outlier?

The common rule is Tukey's: any value more than 1.5 times the interquartile range below Q1 or above Q3. This calculator flags those automatically. It is a convention for spotting candidates, not proof a value is an error.

Sources and review

Last reviewed .