7.07 Comparing data sets

At the start of this chapter we explored the statistical investigation process in which we seek to answer a statistical question. Often we need to make comparisons between different data sets to answer the questions, such as when we need to make a choice between two or more groups (or populations).

For example, if we are looking for the best tree to plant in the school courtyard for shade, we might compare two similar species of tree to determine "Which species of tree grows the fastest?". This is a statistical question and we can collect data by measuring the growth rates of many trees in a nursery.

If it turns out that all of the individual trees of one species grow faster than all of the individuals in the other species then it will be an easy decision. However, that is not always the case–each individual tree grows at a different rate, and there can be overlap between the measurements from both populations.

This is when we need to use statistical methods to analyse and compare data sets.

By comparing the means of central tendency in a data set (that is, the mean, median and mode), as well as measures of spread (range, interquartile range and standard deviation), we can make comparisons between different groups and draw conclusions about our data.

Worked example

The number of minutes spent exercising per day for $10$10 days is recorded for two people who have just signed up for a new gym membership.

Person A: $45,50,50,55,55,60,60,65,65,65$45,50,50,55,55,60,60,65,65,65

Person B: $20,30,45,55,60,60,65,70,70,70$20,30,45,55,60,60,65,70,70,70

(a) Calculate the mean, median, mode, sample standard deviation and range for Person A.

Think: The data set is already in order, which makes finding the median and range much easier. To find the mean we sum the scores and divide it by $10$10 (the number of scores), and then the mode will be the most frequently occurring score. To compute the sample standard deviation, we refer to our calculator.

Do: We see that:

(b) Calculate the mean, median, mode, sample standard deviation and range for Person B.

Think: We will perform the same calculations as before, but using the second data set. Note that this data set is also already ordered.

Do: We see that:

(c) Which person is the most consistent with their exercise, and why?

Think: The person who is most consistent will have scores that are closer together. The two measures of spread that we have found are the range and standard deviation, so we should compare these to see who is most consistent.

Do: Person A has a much smaller range and standard deviation than Person B. In fact, both measures for Person A are less than half those of Person B. We can conclude that Person A is more consistent.

(d) Which person seems to train more overall, and why?

Think: To determine who seems to train more overall, we should consider which measures of central tendency of the two sets are appropriate for comparison

Do: The mode and median for Person B are both larger than for Person A.

However, the mode and median ignores the actual values for the majority of the data set.

If we wanted to see who trained more overall, why not add all the minutes spent at the gym and compare those?

The mean effectively does this, but just divides these values by $10$10 (the number of scores).

So Person A trains more overall.

Reflect: Sometimes two data sets that we want to compare have different sizes. So if we want to compare which data set "performed" typically better, we can't just add up all the scores–the data set with more scores will have an advantage. So we scale each sum by the number of scores in each data set. This gives us a fair comparison of how well each data set performed on average.

There are times that we want to use the mode and median to compare the measures of central tendency. This will be more appropriate when outliers are involved, or if the actual values of the scores are less important.

Practice questions

Question 1

question 2

Question 3

Back-to-back stem plots

We have seen a useful way to compare numerical data sets is parallel box plots as these compare the two sets on the same scale. We can create a similar comparison by creating a back-to-back stem plot. The advantage of a back-to-back stem plot is that the original data is retained and we can calculate the mean, mode and other statistics exactly. A disadvantage is that it is only suitable for small to medium data sets and we can only compare two sets of data at a time.

Two sets of data can be displayed side-by-side using a back-to-back stem plot. In the example below, the pulse rates of $18$18 students were recorded before and after exercise.

Reading a back-to-back stem plot is very similar to reading a regular stem plot.

Referring to the example above:

• The central column displays the stems, with the leaf values on each side.
• The values on the left are the pulse rates of the students before exercise, while the values on the right are their pulse rates after exercise.
• In this example, the fourth row of the plot, $4$4 $3$3 $0$0 $\mid$∣ $8$8 $\mid$∣ $2$2 $2$2 $6$6, displays pulse rates of $80$80, $83$83 and $84$84 before exercise and pulse rates of $82$82, $82$82 and $86$86 after exercise. They are not necessarily the pulse rates of the same students.
• On both sides of the stem column, the leafs are displayed in ascending order with the lowest value closest to the stem.

To create a stem plot, it is usually easier to arrange all of the data values in ascending order, before ordering them in the plot.

Practice questions

Question 4

Question 5