Mean median mode and range: which to use
Mean median mode are three ways to pick one typical value from a list: the total shared out equally, the middle value in order, and the value that turns up most often. The range is the gap between the biggest and smallest values.
Mean median mode can give three different figures from one list. Below they are worked on a short list, then matched to test wording. The add-and-divide basics are on averages; more practice is in the numerical practice hub.
Mean median mode definition: one short list, four answers
Take a short list of call times, in minutes. The same list gives a different answer to each of these four questions, and a test word such as average or typical usually points at just one of them.
- Mean. The total shared out equally. The five calls were 4, 9, 4, 6 and 12 minutes, which add up to 35, and 35 ÷ 5 = 7 minutes each.
- Median. The middle value once the list is in order, here 4, 4, 6, 9, 12. Five values put the middle at position (5 + 1) ÷ 2 = 3, so the median is the third one, 6 minutes.
- Mode. The value that turns up most often. The 4 appears 2 times out of 5, so 2 ÷ 5 = 0.4 of the calls, and 4 minutes is the mode.
- Range. The spread, which is not an average at all. The longest call minus the shortest is 12 − 4 = 8 minutes.
Mean median mode formula: all four worked on one list
The mean needs a total and a count. The median needs the list in order first. The mode needs only a tally, and a list can have two modes or none.
Formulas
Mean = (Total) divided by (Count)
Median = middle, in order
- Total is every value added together, and Count is how many values there are.
- With an even count, the median is halfway between the middle two.
Worked example
A courier logged six delivery times in minutes, in the order they happened: 14, 9, 21, 9, 12 and 13. Find the mean, the median and the range, and name the mode.
Picking the middle of the list as written takes 21 and 9, and (21 + 9) ÷ 2 gives a median of 15. The list has to be put in order first.
- Total: 14 + 9 + 21 + 9 + 12 + 13 = 78
- Mean: 78 ÷ 6 = 13
- Median: in order the list is 9, 9, 12, 13, 14, 21, so the middle pair is 12 and 13, and (12 + 13) ÷ 2 = 12.5
- Range: 21 − 9 = 12
The mean is 13 minutes, the median is 12.5 minutes and the range is 12 minutes. The 9 appears twice and nothing else repeats, so 9 is the mode.
Check it: 6 times 13 is 78, the total you started with.
Mean median mode range: which one does the question want?
Each measure wins on a different job, so read what the table is for before you pick.
- Mean, when the total matters. Five weekly costs of 20, 22, 21, 23 and 19 hundred pounds add up to 105, and 105 ÷ 5 = 21 a week.
- Median, when one value is out of line. Waiting times of 5, 6, 6, 7 and 31 minutes total 55, so the mean is 55 ÷ 5 = 11, yet the middle wait is 6. One long wait lifts the mean 11 − 6 = 5 above what most people sat through, so use the median for pay, prices and waits.
- Mode, when you want the most common. For stock sizes it is the size that sells most. If orders are 3, 5, 7, 8 and 9 units, nothing repeats, each value is 1 of 5, so 1 ÷ 5 = 0.2, and there is no mode to report.
- Range, when the spread is the question. Two reps both average 20 sales a week. Rep A runs from 18 to 22 and rep B from 5 to 35, so B's range is 35 − 5 = 30 against A's 4.
- The add-and-divide method and the missing-value trick sit on the overview: how to calculate an average
Mean median mode examples: a table of values and how often
Test tables often arrive grouped: a value in one column, how many times it happened in the next. The counts are the data, not the rows.
Add the counts to find how many values the table really holds, put the median at the middle of that count, and walk down the running total until you pass it. The mode is the value with the biggest count.
Formula
Position = (Count + 1) divided by (2)
- Count is the sum of the how-many column, not the number of rows.
Worked example
A help desk logged tickets per day over 15 days: 2 tickets on 5 days, 4 tickets on 4 days, 6 tickets on 3 days, 8 tickets on 2 days and 10 tickets on 1 day. What are the mode and the median?
Reading the five rows as five values puts the middle on the third row, 6. The table holds 15 days, so the middle is the 8th day.
- Days in the table: 5 + 4 + 3 + 2 + 1 = 15
- Median position: (15 + 1) ÷ 2 = 8, the 8th day
- Running count: 5 days at 2 tickets plus 4 days at 4 tickets is 5 + 4 = 9 days, which passes the 8th, so the 8th day had 4 tickets
The mode is 2 tickets, the most common day with 5 days, and the median is 4 tickets.
Check it: 5 days at 2 tickets take positions 1 to 5, and 4 more days at 4 tickets take positions 6 to 9, so position 8 is a 4-ticket day.
Mean median mode questions: which measure does the wording ask for?
A question seldom names the measure, so the wording has to. Here are the signals, then a worked question.
Look for the word that names the job: fair share or budget means the mean, typical or middle the median, most common the mode, and spread or gap the range.
Formula
Range = Highest − Lowest
- Highest and Lowest are the largest and smallest values in the list.
Worked example
A table of 11 exam marks reads 52, 58, 61, 61, 64, 67, 70, 72, 75, 79 and 91. The question asks for the range of the marks and the median mark. What are they?
Measuring from the mean instead of from the lowest mark gives a smaller number than the range. The range always runs from the lowest mark to the highest, 91 − 52.
- Range: 91 − 52 = 39
- Median position: with 11 marks, (11 + 1) ÷ 2 = 6, so it is the sixth mark, which is already in order
The range is 39 marks, and the sixth mark, 67, is the median.
Check it: five marks sit below the sixth and five above it.
Want to try these under a clock? The button below opens the numerical practice page. Its 12-question diagnostic runs on a 12-minute clock, and the clock starts only when you press start on that page.
Which other method does your question need?
- Groups of different sizes, each with its own mean: weighted average questions
- Every numerical formula, one worked example each: the formulas reference
Want to try these under a clock? The button below opens the numerical practice page. Its 12-question diagnostic runs on a 12-minute clock, and the clock starts only when you press start on that page.
How we write, check and mark the practice questions is set out on the Methodology page.