About the average calculation
Mean, median and mode are all called "the average" and they answer different questions. On skewed data they diverge sharply: a handful of very large values drags the mean upward while the median barely moves, which is why median income is reported rather than mean income.
Formula
Mean = sum of values ÷ count
Median = the middle value when sorted (average of the middle two if even)
Mode = the most frequently occurring value
Range = maximum − minimum
Standard deviation = √(Σ(x − mean)² ÷ N) population
√(Σ(x − mean)² ÷ (N−1)) sampleWorked example
For 2, 4, 4, 4, 5, 5, 7, 9: the mean is 5, the median is 4.5, the mode is 4, the range is 7 and the population standard deviation is exactly 2.
Frequently asked questions
When should I use the median instead of the mean?
Whenever the data is skewed or contains outliers. Incomes, house prices and response times all have long tails, and a few extreme values pull the mean somewhere no actual data point sits. The median describes the typical case far better.
What is the difference between population and sample standard deviation?
Population divides by N and assumes you have every data point. Sample divides by N−1 and corrects for the fact that a sample underestimates the spread of the population it came from. Use the sample version unless you genuinely have the whole population.
Can data have no mode?
Yes. If every value appears exactly once there is no most-frequent value, and reporting "everything is the mode" would be meaningless. Data can also be bimodal, with two values tied — this calculator lists all of them.
What separators can I use?
Commas, spaces, semicolons or new lines, in any mix. Anything that is not a number is ignored rather than counted as zero, so a stray character will not silently shift your average.
Sources
- Standard descriptive statistics definitions