Analysing data for Edexcel GCSE Maths
This page covers the following topics:
1. Central tendency
2. Spread of data
4. Inter-quartile range
Data can be made easier to interpret be calculating different averages, which are called measures of central tendency. The three main measures of central tendency are the mean, the mode and the median. The mean can be found by calculating the sum of the values of the data and dividing by how many values there are. The median can be found by taking the middle value when all the values are put into order from smallest to largest. When there are an even number of values in the set of data, there will be two middle values, and thus the median can be found by calculating the mean of those two values. The mode of a set of values is the value that appears most often. If the values are separated into classes, the modal class is the most common one.
The range of a set of values is used to describe the spread of the data. The range can be calculated by subtracting the smallest value in the set from the largest value. The range is affected by outliers in the data, which are values which do not follow the pattern of the rest of the data. A more accurate value of the range can be found if it is calculated by removing outliers.
Quartiles are another measure of spread. The lower quartile is the median of the lower half of the data and it is found by taking the (n+1)/4 value, where n is the number of values in the data. The upper quartile is the median of the upper half of the data and it is found by taking the 3(n+1)/4 value. If there are an even number of values, the quartile is found by taking the average of the two middle values.
The interquartile range (IQR) is the difference between the lower and upper quartile. The IQR is a measure of spread which is not affected by outliers, unlike the range of the set of values.
What is the interquartile range of the given set of values?
47, 56, 58, 59, 62, 64, 67
There are 7 values.
For the lower quartile, (7 + 1)/4 = 2, therefore it is the second value.
So the lower quartile is 56.
For the upper quartile, 3(7 + 1)/4 = 6, therefore it is the sixth value.
So the upper quartile is 64.
Therefore, the IQR = 64 − 56 = 8.
Find the median for the following set of values: 52, 68, 59, 61, 70, 55, 72.
We arrange the set of values from smallest to largest: 52, 55, 59, 61, 68, 70, 72. The median is the middle value, therefore the median is 61.
What is the upper quartile of the following set of values: 14, 19, 20, 21, 26, 27, 30, 32.
There 8 values. So, upper quartile = 3(8 + 1)/4 = 6.75. To calculate the upper quartile, the average of the 6th and 7th values must thus be calculated. So, upper quartile = (27 + 30)/2 = 28.5.
The heights of 10 students are recorded as follows: 167, 158, 163, 165, 173, 169, 171, 164, 169, 170. Calculate the mean of their heights. All measurements were taken in centimetres.
Mean = (167 + 158 + 163 + 165 + 173 + 169 + 171 + 164 + 169 + 170)/10 = 166.9 cm.
What is meant by the fact that the upper quartile of a set of values is 45?
This means that the median of the upper half of the set of values is 45.
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