About this tool
This average calculator takes a list of numbers pasted from anywhere, a spreadsheet column, a lab notebook, a set of test scores, and returns the full descriptive-statistics picture rather than a single figure: count, sum, arithmetic mean, median, mode or modes, minimum, maximum, range, population and sample standard deviation and variance, geometric and harmonic means, the first and third quartiles with the interquartile range, any outliers by the 1.5 x IQR rule, mean absolute deviation, standard error, and coefficient of variation, plus the sorted list and a frequency table. A separate panel computes a weighted average from values and weights, the calculation behind course grades, portfolio returns, and blended prices. The parser is forgiving about separators, currency signs, and stray text, so you can paste straight from an email. It is for students, teachers, analysts, and anyone who needs to summarize a batch of numbers correctly.
How it works
Mean = sum / n. Median is the middle value of the sorted list, or the average of the two middle values when n is even. Mode is every value that occurs most often; if no value repeats there is no mode. Population variance = sum of (x - mean)^2 / n and sample variance divides by n - 1 instead (Bessel's correction); standard deviation is the square root of each. Geometric mean = nth root of the product (computed as exp of the mean of logs) and harmonic mean = n / sum(1/x); both require all values to be positive. Quartiles use linear interpolation by default (R type 7, Excel QUARTILE.INC): the p-th quantile sits at position (n - 1) x p in the zero-based sorted list; the exclusive method (R type 6, QUARTILE.EXC) uses (n + 1) x p - 1. IQR = Q3 - Q1, and values below Q1 - 1.5 x IQR or above Q3 + 1.5 x IQR are flagged as outliers. Weighted mean = sum(w x value) / sum(w). Standard error = sample SD / sqrt(n); coefficient of variation = SD / mean.
Frequently asked questions
What is the difference between mean, median, and mode?
The mean is the arithmetic average: add every value and divide by how many there are. The median is the middle value once the numbers are sorted, so half the values lie below it; it is not pulled around by one huge or tiny value the way the mean is, which is why house prices and incomes are usually reported as medians. The mode is the most frequent value and is the only average that works for categories. For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5, the median is 4.5, and the mode is 4.
Should I use population or sample standard deviation?
Use the population formula (divide by n) when your list is the entire group you care about, for example every student in one class. Use the sample formula (divide by n - 1) when the list is a sample drawn from a larger population and you want to estimate that population's spread; dividing by n - 1 corrects the bias that comes from measuring spread around the sample's own mean. Spreadsheet STDEV and STDEV.S give the sample version; STDEV.P gives the population version. For large n the two are nearly identical.
How does the calculator find quartiles and outliers?
By default it uses the linear-interpolation method (R's type 7, Excel's QUARTILE.INC): sort the values, then the first quartile is at position 0.25 x (n - 1) counting from zero, interpolating between neighbors when that is not a whole number. You can switch to the exclusive method used by Excel's QUARTILE.EXC and many textbooks, which gives slightly wider quartiles on small lists. The interquartile range is Q3 - Q1, and any value more than 1.5 IQRs beyond either quartile is listed as an outlier, the same rule box plots use for their whiskers.
When should I use a weighted average, geometric mean, or harmonic mean?
A weighted average is right when some values matter more than others: a final grade where the exam counts 60% and homework 40%, or an average purchase price across lots of different sizes. The geometric mean is right for growth rates and ratios that compound, such as annual investment returns: the geometric mean of 1.10 and 0.90 is about 0.995, showing a small loss, while the arithmetic mean of 1.00 wrongly suggests break-even. The harmonic mean is right for rates over a fixed quantity, such as average speed over equal distances or price-to-earnings ratios.