Choosing the right average for the job matters. Here is a practical guide to when each measure is most appropriate:
| Measure | Best Used When | Avoid When | Real-life Example |
|---|---|---|---|
| Mean (x̅) | Data is roughly symmetrical with no major outliers | Data contains extreme outliers or is skewed | Average test score in a class; average temperature this week |
| Median | Data has outliers or is skewed; ordinal data | You need to use the result in further calculations | Median household income; median house price; median waiting time |
| Mode | Categorical data; finding the most popular item | Data has no repeated values; continuous measurement data | Most popular shoe size; most common blood type; best-selling product |
The Office for National Statistics (ONS) reports median household income rather than mean household income. This is because a small number of extremely wealthy households would pull the mean income significantly upward, giving a misleading impression of what a "typical" UK household earns. The median is unaffected by these extreme values and represents the income of the person right in the middle of the income distribution — a more meaningful measure of typical earnings.
For example, if nine workers earn £25,000 and one CEO earns £2,500,000, the mean salary is £272,500 — but the median is £25,000. The median far better represents what most people earn.
In statistics, there is an important distinction between the sample mean and the population mean.
The population mean (written as μ, the Greek letter "mu") is the true average of every member of a population. For example, the true average height of every adult in the UK. In practice, it is rarely possible to measure every individual, so we estimate the population mean from a sample.
The sample mean (written as x̅, "x-bar") is the average calculated from a subset (sample) of the population, used to estimate μ. Both sample mean and population mean use the same formula (sum ÷ count), but they serve different roles in statistics.
| Symbol | Name | Meaning | Used For |
|---|---|---|---|
| x̅ | x-bar | Sample mean | Average of a sample drawn from a population |
| μ | mu | Population mean | True average of an entire population |
| Σ | Sigma (capital) | Sum of all values | Σx = sum all x values |
| n | n (lowercase) | Sample size | Number of observations in a sample |
| N | N (uppercase) | Population size | Total number in a population |
| M or Md | M | Median | Middle value of ordered data |
| Mo | Mo | Mode | Most frequently occurring value |
| σ | sigma (lowercase) | Population standard deviation | Spread around the population mean |
| s | s | Sample standard deviation | Spread around the sample mean |
When you calculate the mean from a complete dataset in front of you (such as 30 exam scores), that is a sample mean (x̅). The population mean (μ) is a theoretical value you can only know if you have data for every member of the population — which is rarely the case outside of census surveys.
The difference becomes critical when calculating standard deviation: population standard deviation divides by N, while sample standard deviation divides by n−1 (Bessel's correction) to correct for the bias introduced by using a sample rather than the full population.
In mathematics, the word modal simply means "relating to the mode." You will encounter this term in several contexts:
At GCSE level, you will primarily be asked to find the mode (modal value) of a simple list of numbers and identify the modal class from a frequency table. At A-level and beyond, you will explore bimodal and multimodal distributions and their significance in data analysis.
The BBC Bitesize GCSE Maths guide and the AQA GCSE Mathematics specification (component 8300) both list mean, median and mode as core topics under "Statistics and Probability," confirming these are foundational skills tested in national examinations across England, Wales and Northern Ireland.
The mode in maths is the value (or values) that appear most frequently in a dataset. For example, in {3, 5, 5, 7, 9}, the mode is 5 because it appears twice — more than any other value. If no value repeats, the dataset has no mode. If two values are equally common, the dataset is bimodal. The mode is the only average that can be used with non-numerical (categorical) data such as favourite colours or most popular shoe size.
To work out the median: (1) Sort all values from lowest to highest. (2) If n is odd, the median is the value at position (n+1)/2. (3) If n is even, the median is the mean of the values at positions n/2 and n/2+1. Example: sorted data {2, 4, 7, 9, 11} has n=5 (odd), so median = 3rd value = 7. For {2, 4, 7, 9}, n=4 (even), so median = (4+7)/2 = 5.5.
In maths, the mean is the arithmetic average — the sum of all values divided by the count of values. Formula: x̅ = Σx / n. For example, the mean of {10, 20, 30} is (10+20+30)/3 = 60/3 = 20. The mean is the most widely used measure of central tendency and is used in everyday contexts from school reports to national statistics.
The modal class is the class interval (group) that has the highest frequency in a grouped frequency table. It is used when data is grouped into ranges rather than listed as individual values. For example, if a frequency table shows that the 160–170 cm height group has 25 people (the most of any group), then 160–170 cm is the modal class. You cannot find a single modal value from grouped data — only the modal class.
In statistics, the median meaning is the middle value of an ordered dataset that divides it into two equal halves. It is a measure of central tendency that is resistant to outliers. Unlike the mean, one extreme value cannot move the median far from the centre. This makes the median the preferred measure of "typical value" for skewed data such as incomes, house prices and waiting times.
To compute the mean: Step 1 — list all your numbers. Step 2 — add them all together (this is the sum, Σx). Step 3 — count how many numbers there are (this is n). Step 4 — divide the sum by n. The result is the mean. Example: {5, 10, 15, 20, 25} → Sum = 75, n = 5, Mean = 75/5 = 15.
In maths, mode is defined as the value or values that occur with the greatest frequency in a dataset. If all values appear the same number of times, there is no mode. If one value appears more than all others, it is the mode. If two values share the highest frequency equally, the dataset is bimodal. The word "modal" relates to this concept — for example, the "modal class" in grouped data is the class with the highest frequency.
To work out the average (arithmetic mean): add all the numbers together and divide by how many numbers there are. For example, to find the average of {14, 18, 22, 10, 16}: add them (14+18+22+10+16 = 80), count them (5 numbers), divide (80 ÷ 5 = 16). The average is 16. Use our calculator above to do this automatically for any list of numbers.
The sample mean (x̅) is the average of a subset of a population, used to estimate the population mean. The population mean (μ, mu) is the true average of every member of a population. Both are calculated identically (sum ÷ count), but the population mean represents a complete, known dataset while the sample mean is an estimate based on partial data. In standard deviation calculations, the sample version divides by n−1 rather than n to account for estimation error.