Representing Data Numerically (AQA Level 3 Mathematical Studies (Core Maths))
Revision Note
Written by: Naomi C
Reviewed by: Dan Finlay
Mean, Median & Mode
What is the mode?
The mode is the value that appears the most often
The mode of 1, 2, 2, 5, 6 is 2
There can be more than one mode
The modes of 1, 2, 2, 5, 5, 6 are 2 and 5
The mode can also be called the modal value
In some situations there may be no mode
What is the median?
The median is the middle value when you put values in size order
The median of 4, 2, 3 can be found by
ordering the numbers: 2, 3, 4
and choosing the middle value, 3
If you have an even number of values, find the midpoint of the middle two values
The midpoint is the sum of the two middle values divided by 2
The median of 1, 2, 3, 4 is 2.5
2.5 is the midpoint of 2 and 3
What is the mean?
The mean is the sum of the values divided by the number of values
The notation, , is used to represent the mean
The mean of 1, 2, 6 is (1 + 2 + 6) ÷ 3 = 3
The mean can be fraction or a decimal
It may need rounding
You do not need to force it to be a whole number
You can have a mean of 7.5 people, for example!
The mean is often selected because it uses all of the data
How do I calculate the mean from a frequency table?
To find the mean from a frequency table of ungrouped data, use the formula
where is the sum of each data item, , multiplied by its corresponding frequency,
and is the sum of all of the frequencies
To find the mean from a frequency table of grouped data
Use the same formula as for a frequency table
Use the mid-interval value (midpoint) of each group as the value for
How do I know which average to use?
The mode, median and mean are different ways to measure an average
Units for the mean, median and mode are the same as for the data set
In certain situations it is better to use one average over another
For example:
If the data has extreme values (outliers)
Don't use the mean (it's badly affected by extreme values)
If the data has more than one mode
Don't use the mode as it is not clear
If the data is non-numerical, like dog, cat, cat, fish
You can only use the mode
Worked Example
15 students were timed to see how long it took them to solve a mathematical problem. Their times, in seconds, are given below.
12 | 10 | 15 | 14 | 17 |
11 | 12 | 13 | 9 | 21 |
14 | 20 | 19 | 16 | 23 |
(a) Find the mean time, giving your answer to 3 significant figures.
Add up all the numbers (you can add the rows if it helps)
Divide the total by the number of values (there are 15 values)
Write the mean to 3 significant figures
Remember to include the units
The mean time is 15.1 seconds (to 3 s.f.)
(b) Find the median time.
Write the times in order and find the middle value
The median time is 14 seconds
(c) Explain why the median is a better measure of average time than the mode.
Try to find the mode (the number that occurs the most)
There are two modes: 12 and 14
Explain why the median is better
There is no clear mode (there are two modes, 12 and 14),
so the median is better
(d) If a 16th student has a time of 95 seconds, explain why the median of all 16 students would be a better measure of average time than the mean.
The16th value of 95 is extreme (very high) compared to the other values
Means are affected by extreme values
The mean will be affected by the extreme value of 95
whereas the median will not
Worked Example
The frequency table below shows the number of pets owned by 30 students in a class.
Number of pets, | 0 | 1 | 2 | 3 | 4 | |
---|---|---|---|---|---|---|
Frequency, | 5 | 13 | 7 | 4 | 1 |
Work out the mean number of pets owned.
Multiply each data item by its corresponding frequency and add together to find
The sum of the frequencies,, is the total number of students in the class
Use the formula, , to calculate the mean,
Round to 3 an appropriate degree of accuracy
The mean number of pets owned by students in the class is 1.43 (3 s.f.)
Range, Quartiles & Outliers
What are quartiles?
Quartiles are measures of location
Quartiles divide a population or data set into four equal sections
The lower quartile, , splits the lowest 25% from the highest 75%
The median, , splits the lowest 50% from the highest 50%
The upper quartile, , splits the lowest 75% from the highest 25%
There are different methods for finding quartiles, depending on the number of items in the data set,
First, list the items in size order
When finding the median and quartiles from raw data:
The median will be at position
The lower quartile will be at position
The upper quartile will be at position
For larger data sets:
The median will be at position
The lower quartile will be at position
The upper quartile will be at position
The use of rather than will still be accepted however
What are the range and interquartile range?
The range and interquartile range are both measures of dispersion
They describe how spread out the data is
The range is the largest value of the data minus the smallest value of the data
The interquartile range is the range of the central 50% of data
It is the upper quartile minus the lower quartile
The units for the range and interquartile range are the same as the units for the data
The range can be affected by outliers (extreme values)
Outliers will not affect the interquartile range
Examiner Tips and Tricks
If asked to find the range, or the interquartile range, in an exam, make sure you show your subtraction clearly (don't just write down the answer)
What are outliers?
Outliers are extreme data values that do not fit with the rest of the data
They are either a lot bigger or a lot smaller than the rest of the data
Outliers are defined as values that are more than from the nearest quartile
is an outlier if or
Outliers can have a big effect on some statistical measures
Should I remove outliers?
The decision to remove outliers will depend on the context
Outliers should be removed if they are found to be errors
The data may have been recorded incorrectly
For example, the number 17 may have been recorded as 71 by mistake
Outliers should not be removed if they are a valid part of the sample
The data may need to be checked to verify that it is not an error
For example, the annual salaries of employees of a business might appear to have an outlier but this could be the director’s salary
Worked Example
Find the range and interquartile range for the data set given below.
43 29 70 51 64 43 44
Find the range by subtracting the minimum value from the maximum value
Range = 41
Arrange the values from smallest to largest
The lower quartile will be at position
The upper quartile will be at position
Subtract the lower quartile from the upper quartile to find the interquartile range
IQR = 21
Standard Deviation
What is standard deviation?
The standard deviation, , is a measure of dispersion
It describes how spread out the data is in relation to the mean
If greater the value of the standard deviation, the more spread out the data is
The units for the standard deviation are the same as the units for the data
How is standard deviation calculated?
The standard deviation is the square root of the mean of the squares of the differences between the values and the mean
The formula used in this course for standard deviation is the standard deviation for a sample
You can calculate the standard deviation of a small data set by hand
You can also enter the data to your calculator and use the stats calculation options to calculate the standard deviation of a data set
Examiner Tips and Tricks
If you use your calculator to find the standard deviation, make sure that you find the standard deviation for a sample, , and not the standard deviation for a population,
Worked Example
Find standard deviation for the data set given below.
43 29 70 51 64 43
Method 1: Calculator
You can calculate the standard deviation using your calculator
Input all of the values into a spreadsheet
Select the statistics calculation option and find the value for the standard deviation
Round appropriately
15.1 (to 1 d.p.)
Method 2: By hand
To calculate the standard deviation by hand, use the formula,
Start by finding the mean,
Find the difference between each data item and the mean, square it and add the results together,
Divide the result by 1 less than the number of data items,
Finally take the square root of the result
Round appropriately
15.1 (to 1 d.p.)
Last updated:
You've read 0 of your 5 free revision notes this week
Sign up now. It’s free!
Did this page help you?