Mean, Median, Mode and Standard Deviation Explained
What each summary statistic really tells you, when the average misleads, why n − 1 sits in the denominator, and how to read quartiles and outliers.
Five numbers can describe a data set well enough to spot what’s wrong with it. They can also bury the problem completely. It depends entirely on which five you look at.
The mean has one obvious weakness
Put nine people earning $40,000 in a room with one person earning $5 million. The average salary is $536,000. Nobody in that room earns anything remotely close to that figure, and if you reported it as “typical,” you’d be lying with an accurate calculation.
The median handles this fine. Sort the ten salaries, take the middle, and you get $40,000. That’s the number that describes the room.
So the rule of thumb: if your data has a long tail — incomes, house prices, page load times, waiting times in a clinic — the median is usually the honest summary. If it’s roughly symmetric, the mean is fine and does more work, because most statistical machinery is built on top of it.
Mode gets ignored, which is a shame
Mean and median only work on numbers. Mode works on anything you can count: the most common shoe size, the most-picked answer, the browser that shows up most in your logs. It’s also the only one of the three that has to be a value actually present in the data.
A set can have two modes, or five, or none at all. “No mode” is a real answer, not an error, and it just means nothing repeats.
Standard deviation, and that annoying n − 1
Two classes both average 70% on a test. In the first, every score is between 66 and 74. In the second, half the class scored 45 and half scored 95. Identical mean, completely different situation. Standard deviation is the number that tells them apart.
Now the part that trips people up. Sample standard deviation divides by n − 1 instead of n, and it looks arbitrary until you see why. Your sample has its own mean, and that mean sits by construction closer to your sample’s values than the true population mean does. Squared deviations come out slightly too small every time. Dividing by n − 1 nudges the estimate back up. It’s called Bessel’s correction, and if the numbers you’re holding are a sample of something bigger, it’s the version you want.
Why summary statistics need a plot next to them
In 1973 the statistician Francis Anscombe published four data sets that share the same mean of x, the same mean of y, the same variance, the same correlation to two decimal places, and the same fitted regression line.
Plot them and they’re nothing alike. One is a tidy linear relationship. One is a clean curve. One is a perfect line with a single point yanking it off course. One is a vertical stack of points at a single x value with one lone point far to the right doing all the work.
Every summary number agreed. The pictures didn’t. Anscombe’s quartet is still the fastest argument for looking at your data before you summarise it.
Quartiles and the 1.5 rule
Q1 is the value a quarter of the way through your sorted data, Q3 is three quarters through, and the gap between them — the interquartile range — covers the middle half. It’s a measure of spread that ignores the tails entirely, which is exactly what you want when a couple of extreme values are shouting over everything else.
The convention of flagging anything beyond 1.5 × IQR from either quartile comes from John Tukey’s Exploratory Data Analysis in 1977. It’s a heuristic, not a verdict. A flagged point might be a typo, a broken sensor, or the single most interesting observation you have. The calculator marks it; deciding what it means is your job.
Paste your numbers into the Statistics Calculator and you’ll get all of it — mean, median, mode, quartiles, both standard deviations — without a spreadsheet formula in sight.