Statistics Calculator
Paste a list of numbers to get the mean, median, mode, sample and population standard deviation (with every step shown), variance, range, quartiles and standard error, plus z-scores and weighted averages.
Statistics calculator
Standard deviation step by step
Average calculator and median calculator
The calculator above gives the average (mean) and median of any list. For a weighted average, such as grades with different weights:
Standard deviation formula in statistics
Standard deviation measures how spread out numbers are around their mean:
- Sample standard deviation: s = ā( Ī£(x ā xĢ)Ā² Ć· (n ā 1) ). Use it when your numbers are a sample from a bigger group (most real data).
- Population standard deviation: Ļ = ā( Ī£(x ā Ī¼)Ā² Ć· N ). Use it when you have every member of the group.
The stdev formula in spreadsheets follows the same split: STDEV.S (or STDEV) is the sample formula and STDEV.P the population formula.
How to calculate mean and standard deviation (worked example)
For 2, 4, 4, 4, 5, 5, 7, 9: the sum is 40 and n = 8, so the mean is 5. Subtract the mean from each value and square it: 9, 1, 1, 1, 0, 0, 4, 16, which add up to 32. Population variance is 32 Ć· 8 = 4, so Ļ = 2; sample variance is 32 Ć· 7 = 4.571, so s = 2.138. The table in the calculator shows these steps for your own numbers, which is what a standard deviation with mean calculator (or standard deviation calculator using mean) does: it works from the mean to each deviation. To calculate SD quickly by hand for a large list, use Ī£xĀ² ā nĀ·xĢĀ² for the sum of squares.
Average calculator and median calculator
The average (mean) is the sum Ć· count; the median is the middle value when sorted (or the mean of the two middle values); the mode is the most common value. For 12, 15, 15, 20, 38 the average is 20, the median 15 and the mode 15: one large value pulls the average up, which is why incomes and home prices are usually reported as medians. Quartiles here use the median-of-halves method (the TI-84 method); Excel's QUARTILE.INC can give slightly different Q1 and Q3.
Also searched as: standard dev calculator, calculate standard deviation calculator.
Common questions
How do I calculate standard deviation?
Find the mean, square each value's distance from it, add them, divide by n ā 1 (sample) or n (population), then take the square root.
What is the standard deviation of 2, 4, 4, 4, 5, 5, 7, 9?
Population Ļ = 2; sample s ā 2.138. The mean is 5.
Should I use sample or population standard deviation?
Sample (n ā 1) unless your data include every member of the group.
What's the difference between average and median?
The average is the sum Ć· count; the median is the middle value. Outliers move the average much more.
What is a z-score?
(Value ā mean) Ć· standard deviation: how many standard deviations a value is from the mean.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods: Measures of scale
- OpenStax: Introductory Statistics 2e
Related
General information, not financial, tax or legal advice. Check important numbers with the official source before acting on them.