Why summarise at all
A list of measurements is hard to reason about. Two questions usually matter: what is a typical value, and how much do the values vary? The first is answered by an average, the second by a measure of spread. Together they turn a column of numbers into something you can compare and communicate.
Here is the data set used throughout. Imagine eight repeated readings of some quantity, in arbitrary units:
2, 4, 4, 4, 5, 5, 7, 9
These are illustrative numbers chosen to keep the arithmetic simple.
The mean
The mean is the sum divided by the count.
Sum = 2 + 4 + 4 + 4 + 5 + 5 + 7 + 9 = 40. There are 8 values, so the mean is 40 / 8 = 5.
The mean uses every value, which is its strength and its weakness: one extreme reading pulls it noticeably.
The median
The median is the middle value when the data are sorted. The list above is already sorted. With eight values there is no single middle, so take the two central ones (the 4th and 5th, which are 4 and 5) and average them: (4 + 5) / 2 = 4.5.
With an odd number of values, the median is simply the middle one. The median ignores how far the extremes lie, which makes it resistant to outliers.
The mode
The mode is the most frequent value. Here 4 appears three times, more than anything else, so the mode is 4. It is mainly useful for counts and categories, and less so for measured quantities where repeats are rare.
Range
The range is the largest value minus the smallest: 9 − 2 = 7. It is easy to compute, but it depends only on two readings and says nothing about the values in between.
Standard deviation
The standard deviation describes a typical distance of the values from the mean. It takes four steps.
- Subtract the mean (5) from each value: −3, −1, −1, −1, 0, 0, 2, 4.
- Square each result: 9, 1, 1, 1, 0, 0, 4, 16.
- Add the squares: 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32.
- Divide by the count and take the square root.
Dividing by 8 gives 32 / 8 = 4, and the square root is 2. This is the population standard deviation, appropriate when the eight values are the entire group of interest.
When the values are a sample used to estimate a larger set of possible readings, as in most testing, it is common to divide by one less than the count. That gives 32 / 7 ≈ 4.571, and the square root is about 2.14. This is the sample standard deviation. For small data sets the two differ noticeably, and for large ones hardly at all. Spreadsheets and calculators often have both functions, so check which one you are using and say which you reported.
The squared quantity from step 4 before the root, 4 or about 4.571, is called the variance. Its units are the square of the original units, which is why the root is taken for reporting.
Reading the result
A mean of 5 with a standard deviation near 2 says the readings typically sit within a couple of units of 5. How to interpret that depends on the process. For data that follow a roughly bell-shaped pattern, a large share of values often lies within one standard deviation of the mean, but that is a tendency and not a guarantee, and many real data sets are not bell-shaped.
Always state the number of readings alongside the mean and standard deviation. A figure calculated from three readings is far less reliable than one calculated from three hundred, a point that connects directly to measurement uncertainty.
When the mean misleads
Add a ninth reading of 100 to the data. The sum becomes 140, so the mean becomes 140 / 9 ≈ 15.6. The sorted list is now 2, 4, 4, 4, 5, 5, 7, 9, 100, and the median is the fifth value, which is 5.
One extreme reading, perhaps a recording error or a real event, more than tripled the mean while the median barely moved. The standard deviation grows enormously too. This does not mean the mean is wrong. It means you should look at the data before choosing a summary, and investigate the outlier before deciding whether to keep it. Do not delete a value simply because it is inconvenient.
Practical habits
- Look at the numbers or a plot first. A simple chart often shows clusters, trends and outliers that summary figures hide, which is part of good data visualization.
- Report both centre and spread. A mean with no spread is half a story.
- Keep sensible precision. Quote results with a suitable number of significant figures rather than every digit a tool prints.
- Check by hand once. Computing a small example manually, as above, helps you trust the spreadsheet formula you will use later.
- Watch for changing conditions. If readings drift over time, one mean for the whole set hides that.
These tools are the start of engineering data analysis. They describe data but do not prove anything about causes. For decisions involving safety, quality or regulation, follow the statistical methods required in your industry and involve someone qualified.