The formula this page uses
x̄ = Σx / n s = √[ Σ(x − x̄)² / (n − 1) ] z = (x − x̄) / s
Statistics Calculator
Solve statistics problems with clear steps, notation, and a final check. Choose a focused operation below to calculate, inspect the method, and connect the result to a visual model.
The visual updates with your calculation.
Result—Choose a mode, enter the known values, and calculate.
Show your work Step-by-step method
- The calculation method and verification will appear here.
Statistics
Statistics Calculator explained
The short version
- The mean is the balance point of the data; the median is the value in the middle after sorting.
- Standard deviation says how far a typical value sits from the mean, in the same unit as the data.
- A z-score converts any value into 'how many standard deviations from average', which is what a percentile table needs.
What each part means
SymbolWhat it means
x̄ — Sample meanThe average, in the same unit as the data.
n — Sample sizeHow many values you have. n − 1 is used for a sample, n for a whole population.
s — Sample standard deviationTypical distance from the mean, in the data's unit.
z — z-scoreA pure number with no unit: how many standard deviations above or below the mean a value falls.
Show your work: a full example
- The data set4, 8, 9, 9, 15 with n = 5
- Add and divide for the mean(4 + 8 + 9 + 9 + 15) ÷ 5 = 45 ÷ 5 = 9
- Distance of each value from 9−5, −1, 0, 0, 6
- Square each distance and add25 + 1 + 0 + 0 + 36 = 62
- Divide by n − 1 = 4 for the sample variance62 ÷ 4 = 15.5
- Square-root for the standard deviations = √15.5 = 3.937
- z-score of the value 15(15 − 9) ÷ 3.937 = 1.524, so 15 sits about 1.5 standard deviations above average
A second, different case
- A different case: one outlier joins4, 8, 9, 9, 15, 51 with n = 6
- New mean(45 + 51) ÷ 6 = 96 ÷ 6 = 16
- New median: average the middle two of the sorted list(9 + 9) ÷ 2 = 9
- Compare what movedthe mean jumped from 9 to 16, the median did not move at all
- Population standard deviation of the original five, dividing by n√(62 ÷ 5) = √12.4 = 3.521
- Sample standard deviation of the same five, dividing by n − 1√(62 ÷ 4) = √15.5 = 3.937
- Read the differencedividing by the smaller number n − 1 always gives the larger answer
Copy-ready example
4, 8, 9, 9, 15 with n = 5
The data set
Questions about the Statistics Calculator
Do I divide by n or by n − 1?
Divide by n − 1 when your numbers are a sample drawn from something bigger, which is almost every homework question. Divide by n only when the numbers really are everyone: all 30 students in your class, all 12 months of last year.
Should I report the mean or the median?
Use the median when a few extreme values would drag the average around, which is why house prices and salaries are reported as medians. Use the mean when the data is roughly symmetric and you plan to do more arithmetic with it.
What does a negative z-score tell me?
That the value is below the mean. The sign carries the direction and the size carries the distance, so z = −2.0 means two standard deviations below average, or roughly the bottom 2% of a normal distribution.
How many values do I need before a standard deviation is meaningful?
With fewer than about 5 values, one number can change the answer completely, as the outlier example above shows. Small samples are still calculable, but quote them alongside the data itself rather than on their own.
Calculate, interpret, verify.
This workspace keeps the formula and the meaning together. Decimal results are rounded for display; retain full precision when you continue a calculation.
Enter known values
Match each input to the quantities in the problem and keep units consistent.
Use the relationship
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Read the visual
Use the diagram, plot, or data display to check scale, direction, and plausibility.
Practice unaided
Move to targeted questions once you can explain why the method applies.
Learn the mathematics
Understand statistics behind this calculator
A calculator confirms an answer. Working the method yourself is what makes the next problem faster.