Key takeaways
- Mean = sum of values ÷ number of values. Median = middle value. Mode = most common value.
- The median resists outliers. The mean does not.
- Range and standard deviation measure how spread out the data is.
- Population standard deviation divides by n. Sample standard deviation divides by n − 1.
In this guide
What is statistics?
Statistics is the maths of data. It helps you collect numbers, summarise them and draw fair conclusions from them.
It has two big parts. Descriptive statistics summarise data you already have, such as the average mark in a class. Inferential statistics use a sample to say something about a bigger group, such as an election poll.
Why does statistics matter?
Numbers are used to persuade you every day: in news reports, adverts, sports and school results. Knowing which average was used, and how spread out the data is, helps you see what the numbers really say.
Salaries of 30, 32, 35, 38, 40 and 500 thousand have a mean of 112.5 thousand, but a median of 36.5 thousand. Same data, very different story.
The key measures
| Measure | What it tells you | How to find it |
|---|---|---|
| Mean | The typical value (the 'average') | Add all values, divide by how many |
| Median | The middle value | Sort the data, take the middle one |
| Mode | The most common value | Count which value appears most |
| Range | The total spread | Largest minus smallest |
| Standard deviation (σ) | Typical distance from the mean | σ = √(Σ(x − μ)² ÷ n) |
In the standard deviation formula, Σ means add them all up, x is each value, μ (mu) is the mean and n is the number of values.
How to calculate them step by step
Let's use this data set of 8 values: 2, 4, 4, 4, 5, 5, 7, 9.
- 1Mean. Add them:
2 + 4 + 4 + 4 + 5 + 5 + 7 + 9 = 40. Divide by 8:mean = 5. - 2Median. The data is already sorted. With 8 values, the median is halfway between the 4th and 5th:
(4 + 5) ÷ 2 = 4.5. - 3Mode. The value 4 appears three times, more than any other, so
mode = 4. - 4Range.
9 − 2 = 7. - 5Standard deviation. Subtract the mean from each value and square it: 9, 1, 1, 1, 0, 0, 4, 16. These add to 32. Divide by 8 to get the variance, 4. The square root gives
σ = 2.
You can check each of these with the median calculator, mode calculator and standard deviation calculator.
Population or sample?
If your data is the whole group (every student in the class), divide by n. That is the population standard deviation, and it is what the standard deviation and variance calculator on this site use. If your data is a sample from a bigger group, divide by n − 1 instead. For the data above that gives √(32 ÷ 7) ≈ 2.14.
Common mistakes to avoid
- Using the mean when there are outliers. One extreme value can drag the mean far away from most of the data.
- Forgetting the square root. Variance is σ². Standard deviation is its square root.
- Mixing up sample and population. Check which one your teacher or textbook wants.
- Thinking there is always one mode. A data set can have two modes, or none.
Tips and tricks
- Quick check for the mean: it must lie between the smallest and largest values.
- Weighted averages are for when some values count more, like exam marks worth different amounts. Use the weighted average calculator.
- Standard deviation is never negative. It is 0 only when every value is the same.
Where you'll use it in real life
- School: your average mark, and how your score compares with the class.
- Sport: batting averages, goals per match, consistency of a player.
- Business: average sales per day and how much they vary.
- Health: normal ranges for blood pressure or growth charts.
Quick summary
Use the mean for a typical value, the median when there are outliers and the mode for the most common value. Measure spread with the range or the standard deviation, and know whether you need the population or sample version. The free calculators linked above show each step.
Frequently asked questions
When should I use mean vs median?
Use the median when the data has outliers or is lopsided, like incomes, because it is not pulled by extreme values. Use the mean when the data is fairly balanced.
What does standard deviation tell me?
It tells you how far values typically are from the mean. A small standard deviation means the data is bunched together. A large one means it is spread out.
What is the difference between variance and standard deviation?
Variance is the average of the squared distances from the mean. Standard deviation is the square root of the variance, so it is in the same units as the data.
Can a data set have more than one mode?
Yes. If two values tie for most common, the data set has two modes. If every value appears once, there is no mode.
