Paste a list of numbers and it computes the usual descriptive statistics at once: count, sum, mean, median and mode, minimum, maximum and range, quartiles by the method you choose, the interquartile range and outliers, variance and standard deviation for a sample or a whole population, the standard error of the mean, skewness, excess kurtosis, the coefficient of variation and a confidence interval for the mean at 90, 95 or 99%. It draws a box plot and a histogram and lists a z-score for every distinct value. Data with frequencies ("value count" on each line) is expanded for you.
The second tab takes pairs of numbers and gives Pearson's correlation r, R², the least-squares line with its equation, the standard error of the estimate, a scatter plot with the line and every residual. The third tab answers probability questions about the normal, binomial and Poisson distributions. Everything is computed in the page; the data is not sent anywhere.
Which numbers to look at
The mean uses every value and is pulled by extreme ones; the median is the middle value and ignores how extreme the ends are. With the sample data in the box, 12, 15, 11, 18, 22, 15, 14, 30, 13, 16, 17, 15 and 45, the mean is 18.69 but the median is 15: the 30 and the 45 pull the mean up. When the mean and median differ a lot, report the median (and the quartiles), or report both. The mode is the most common value, 15 here (three times); a list where every value occurs equally often has no mode.
The standard deviation measures spread in the data's own units. Choose sample when your numbers are a sample from something larger (almost always), which divides by n − 1; population divides by n and is right only when you have every member. For the sample above s = 9.31; without the two outliers it falls to 3.04. The standard error of the mean (s/√n) measures how precisely the mean is known, not how spread the data is.
Quartiles and outliers: the method matters
There is no single definition of a quartile, and calculators disagree on small data sets. Three are offered:
- Tukey's hinges: the medians of the lower and upper halves (including the median when n is odd). This is what most school textbooks and box plots use.
- Inclusive: linear interpolation at position (n − 1)p + 1, as in Excel's QUARTILE.INC, R's default and NumPy.
- Exclusive: position (n + 1)p, as in Excel's QUARTILE.EXC and Minitab.
For 2, 4, 4, 4, 5, 5, 7, 9 the third quartile is 6, 5.5 or 6.5 depending on the method. Outliers are values beyond 1.5 × IQR from the quartiles (Tukey's fences); the box plot's whiskers stop at the most extreme values inside the fences and the outliers are drawn as circles. An outlier is a value to check, not one to delete.
Shape, intervals and z-scores
Skewness (sample G1, as in Excel's SKEW) is positive when the long tail is on the right. Excess kurtosis (sample G2, Excel's KURT) is 0 for a normal distribution and positive for heavy tails. Both are unreliable with small samples.
The confidence interval is mean ± t × SEM, with t from Student's t distribution with n − 1 degrees of freedom; the tool computes t itself (2.262 for n = 10 at 95%) rather than using a normal 1.96, which is too narrow for small samples. It assumes independent values from a roughly normal population or a large sample. A z-score says how many standard deviations a value is from the mean.
Regression and distributions
Pearson's r measures the strength of a straight-line relationship, from −1 to 1; R² is the share of the variance in y that the line explains. Correlation does not show cause, and a curved relationship can have r near 0, so look at the scatter plot. The normal panel gives the density, P(X ≤ x), P(X ≥ x) and the inverse (the x for a given probability); the special functions are implemented here and checked against SciPy to about 1e-12. Binomial gives the chance of exactly, at most or at least k successes in n trials; Poisson the same for counts at an average rate λ. For a worked comparison of mean, median and standard deviation, read which average to report; for counting problems, the combinatorics calculator is next door, and the function grapher plots curves.
