2.3 Comparing distributions and making sensible conclusions
There are advantages and disadvantages of using different measures of spread. See the table below.
| Average | Advantages | Disadvantages |
|---|---|---|
| Mean | Takes all values from the data set into consideration | Can be skewed by very large or very small values in the data set |
| Median | Very large or very small values do not affect it | It can be very slow to calculate for a large data set |
| Mode | The only average which can be used for non-numerical data | There could be more than one mode There may be no mode It might not represent the data set |
When analysing data, it is important to consider which measures of spread will be most suitable.
Activity _unit9.2.3 Activity 10 Calculating measures of spread
Below are the English and Maths test scores for 10 school children.
Find the mean, median, mode and range of the English scores. The Maths measures have already been calculated.
| Maths | 77 | 78 | 76 | 78 | 78 | 76 | 80 | 79 | 78 | 80 |
| English | 67 | 73 | 101 | 68 | 66 | 85 | 69 | 86 | 101 | 64 |
|
Maths | English |
| Mean | 78 | |
| Median | 78 | |
| Mode | 78 | |
| Range | 4 |
Discussion
|
Maths | English |
| Mean | 78 | 78 |
| Median | 78 | 71 |
| Mode | 78 | 101 |
| Range | 4 | 37 |
Activity _unit9.2.4 Activity 11 Reflecting
If you were to compare the scores in the two subjects, English and Maths, which measure of average would you use and why?
Discussion
Range: the range of scores in English (37) is far greater than that in Maths (4).
Mean: the mean score in each subject is 78, which implies that the scores of the students are more-or-less identical in English and Maths. But looking at the actual scores, you can see that this is not the case.
Median: if you compare the medians (71 and 78), you might assume that the students generally scored less in English (which is partly true, but there are also some much higher scores there too).
Mode: if you just state the modal score for each subject (101 and 78), you have no information about the scores of the other students.
So, which is best? It seems that to give maximum information, a combination of the median and the range would be best.
In summary, English has a median score of 71 and a range of 37, and Maths has a median score of 78 and a range of 4.
Making sensible conclusions
It is important to be aware that the way data is sampled and analysed will affect the conclusions we make.
Calculating only one measure of central tendency may result in a very different conclusion being formed than if a different measure was calculated. Generally, using more than one average along with the range will give learners a clearer idea of the data set they are exploring.
Misleading data in the world around us
Newspapers, adverts and political campaigns often use statistics to get their point across. As we saw in the example of the English and Maths scores, one statistical measure on its own will not give us a fair understanding of the data set.
Having data sense is about being aware of how data is collected and how statistics are calculated in order to make sense of figures presented to us and to avoid being misled by campaigns published by advertising companies and political parties. Knowing how to read and understand statistics allows us to make well informed decisions.
Similarly, the way graphs and charts are presented to us can sometimes be misleading and the choice of chart or graph can lead us towards particular conclusions.
We will discuss this in section 4 of this week.