Mean, Median and Standard Deviation

Three summary statistics, three different questions. Choosing the wrong one is how completely honest data ends up telling a misleading story.

Three numbers answering three different questions

Mean, median and standard deviation are usually taught together and then used interchangeably, which is where the trouble starts. They answer genuinely different questions, and picking the wrong one is how honest data produces misleading claims.

• Mean — if the total were shared out equally, what would each share be?
• Median — what does the middle of the group look like?
• Standard deviation — how far from typical is a typical value?

A worked example

Ten salaries on a team, in thousands: 42, 45, 47, 48, 52, 55, 58, 61, 64, 68.

Mean and median sit within $500 of each other, which tells you the distribution is roughly symmetric. When those two numbers agree, either one is a fair summary.

Now add one founder

Replace the top earner with someone on $480,000 and nothing else changes:

The mean rose by 76% and the median did not move at all. Nine of the ten people now earn less than the average. This is precisely why income, house prices and response times are reported as medians — the mean is dragged by the tail, while the median only cares about position.

The flip side: the mean is the right answer when you care about the total. Payroll budget depends on the mean, not the median. The average calculator returns mean, median and mode together so you can see whether they disagree.

What standard deviation actually measures

It is the typical distance from the mean, computed in a way that stops positive and negative gaps cancelling out.

The squaring is doing two jobs: it removes the sign, and it penalises large deviations disproportionately. That is why one extreme value can move the standard deviation far more than it moves the mean, as the salary example showed.

Population or sample? The n − 1 question

Step 3 hides a decision. Divide the sum of squares by n and you get the population standard deviation. Divide by n − 1 and you get the sample standard deviation.

Take the set 2, 4, 4, 4, 5, 5, 7, 9. The mean is 5, the median is 4.5, the population SD is exactly 2, and the sample SD is 2.138.

The n − 1 adjustment exists because a sample’s own mean sits closer to its own values than the true population mean does, so the raw calculation systematically underestimates spread. Dividing by the smaller number corrects the bias. With 10 values the difference is about 5%; with 1,000 it is negligible. When in doubt, use the sample version — it is the default in most software and the conservative choice. The standard deviation calculator reports both.

The 68-95-99.7 rule, and its limits

For data that follows a normal distribution, roughly 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three. That is a genuinely useful shortcut — it turns a standard deviation into an intuition about range.

It only holds for normal distributions. Applied to salary data, web response times, insurance claims or anything else with a long right tail, it will badly understate how often extreme values occur. Before leaning on the rule, check whether mean and median agree. When they diverge, the distribution is skewed and the rule does not apply.

For turning these differences into percentages, see percentage increase and decrease.