People use the word 'average' for all sorts of things: the typical weekly shop, a pupil's exam results or the price of a house in a neighbourhood. But there is more than one mathematical average, and using the wrong one can paint a misleading picture.
The arithmetic mean is the familiar add-and-divide calculation. The median is the middle value once numbers have been ordered. The mode is the most common value. For straightforward homework the mean may be all you need; for real-world comparisons it helps to know why the other measures exist.
Calculate the arithmetic mean
To find the mean of a set of numbers, add them together and divide by how many values there are. The formula is: mean = total of all values ÷ number of values.
Take the test scores 64, 72, 80 and 84. Their total is 300. There are four scores, so the mean is 300 ÷ 4 = 75.
The answer is not necessarily one of the original numbers. In this case none of the four scores is 75, and that is perfectly normal.
Count the values accurately
A very common mistake is calculating the correct total and dividing by the wrong number of observations. If you have five weekly expenses, the divisor is five even if two weeks had the same amount.
Suppose you recorded £10, £12, £12, £15 and £21. The total is £70, and the mean is £70 ÷ 5 = £14.
Do not divide by four because there are only four distinct amounts. Repeated values count separately because they represent separate observations.
Find the average with decimals
The same method works for decimal values. Suppose three measurements are 1.2m, 1.5m and 1.8m. Their total is 4.5m; dividing by three gives a mean of 1.5m.
Write the units in the answer. An average of 1.5 means very little without knowing whether the numbers describe metres, kilograms or hours.
Take care with rounding. Keep sufficient precision during addition and division, then round only the final answer to a level appropriate for the measurement.
Understand the median
The median is the middle value when the numbers are arranged from smallest to largest. Consider 2, 5, 7, 9 and 20. The median is 7 because it is the third of five ordered numbers.
With an even number of values, find the mean of the two middle ones. For 2, 5, 7 and 9, the two middle values are 5 and 7, so the median is (5 + 7) ÷ 2 = 6.
The median often gives a more useful description of a typical price or income when a few very large numbers would pull the mean upward.
Find the mode
The mode is the value that occurs most often. In the list 3, 4, 4, 5, 6, the mode is 4 because it appears twice.
A set may have more than one mode if several values tie for the highest frequency. It may also have no mode under a convention that requires an actually repeated value.
Modes are handy for non-numerical categories too. You can ask which shoe size or product colour appears most often even when an arithmetic mean would make no sense.
What does the range tell you?
The range is not an average, but it tells you how spread out the data is. Subtract the smallest value from the largest.
For scores 64, 72, 80 and 84, the range is 84 - 64 = 20. Two groups can have the same mean but very different ranges.
An average alone can hide variation. When comparing performance, prices or travel times, looking at both a measure of central tendency and the spread is often more informative.
See how outliers affect the mean
Imagine five house prices, in thousands of pounds: 180, 190, 200, 210 and 920. The total is 1700, so the mean is £340,000.
The median, however, is £200,000. One unusually expensive property raises the mean dramatically while the middle sale remains much closer to the majority.
Neither figure is inherently wrong. The mean describes the total divided equally across sales; the median describes the central sale price. Use the one that answers your question.
Calculate a weighted average
Sometimes observations do not count equally. Suppose coursework is worth 40% of a grade and an exam is worth 60%. A coursework score of 80 and exam score of 70 do not produce a simple mean of 75 if the weights apply.
Calculate 80 × 0.40 = 32 and 70 × 0.60 = 42. Add them to get a weighted result of 74.
The weights must total 100%, or 1, for this simple form. With other weights, multiply each value by its weight, add the results and divide by the sum of weights.
Average speed needs special care
Suppose you drive 60 miles at 30mph and return the same 60 miles at 60mph. The simple mean of the speeds is 45mph, but that is not your overall average speed.
The outward trip takes two hours, the return one hour. You travel 120 miles in three hours, so average speed is 120 ÷ 3 = 40mph.
For average speed, use total distance divided by total elapsed time. Speeds cannot always be averaged directly, particularly when journey durations differ.
How to calculate an average in a spreadsheet
In Excel or another spreadsheet, put the observations in a column. The function AVERAGE can calculate the arithmetic mean of a suitable numerical range.
For example, =AVERAGE(B2:B6) works for five entries in cells B2 to B6. Check that the range includes the correct records and does not accidentally include a heading or unrelated total.
Blank cells and text may be treated differently from numeric zeroes. A genuinely recorded zero score is part of the average; a missing score may need separate handling depending on the analysis.
Mean versus median in everyday life
A mean is useful when every observation should contribute proportionally, such as mean test score across equally weighted tests.
A median often describes a typical household income or sale price more robustly because extreme values have less influence. A mode can be ideal when you want the most frequently ordered shirt size.
The choice is about the question, not which statistic produces the most attractive headline. If you are publishing data, label the measure accurately.
Check for impossible or misleading answers
For an ordinary arithmetic mean of finite numbers, the result should lie between the smallest and largest values. If your four exam scores range from 60 to 85 and you get 112, something went wrong.
Make sure all entries use the same unit. Averaging £30 with 25 dollars without currency conversion is not meaningful.
Do not include subtotals twice. Adding a column's total as though it were another observation gives a much larger and incorrect mean.
Three exercises to try
Find the mean of 6, 8 and 10: total 24 divided by three gives 8. Find the median of 2, 4, 8 and 10: the middle values are 4 and 8, giving 6.
Find the mode of 1, 2, 2, 3, 3, 3 and 4: it is 3. These three measures answer different questions about the same data.
The rule worth remembering
When somebody asks you to 'calculate the average', begin by checking whether they mean the arithmetic mean. Add the values, count the observations and divide.
Then ask whether the mean genuinely describes what they care about. The median or mode may tell a clearer story, especially when the data contains very unusual values.