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Descriptive Statistics, Explained

After reading this you will be able to summarize a batch of numbers with a mean, a median, quartiles and a spread, read the shape of the data from its skewness, and spot outliers with a rule you can check by hand.

What descriptive statistics tell you

Descriptive statistics compress a list of numbers into a handful of values that answer three questions: where is the center, how spread out are the values, and what is the shape of the distribution. That is all. They describe the data you have. They do not, on their own, make claims about a larger population or test a hypothesis.

Take ten exam scores: 72, 85, 91, 68, 77, 85, 94, 62, 88, 79. The raw list is hard to reason about. Reduce it and the picture sharpens: the mean is 80.1, the median is 82, the lowest score is 62 and the highest is 94. Now you know the class scored in the low 80s with a 32 point range and no wild extremes. That summary took four numbers.

When to use them, and when not

Reach for descriptive statistics first, before any model or test. They catch data entry errors, reveal skew, and tell you which later methods are safe to use. A t test assumes roughly symmetric data; skewness of 2.5 warns you off it before you run anything.

They are the wrong tool when your question is about a population you did not fully measure. If you sampled 40 customers and want to state the average spend of all 12,000 customers with a margin of error, you need a confidence interval, not just the sample mean. Descriptives also mislead on categorical data. The mean of ZIP codes is meaningless. For counts of categories, use entropy or diversity measures instead.

A single number never replaces looking at the data. Anscombe's quartet is four datasets with identical means, variances and correlations that look completely different when plotted. Always pair a summary with a chart.

The formulas and the intuition

Start with the center. The mean adds every value and divides by the count.

\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i

Here \bar{x} is the mean, n is how many values you have, and x_i is the value at position i. The mean balances the data like a seesaw. Move one value far to the right and the balance point slides with it.

The median is the middle value once the numbers are sorted. With an even count it is the average of the two middle values. It ignores how far away the extremes sit, so it barely moves when a single value blows up.

For spread, the sample variance measures the average squared distance from the mean.

s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2

The divisor is n-1, not n. This is Bessel's correction. It corrects the tendency of a sample to underestimate the true spread, because the sample mean sits closer to the sample points than the unknown population mean does. The standard deviation s is the square root of the variance, which puts the spread back in the original units.

Skewness measures asymmetry. Zero means symmetric. Positive means a longer right tail (a few large values), negative means a longer left tail.

g_1 = \frac{1}{n} \sum_{i=1}^{n} \left( \frac{x_i - \bar{x}}{s} \right)^3

Cubing the standardized deviations keeps the sign, so a value far above the mean adds a large positive term while a value far below adds a large negative one.

A worked example with the demo data

Ten exam scores, from raw list to summary

Use the demo data 72, 85, 91, 68, 77, 85, 94, 62, 88, 79.

  1. Sort the values: 62, 68, 72, 77, 79, 85, 85, 88, 91, 94.
  2. Mean: the sum is 801, and 801 / 10 = 80.1.
  3. Median: with 10 values the middle two are the 5th and 6th, 79 and 85. Their average is 82.
  4. Variance: sum the squared deviations from 80.1. That total is 936.9, and 936.9 / 9 = 104.1.
  5. Standard deviation: \sqrt{104.1} \approx 10.2.
  6. Quartiles by linear interpolation: Q_1 = 72.25, Q_3 = 88.75, so the interquartile range is IQR = 16.5.

The median (82) sits above the mean (80.1). That gap is small and points to a mild left skew, driven by the two low scores 62 and 68. The computed skewness is about -0.36, which confirms it.

The median bar sits slightly right of the mean bar, the visual signature of mild left skew.

Watch an outlier drag the mean

The clearest way to feel the difference between mean and median is to move one point and watch. In the demo data the mean and median are only 1.9 apart. Push the top score up and the gap opens fast.

With the demo scores the mean is 80.1 and the median is 82. Raise the highest value from 94 to 194 and the mean climbs to 90.1 while the median stays at 82. The mean follows the extreme; the median does not.

Reading the results

Compare the mean and median first. When they are close, the data is roughly symmetric and either center is a fair summary. When they differ, the mean has been pulled by a tail or an outlier, and the median is the safer headline number. House prices are the classic case: a few mansions lift the mean rent well above what a typical household pays, so reports quote the median.

Read the spread against the center. A standard deviation of 10.2 on a mean of 80.1 is a coefficient of variation of about 13%. That is moderate. The same standard deviation on a mean of 15 would be a coefficient of variation of 68%, meaning the values scatter widely relative to their size.

Range
Max minus min. Fast to read, but one bad value ruins it. Here 94 - 62 = 32.
Interquartile range
Q3 minus Q1, the spread of the middle half. Ignores the tails, so it is robust. Here 16.5.
Skewness
Sign and size of asymmetry. Below -1 or above +1 is strongly skewed.

Common mistakes

Do not mix up population and sample variance. Dividing by n gives the population variance; dividing by n-1 gives the sample variance. For the demo data that is 93.7 versus 104.1, an 11 percent difference. Most software and this calculator report the sample version.

Two more traps catch people often. First, quartiles are not defined by a single universal rule. This calculator uses linear interpolation, the default in most software, so its Q1 of 72.25 may differ from a textbook that uses a different method. Compare quartiles only when both were computed the same way.

Second, do not delete outliers on sight. The 1.5 \times IQR rule flags candidates, not confirmed errors. A flagged point can be a genuine, informative value. Investigate before you remove anything, and report what you did.

Related tools

Once you have a summary, these calculators go further on specific questions. To test whether a flagged point is a real outlier, use the Outlier Detector, which adds Grubbs' test and z-scores to the Tukey fences shown here. To find the exact value at a given percentile or the rank of a value, use the Percentile calculator. To measure how two variables move together, use the Covariance calculator. For category counts rather than numbers, the Shannon entropy calculator reports diversity and evenness. And to check whether a large list of figures shows the digit pattern expected of natural data, try the Benford's law checker.

Frequently asked questions

Should I report the mean or the median?

Report the median when the data is skewed or has outliers, because it resists them. Report the mean for symmetric data, or when you need a value that later sums or averages correctly. When the two differ noticeably, showing both is honest.

Why does the variance divide by n minus 1?

Because the sample mean is fitted from the same data, the points sit closer to it than to the true population mean. Dividing by n-1 instead of n corrects that bias so the sample variance estimates the population variance without systematically running low.

What counts as strong skewness?

As a rough guide, skewness between -0.5 and 0.5 is nearly symmetric, between 0.5 and 1 in absolute value is moderate, and beyond 1 is strong. The demo data at -0.36 is only mildly skewed.

How many values do I need for these numbers to mean anything?

Mean and median are informative with even a handful of points. Skewness and quartiles are unstable below about 20 values, because a single point shifts them a lot. Treat shape statistics from tiny samples as hints, not conclusions.

Does the calculator use population or sample formulas?

It reports the sample standard deviation and variance, dividing by n-1, which matches the default in Excel, R and most statistics software.

Try it yourself in the tool

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