Measures of Dispersion for Ungrouped Data
How to Find the mean, median and mode
Use this to summarise a set of ungrouped data with its three measures of central tendency.
Before you start
- Adding whole numbers and dividing to get an average
- Ordering numbers from smallest to largest
- Counting how many times each value repeats
When to use it
Use this to summarise a set of ungrouped data with its three measures of central tendency.
The steps
- For the mean, add all values and divide by how many there are.
- For the median, arrange the values in order and take the middle one.
- If there is an even number of values, average the two middle ones.
- For the mode, find the value that appears most often.
- State all three clearly, with units if the data has them.
Worked example
Seven students scored 3, 8, 5, 8, 7, 10 and 8 marks in a quiz. Find the mean, median and mode.
- For the mean, add all seven values and divide: (3 + 8 + 5 + 8 + 7 + 10 + 8) ÷ 7 = 49 ÷ 7 = 7.
- For the median, arrange in order: 3, 5, 7, 8, 8, 8, 10.
- There are 7 values (odd), so take the middle (4th) value = 8; no averaging is needed.
- For the mode, find the value that appears most often: 8 appears three times, more than any other.
- State all three with units: mean = 7 marks, median = 8 marks, mode = 8 marks.
A second example, with a twist
There is an even number of values, so the median is the average of the two middle ones, and the data has two modes (bimodal). Eight pupils read 2, 3, 3, 5, 5, 6, 7 and 9 books over the holidays.
Find the mean, median and mode.
- For the mean, add all eight values and divide: (2 + 3 + 3 + 5 + 5 + 6 + 7 + 9) ÷ 8 = 40 ÷ 8 = 5.
- For the median, arrange in order (already ordered): 2, 3, 3, 5, 5, 6, 7, 9.
- There are 8 values (even), so average the two middle ones (4th and 5th): (5 + 5) ÷ 2 = 5.
- For the mode, 3 appears twice and 5 appears twice, both more than any other value, so the data is bimodal.
- State all three with units: mean = 5 books, median = 5 books, modes = 3 and 5 books.
Formulae you may need
Formula pages
Practise this in a KBAT problem
Frequently asked questions
What if two values tie for the mode?
Then the data has two modes and is called bimodal, list both values. If three or more tie it is multimodal.
And if every value appears the same number of times, there is no mode at all; just say 'no mode'.
Which average should I actually report?
Report all three when the question asks. Each describes the data differently: the mean uses every value, the median is the middle and resists extreme values, and the mode is the most common.
Together they give a fuller picture of the data.
Do I need to order the data to find the mean?
No. Ordering only matters for the median, where you need the middle value.
The mean just needs the total of all values and the count, so you can add them in any order. Order the list before finding the median, though.