Statistics Calculator
Mean · Median · Mode · Standard deviation · Quartiles · Outliers
📈 Paste or type your numbers to see the mean, median, mode, standard deviation, quartiles, outliers, and a histogram.
What Does a Statistics Calculator Do?
A statistics calculator summarizes a list of numbers with descriptive statistics. These describe three things about the data: its center (mean, median, mode), its spread (range, variance, standard deviation, interquartile range), and its shape (skewness, kurtosis, outliers).
Paste test scores, measurements, survey answers, prices, or lab readings, and this calculator returns every common summary at once, with the standard deviation worked out step by step and a histogram of the distribution. It is useful for homework, checking spreadsheet formulas, quality control readings, and getting a quick feel for any data set before deeper analysis.
How to Use the Calculator
- Enter your data. Type or paste numbers separated by commas, spaces, semicolons, or new lines. A column copied from Excel or Google Sheets works directly. Negative numbers and decimals are accepted.
- Choose sample or population. Pick Sample when your numbers are a subset of a larger group, which is the usual case. Pick Population only when you have every member of the group.
- Choose a quartile method. Inclusive matches Excel QUARTILE.INC and Google Sheets QUARTILE. Exclusive matches QUARTILE.EXC and the method in the NIST handbook.
- Read the results. The mean, median, standard deviation, and mode appear at the top, followed by the shape verdict, the calculation steps, a histogram, and a full summary table.
Statistics Formulas with a Worked Example
Mean vs Median vs Mode
| Measure | How to find it | Best for |
|---|---|---|
| Mean | Sum ÷ count | Symmetric data without outliers |
| Median | Middle sorted value | Skewed data such as incomes or house prices |
| Mode | Most frequent value | Repeated values, categories, shoe sizes |
| Geometric mean | nth root of the product | Growth rates and ratios (positive values only) |
For 2, 4, 4, 4, 5, 5, 7, 9 the median is (4 + 5) ÷ 2 = 4.5 and the mode is 4. In skewed data the mean is pulled toward the long tail, so when the mean sits well above the median the data are usually skewed right.
How to Calculate Standard Deviation by Hand
Follow the deviation table below for the data set 2, 4, 4, 4, 5, 5, 7, 9. The calculator shows the same steps for your own numbers.
| x | x − x̄ | (x − x̄)² |
|---|---|---|
| 2 | −3 | 9 |
| 4 (three times) | −1 | 1 each, 3 total |
| 5 (twice) | 0 | 0 |
| 7 | 2 | 4 |
| 9 | 4 | 16 |
| Sum | 0 | 32 |
- Find the mean: 40 ÷ 8 = 5.
- Subtract the mean from each value. The differences always add to zero, which is a useful check.
- Square each difference and add them to get the sum of squares, 32.
- Divide by n − 1 = 7 for a sample (4.5714) or by n = 8 for a population (4).
- Take the square root: s = 2.1381 or σ = 2.
Quartiles, Interquartile Range, and Outliers
Quartiles split sorted data into four equal parts. Q1 is the 25th percentile, Q2 is the median, and Q3 is the 75th percentile. The interquartile range (IQR = Q3 − Q1) measures the spread of the middle half and ignores the extremes.
The two methods disagree on whether 9 is an outlier, which shows why small data sets need care. Values beyond the outer fences (3 × IQR) are extreme outliers.
Skewness and Kurtosis
Skewness measures lack of symmetry. Zero means symmetric, positive means a longer right tail, and negative means a longer left tail. For a sample, the calculator uses the adjusted Fisher Pearson coefficient G₁; for a population, it uses g₁ = m₃ ÷ m₂^1.5.
A common rule of thumb treats |skewness| below 0.5 as roughly symmetric, 0.5 to 1 as moderately skewed, and above 1 as highly skewed. It is a guide, not a formal test.
Excess kurtosis compares the tails with a normal distribution, which scores 0. Positive values suggest heavier tails and more outliers; negative values suggest lighter tails. Sample skewness needs at least 3 values and sample kurtosis at least 4.
Interpreting Results and Common Mistakes
Standard deviation in context. For roughly bell shaped data, about 68% of values fall within one standard deviation of the mean and about 95% within two. The default scores have mean 81.1 and s = 9.04, which puts most scores between about 72 and 90.
Reading the histogram. The calculator groups values into bins using Sturges' rule, k = ⌈log₂ n⌉ + 1, capped at 20 bins. Ten values give 5 bins and 1,000 values give 11. A single central peak with even sides suggests symmetric data, a long run of short bars on one side shows skew, and two separate peaks suggest the data mix two groups that may be better analyzed separately. The darker bar holds the mean.
- Wrong denominator. Excel STDEV.S and STDEV.P give different answers. Match the calculator's sample or population setting to the function you are checking.
- Standard deviation vs standard error. SD describes the spread of individual values; SE describes how precisely the mean is estimated and shrinks as n grows.
- Different quartiles in different software. Statistical packages use several quartile definitions, so compare using the same method.
- Removing outliers automatically. An outlier can be a typing error or a real observation. Investigate before deleting it.
- Mixing units. Variance is in squared units (cm²), while standard deviation is in the original units (cm).
Frequently Asked Questions
Sources & References
- NIST/SEMATECH. Engineering Statistics Handbook, 1.3.5.1 Measures of Location (mean, median, mode). itl.nist.gov
- NIST/SEMATECH. Engineering Statistics Handbook, 1.3.5.6 Measures of Scale (variance with n − 1, standard deviation, range, IQR). itl.nist.gov
- NIST/SEMATECH. Engineering Statistics Handbook, 1.3.5.11 Measures of Skewness and Kurtosis (g₁, adjusted G₁, excess kurtosis). itl.nist.gov
- NIST/SEMATECH. Engineering Statistics Handbook, 7.1.6 What are outliers in the data? (inner and outer fences, (n + 1) quartiles). itl.nist.gov
Results are for educational purposes. Verify important analyses with statistical software. Methodology reviewed: 2026.