Mean, Median, Mode and Range
The mean, median and mode are types of average. The range gives a measure of the spread of a set of data.
This section revises how to calculate these measures for a simple set of data. It then goes on to look at how the measures can be calculated for a table of data.
Calculating the Mean, Median, Mode and Range for simple data
The table below shows how to calculate the mean, median, mode and range for two sets of data. Set A contains the numbers 2, 2, 3, 5, 5, 7, 8 and Set B contains the numbers 2, 3, 3, 4, 6, 7.
Measure Set A 2, 2, 3, 5, 5, 7, 8 Set B 2, 3, 3, 4, 6, 7 The Mean To find the mean, you need to add up all the data, and then divide this total by the number of values in the data. Adding the numbers up gives: 2 + 2 + 3 + 5 + 5 + 7 + 8 = 32
There are 7 values, so you divide the total by 7: 32 ÷ 7 = 4.57...
So the mean is 4.57 (2 d.p.) Adding the numbers up gives: 2 + 3 + 3 + 4 + 6 + 7 = 25
There are 6 values, so you divide the total by 6: 25 ÷ 6 = 4.33...
So the mean is 4.33 (2 d.p.) The Median To find the median, you need to put the values in order, then find the middle value. If there are two values in the middle then you find the mean of these two values. The numbers in order: 2 , 2 , 3 , (5) , 5 , 7 , 8
The middle value is marked in brackets, and it is 5.
So the median is 5 The numbers in order: 2 , 3 , (3 , 4) , 6 , 7
This time there are two values in the middle. They have been put in brackets. The median is found by calculating the mean of these two values: (3 + 4) ÷ 2 = 3.5
So the median is 3.5 The Mode The mode is the value which appears the most often in the data. It is possible to have more than one mode if there is more than one value which appears the most. The data values: 2 , 2 , 3 , 5 , 5 , 7 , 8
The values which appear most often are 2 and 5. They both appear more time than any of the other data values.
So the modes are 2 and 5 The data values: 2 , 3 , 3 , 4 , 6 , 7
This time there is only one value which appears most often - the number 3. It appears more times than any of the other data values.
So the mode is 3 The Range To find the range, you first need to find the lowest and highest values in the data. The range is found by subtracting the lowest value from the highest value. The data values: 2 , 2 , 3 , 5 , 5 , 7 , 8
The lowest value is 2 and the highest value is 8. Subtracting the lowest from the highest gives: 8 - 2 = 6
So the range is 6 The data values: 2 , 3 , 3 , 4 , 6 , 7
The lowest value is 2 and the highest value is 7. Subtracting the lowest from the highest gives: 7 - 2 = 5
So the range is 5
Calculating the Mean, Median, Mode and Range for a table of data
Sometimes we are given the data in a table. The methods for calculating mean, median, mode and range are exactly the same, but we need to think carefully about how we carry them out. In this section we will use one set of data in a table and calculate each measure in turn.
Example A dice was rolled 20 times. On each roll the dice shows a value from 1 to 6. The results have been recorded in the table below:
Value
Frequency
1
3
2
5
3
2
4
4
5
3
6
3
The frequency is the number of times each value occured. For example, the value 1 was rolled 3 times, the value 2 was rolled 5 times and so on...
When we want to think about calculating the measures for this data set, it can be helpful to think about what the numbers would look like if we wrote them out in a list: 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6
We could just calculate the mean, median, mode and range from this list of data, using the methods described in the first part of this section. The problem is that if there were hundreds of values in the table then it would take a long time to write out the list of data and even longer to do the calculations. It would be better if we could work directly from the table to calculate the measures. The method for doing this is shown below. Finding the mean from a table of data
Value
Frequency
1
3
2
5
3
2
4
4
5
3
6
3
We know that if we write the example data in a list it looks like this: 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6
Normally we would add up the data and divide the total by the number of values: The total is 1+1+1 + 2+2+2+2+2 + 3+3 + 4+4+4+4 + 5+5+5 + 6+6+6 = 68 The number of values is 20, so the mean is 68 ÷ 20 = 3.4
We could have found these figures more easily! To get the total, we have added up 3 lots of "1", 5 lots of "2", 2 lots of "3", 4 lots of "4", 3 lot of "5" and 3 lots of "6". This is the same calculation as 3×1 + 5×2 + 2×3 + 4×4 + 3×5 + 3×6 = 68. We have multiplied each value by its frequency and added up the results to get the total of all the values. We can also get the "number of values" more easily by simply adding up all the frequencies: 3 + 5 + 2 + 4 + 3 + 3 = 20
So how do we do this in a table?
Firstly, you need to add an extra column in the table: This is where you multiply each value by its frequency. For example, the value 5 has a frequency of 3, so we multiply 5 by 3 to get 15.
Secondly, you need to calculate two important totals: (1) add up the values in the frequency column to find out the number of data values. In this case there are 20 values. (2) add up the values in the value × frequency column to find out the total of all the data values. In this case the total is 68.
Finally, you need to calculate the mean: To do this, divide the total of all the data values by the number of data values. In this case you need to divide 68 by 20, giving 3.4. Value
Frequency
Value × Frequency
1
3
1 × 3 = 3
2
5
2 × 5 = 10
3
2
3 × 2 = 6
4
4
4 × 4 = 16
5
3
5 × 3 = 15
6
3
6 × 3 = 18
Totals
20
68
This method of calculating the mean for a table of data is exactly the same as the one used with a list of data. We have still added up all the values and divided by the number of values, but this way is a bit more efficient!
Finding the median from a table of data
Value
Frequency
1
3
2
5
3
2
4
4
5
3
6
3
We know that there are 20 data values in our table. If you imagine the 20 values written out, there would be two values in the middle. These would be the 10th and 11th values, and the median would be the mean of these two "middle values".
From the list below we can see that the "middle values" are 3 and 4: 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6 The median would therefore be (3+4)÷2 = 3.5
So how do we do this from a table? Because there are 20 values, we know that we need to find the mean of the 10th and 11th values. To find these values we need to count through the table until we get to them. Look at the table. The value "1" has a frequency of 3, so the first three values in the table are "1"s. The value "2" has a frequency of 5, so the next 5 values are all "2"s. This takes us up to the 8th value. The next 2 values are "3"s, which takes us up to the 10th position in the data, so the 10th value must be a "3". The next 4 values are "4"s, so the 11th value must be a "4".
We can now see that the 10th and 11th values are a "3" and a "4", so the median is 3.5.
Finding the mode and range from a table of data
Value
Frequency
1
3
2
5
3
2
4
4
5
3
6
3
Finding the mode is much easier from a table, because the frequency column tells us how many times each value occured. We can find the value which occured the most often by looking for the value with the highest frequency. In this case we can see that the value with the highest frequency is "2". The mode of this set of data is therefore 2
Finding the range is also easy from a table. To find the highest and lowest data values, you simply look for the highest and lowest values in the values column. In this case the lowest value is "1" and the highest value is "6", and 6 - 1 = 5. The range of this set of data is therefore 5
6