Three averages, and why they disagree
When the mean misleads, why the median is usually the honest number for money, and what a standard deviation adds that no average can.
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In short
- Mean, median and mode are three different questions. Reporting one as "the average" without saying which is where most statistical misdirection starts.
- The mean is dragged by outliers. One large value in a small set moves it a long way and moves the median not at all.
- For income, house prices, wealth and waiting times, the median is almost always the more representative figure.
- A data set can have no mode, one mode, or several. It is the only average that works on non-numeric data.
- Two sets can share a mean and be entirely different. Standard deviation is what tells them apart.
"Average" is not one thing. It is at least three, and which one is being quoted changes the answer — sometimes by a great deal.
Mean. Add everything and divide by the count.
Median. Sort the values and take the middle one.
Mode. The value that appears most often.
Where they part company
Consider the monthly salaries in a small firm, in thousands of rupees:
25, 28, 30, 30, 32, 35, 40, 45, 500
| Measure | Value |
|---|---|
| Mean | ₹85,000 |
| Median | ₹32,000 |
| Mode | ₹30,000 |
The mean is higher than eight of the nine salaries. It is arithmetically correct and it describes nobody in the room. One founder's salary of ₹5,00,000 has pulled it above every ordinary employee.
The median is ₹32,000 — the middle person — and it is unmoved by how large that one salary is. Raise it to ₹50,00,000 and the mean becomes ₹5,85,000 while the median stays exactly ₹32,000.
That resistance to extreme values is what makes the median the honest measure for skewed data, and income, wealth, house prices, waiting times and web page load times are all strongly skewed.
Which to use
Use the mean when values are roughly symmetric and there are no extreme outliers: heights, exam scores, temperatures, manufacturing tolerances. It uses every data point, which the median does not, and it is what almost every further statistical technique is built on.
Use the median when the distribution is skewed or when outliers exist and are real rather than errors. Income and property prices are the classic cases. A country's median household income tells you about a typical household; its mean tells you partly about its billionaires.
Use the mode for categorical data where arithmetic is meaningless — the most common blood group, the best-selling shoe size, the most frequent complaint category. It is the only one of the three that works on words.
The mode is also the only one that may not exist, or may exist several times over. 2, 2, 5, 5, 9 has two modes. 1, 2, 3, 4 has none.
Why the news uses one and not the other
The choice is frequently rhetorical.
"Average salary at this company is ₹85,000" is true of the firm above and creates a false impression. "Median salary is ₹32,000" is true and creates an accurate one. Both are defensible; only one is informative.
The reverse manoeuvre also exists. Where a distribution is skewed the other way — a few very low values among many high ones — the mean understates and quoting it looks modest. Whenever a single figure is doing persuasive work, ask which average it is and what the distribution looks like.
A useful diagnostic: if the mean and median are far apart, the data is skewed and the mean is not describing a typical case. If they are close, either will do.
What no average tells you
Two data sets:
A: 48, 49, 50, 51, 52
B: 10, 30, 50, 70, 90
Both have a mean of 50 and a median of 50. They are not remotely the same data.
Range is the crudest measure of spread — 4 against 80 here — and is entirely determined by the two most extreme values.
Standard deviation measures typical distance from the mean, using every point. For A it is about 1.58; for B, 31.62. That difference is the whole story the averages missed.
Interquartile range — the span from the 25th to the 75th percentile — is the median's companion: robust to outliers, and the right partner when the median is the right centre.
A mean without a standard deviation, or a median without an interquartile range, is half a description. Any summary of data that quotes one number should prompt the question of how spread out it was.
The weighted mean
A plain mean assumes every value counts equally. Often they do not.
If a shop sells 10 items on Monday at a 90% success rate and 1,000 on Tuesday at 50%, the simple average of the two percentages is 70%. The true overall rate is:
(9 + 500) ÷ 1,010 = 50.4%
The weighted mean multiplies each value by the size of its group before averaging. It is the correct method whenever the groups differ in size, and forgetting it is one of the most common errors in reporting departmental, regional or period-by-period figures.
Grade point averages, portfolio returns and index values are all weighted means. So is any average of averages, whether or not the person computing it realises.
Reading a percentile
Percentiles generalise the median, which is simply the 50th.
Being at the 90th percentile means 90% of the data lies below you. This is how exam ranks, growth charts and salary benchmarks are reported, and it works well precisely because it makes no assumption about the shape of the distribution.
Note that percentiles are not evenly spaced in value. Moving from the 50th to the 60th percentile in a competitive exam may require a handful of extra marks; moving from the 95th to the 99th may require a great many, because the candidates are packed tightly in the middle and thinly at the top.
Skew, and reading a distribution's shape
The relationship between the three averages describes the shape of the data without needing to plot it.
Symmetric. Mean, median and mode land in roughly the same place. Heights, measurement errors and exam scores usually look like this.
Right-skewed, with a long tail of high values: mean above median above mode. Income, wealth, house prices, city populations and response times all take this shape, which is why the mean is above the median in every one of them.
Left-skewed, with a tail of low values: mean below median below mode. Age at death in a developed country, or scores on a test most people pass easily.
Comparing the mean and median therefore tells you something before you look at anything else. If they differ substantially, the data has a tail, and a single number will not describe it well whichever one you choose.
Outliers: remove or keep?
An outlier is a value far from the rest, and the decision about what to do with it is a judgement rather than a calculation.
Keep it when it is real. The ₹5,00,000 salary in the example is genuine data about that firm, and deleting it would misrepresent the payroll. Use the median alongside the mean rather than discarding anything.
Investigate it when it might be an error. A recorded height of 17 cm is a data entry mistake, not a person.
Never remove it because it is inconvenient. Dropping values that spoil a conclusion is the most common form of quiet data manipulation, and it is detectable — the sample size changes.
The robust approach is to report both the mean and the median and let the difference between them do the talking. Where they diverge, the divergence is itself the finding.
What this calculator assumes
- Values may be separated by commas, spaces or line breaks, and non-numeric entries are ignored.
- The median of an even-sized set is the mean of the two middle values, which is the standard convention.
- All modes are reported where several values tie, and no mode is reported when every value appears once.
- Standard deviation is the sample form, dividing by n − 1, which is correct when the data is a sample of a larger population.
- Values are treated as unweighted. For a weighted mean, repeat each value according to its weight.