Comparing data sets
In a nutshell
Data can be compared in lots of different ways, most commonly using the mean, median, mode and range. Diagrams can also be used to visually make comparisons and make comments about the way data is spread out.
Using the mean, median, mode and range to compare data
The simplest way to compare data is by using different averages. To make comparisons, use the following steps:
procedure
1.
 Make an observation about an average.

2.
 Explain what it means in terms of comparison.

Example 1
The heights reached by students and teachers as they jump in the air were recorded and averages were taken:
Teachers
Mean: $185cm$
Median: $184cm$
Range: $33cm$  Students
Mean: $132cm$
Median: $122cm$
Range: $43cm$

Compare the distributions of the heights reached by the teachers and the students.
Comparing averages.
1. Make an observation.
"The mean and median height reached by the teachers was higher than that reached by the students."
2. Explain what it means.
"Therefore, the teachers generally jumped higher."
Comparing ranges.
1. Make an observation.
"The range of the heights reached by students was greater than the teachers.'"
2. Explain what it means.
"Therefore, the heights reached by students were more varied or spread out."
Tip: It's important to include the context of the question in your answer when comparing data.
Comparing data using box plots
Box plots are a common way for sets of data to be compared with one another. It is very easy to read off quantities such as the median and interquartile range from them. When describing comparisons using box plots, use the same methods as above.
Example 2
Take a look at the following box plots showing the heights of different plants in Amy and Daniel's gardens.
Compare the distribution of the heights of their plants.
Comparing distribution using averages.
1. Make an observation.
"The median height of Amy's plants was higher than the median height of Daniel's."
2. Explain what it means.
"Therefore, generally, Amy's plants were taller."
Comparing spreads using ranges and interquartile ranges.
1. Make an observation.
"The range and interquartile range of Amy's plants were smaller than the range and IQR of Daniel's.'"
2. Explain what it means.
"Therefore, the spread of Amy's plants is smaller. There is less variation in the heights of her plants."
Note: Box plots DO NOT tell you the frequency. In the example above, there is nothing to suggest that either person has more plants than the other.
Is there evidence to suggest...?
A common question when comparing sets of data or looking at outcomes will ask if there is any evidence to suggest a certain comment. Averages and ranges can be used as evidence in these situations.
Example 3
Amy has been using a new brand of fertiliser to help her plants grow, whilst Daniel has used nothing. Is there any evidence to suggest that the fertiliser has worked?
"Yes there is evidence to support the fertiliser working. The median height of Amy's plants was higher."
Based on the evidence, the fertiliser manufacturer says that its fertiliser increases plant growth. Comment on this claim.
"The fertiliser might have increased plant growth, however the growth of Amy's plants could also have been caused by other factors, such as her garden having more sunlight or because she has taller varieties of plants."