Management Accounting · Summarising and analysing data

Mean, Median and Mode: Measures of Central Tendency

Updated 11 October 2026 · Fact-checked

A measure of central tendency is one value that represents a whole data set. The mean is the sum of values divided by their number. The median is the middle value when data is ordered. The mode is the most frequent value. For grouped data, use class mid-points and frequencies.

Understand Measures of Central Tendency

A set of numbers is hard to read. An average gives you one figure that stands for the whole set. In management accounting you use it to summarise things like daily output, unit costs or delivery times.

There are three main averages. The mean uses every value, so it is the most complete. The median is the middle item once the data is in order. The mode is the value that occurs most often.

They can give different answers. Take the values 2, 3, 3, 4, 28. The mean is 8, the median is 3 and the mode is 3. The large value 28 pulls the mean up. The median ignores it. So the median is better when data has extreme values (outliers).

A weighted average is used when some values matter more than others. A cost of $10 on 900 units should count for more than a cost of $20 on 100 units. You multiply each value by its weight, add up, then divide by the total weight.

For grouped data you only know how many items fall in each class, not the exact values. You assume each item sits at the class mid-point. The results are therefore estimates.

Key formulas to remember

Arithmetic mean (raw data): x̄ = Σx ÷ n; Add all values, divide by the number of values.
Mean (frequency or grouped data): x̄ = Σfx ÷ Σf; For grouped data, x is the class mid-point. The result is an estimate.
Class mid-point: Mid-point = (lower limit + upper limit) ÷ 2; Check class boundaries are continuous before using this.
Median position (raw data, ordered): Position = (n + 1) ÷ 2; If the position is a half, take the average of the two middle values.
Median (grouped data): Median = L + [(n ÷ 2 − cf) ÷ f] × w; L = lower boundary of median class, cf = cumulative frequency before it, f = its frequency, w = class width.
Mode (grouped data): Mode = L + [d1 ÷ (d1 + d2)] × w; L = lower boundary of modal class, d1 = modal frequency minus previous frequency, d2 = modal frequency minus next frequency. Equal class widths assumed.
Weighted average: Weighted mean = Σwx ÷ Σw; w is the weight, such as units, hours or proportion.

How to solve Measures of Central Tendency questions

Use this method for any question on averages. Read first which average is asked for and whether the data is raw, frequency or grouped.

1Identify the average required: mean, median, mode or weighted average.
2Identify the data type: a list of values, a frequency table, or classes (grouped).
3For grouped data, find each class mid-point. Use these as x.
4For the mean, add columns for fx. Total f and fx, then divide Σfx by Σf.
5For the median, order the data or build cumulative frequencies. Find the position and read the value, or interpolate with the formula.
6For the mode, pick the most frequent value, or the modal class and interpolate if asked.
7For a weighted average, multiply each value by its weight, total these, and divide by the total weights.
8Check the answer is sensible: the mean must lie between the lowest and highest value. Give the unit and the rounding asked for.

Quickest way: Fast checks for objective test questions

When to use it: Use this in Section A when you have about 2 to 2.5 minutes per two-mark question and a calculator on screen.

Raw data: use the calculator to total and divide. Write the sum down so you can re-check.
Weighted average: divide by total weights, not by the number of items.
Median of raw data: sort first. With an even count, average the two middle values.
Grouped data: build a small three-column table of f, x and fx. Do not skip it.
Eliminate options: the mean must sit between the smallest and largest value, and the median of a class table must lie inside the median class.

Common mistakes in Measures of Central Tendency

Taking a simple average when a weighted average is needed.
The values look easy to add and divide.
Fix: If items have different quantities or importance, multiply by weights and divide by total weights.
Finding the median without ordering the data.
Students pick the middle number of the list as given.
Fix: Always sort from smallest to largest first.
Dividing Σfx by the number of classes instead of Σf.
Confusing the number of rows with the number of items.
Fix: Divide by the total frequency, Σf.
Using class limits instead of mid-points for grouped data.
The limits are the numbers shown in the table.
Fix: Calculate the mid-point of each class and use that as x.
Saying the mean is always the best average.
It uses all the data, so it seems the most reliable.
Fix: Outliers distort the mean. For skewed data the median is often more representative. The mode suits the most common or popular item.
Forgetting that a data set can have no mode or more than one.
Students expect exactly one answer.
Fix: If every value occurs once, there is no mode. If two values tie, the data is bimodal.

Worked examples

Example 1

A factory records the number of machine breakdowns per week over 10 weeks: 2, 5, 3, 3, 7, 4, 3, 6, 4, 8. Calculate the mean, median and mode.

Show the solution

Mean: total = 2 + 5 + 3 + 3 + 7 + 4 + 3 + 6 + 4 + 8 = 45. Divide by 10 = 4.5.
Median: order the data: 2, 3, 3, 3, 4, 4, 5, 6, 7, 8.
Position = (10 + 1) ÷ 2 = 5.5, so average the 5th and 6th values: (4 + 4) ÷ 2 = 4.
Mode: 3 occurs three times, more than any other value. Mode = 3.

Answer: Mean = 4.5 breakdowns, median = 4, mode = 3.

Example 2

A company buys a material in three batches: 200 kg at $5.00 per kg, 300 kg at $5.50 per kg and 500 kg at $6.00 per kg. Calculate the weighted average price per kg.

Show the solution

Multiply each price by its quantity: 200 × 5.00 = 1,000; 300 × 5.50 = 1,650; 500 × 6.00 = 3,000.
Total cost = 1,000 + 1,650 + 3,000 = $5,650.
Total weights (kg) = 200 + 300 + 500 = 1,000.
Weighted average = 5,650 ÷ 1,000 = $5.65 per kg.
Check: the simple average of the prices would be (5.00 + 5.50 + 6.00) ÷ 3 = $5.50, which is wrong because it ignores quantities.

Answer: The weighted average price is $5.65 per kg.

Exam tips

Read the question for the word 'estimate' or 'grouped'. It tells you to use mid-points.
In multiple response questions, check each statement about advantages and disadvantages separately. Mean uses all data but is affected by outliers; median ignores extremes; mode is easy but may not exist or may not be unique.
Number entry questions often state the rounding. Keep full figures in your calculator until the end.
If a question gives costs and quantities, think weighted average straight away.
Use the sense check: a mean outside the range of the data is always wrong.

Practice questions from Summarising and analysing data

Measures of Central Tendency in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

CFA Level IMeasures of Central Tendency

Measures of Central Tendency: frequently asked questions

How do I calculate the mean of grouped data?

Find the mid-point of each class and multiply it by the class frequency. Add these to get Σfx, then divide by Σf. The answer is an estimate because the exact values are not known.

When should I use the median instead of the mean?

Use the median when the data contains extreme values or is skewed. The median is not affected by outliers, so it often describes a typical value better. The mean is preferred when data is fairly symmetrical and you need to use every value.

What is the weighted average formula in management accounting?

Weighted mean = Σwx ÷ Σw, where w is the weight and x is the value. Weights are often units, hours or proportions. It is used for average prices, costs and rates.

Can a data set have more than one mode?

Yes. If two or more values share the highest frequency, the data has more than one mode. If every value occurs only once, there is no mode.