Three averages, three different questions
"Average" almost always means the mean, but the mean is the wrong answer surprisingly often. It is pulled hard by extreme values: put one billionaire in a room of ten teachers and the mean salary is meaningless while the median barely moves.
- Use the mean for symmetric data with no wild outliers — test scores, measurement repeats.
- Use the median for skewed data — income, house prices, response times, anything with a long tail.
- Use the mode for categories and for genuinely discrete data — the most common shoe size, the most frequent rating.
Spread: variance and standard deviation
Two datasets can share a mean and be nothing alike. Standard deviation measures the typical distance from the mean, in the same units as the data, which is why it is quoted far more often than variance.
The n − 1 in the sample formula is Bessel's correction. A sample's own mean sits closer to its own points than the true population mean does, so dividing by n systematically underestimates the spread. Dividing by n − 1 corrects the bias. Use the sample formula unless your data genuinely is the entire population.
For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three.
Quartiles and outliers
Quartiles cut sorted data into four equal parts. Q1 is the 25th percentile, Q2 is the median, Q3 is the 75th. The interquartile range (Q3 − Q1) covers the middle half of the data and is immune to extreme values, which makes it a robust measure of spread.
The standard outlier rule flags anything below Q1 − 1.5×IQR or above Q3 + 1.5×IQR. That is a prompt to investigate, not permission to delete — outliers are sometimes errors and sometimes the most important observation in the set.
When the mean and median diverge sharply, the distribution is skewed. Comparing the two is the fastest one-second check on whether an average is telling you the truth.