10.05 Comparing data sets

A back-to-back stem-and-leaf plot is very similar to a regular stem-and-leaf plot, in that the "stem" is used to group the scores and each "leaf" indicates the individual scores within each group.

This allows us to compare the shape and location of the two distributions side by side.

Examples

Consider the following histograms that show the height of students in two basketball teams. We know that one graph represents a team made up of Year 12 students and the other represents a Year 8 team. Which one corresponds to the Year 12 team?

Team A:

Team B:

We can still compare these distributions, even though there is clearly a different number of students in the teams, because we are only interested in the shape and location of the data.

In our comparison we want to mention the most significant differences, and also describe relevant characteristics that are the same, or similar.

In this case we can observe these important similarities and differences:

• In this case we can observe these important similarities and differences:

• The modal class for Team A of 170-175 cm is much higher than the modal class 150-155 cm for Team B.

• If we ignore the outlier values in the 195-200 class for Team A, then the range of both distributions is similar, at 35 cm for Team A and 30 cm for Team B.

• Student heights have greater spread overall for Team A.

• The heights for Team A appear to be concentrated around the modal class so we can say that the a clustered at 170-180 cm.

Based on these observations, we could confidently say that Team A is the team of Year 12 students. In this case, the decision is clear because of the difference in the height for the modal class, which we would expect to be significantly higher for the older students.

It is important to be able to compare data sets because it helps us make conclusions or judgements about the data. For example, suppose Jim scores \dfrac{5}{10} in a geography test and \dfrac{6}{10} in a history test. Based on those marks alone, it makes sense to say that he did better in history.

But what if everyone else in his geography class scored \dfrac{4}{10}, while everyone else in his history class scored \dfrac{8}{10}? Now we know that Jim had the highest score in the class in geography, and the lowest score in the class in history. With this extra information, it makes more sense to say that he did better in geography.

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

Examples

Example 2

A science class with 20 students, was given two different 10 question True/False tests, one about dinosaurs and one about nanotechnology. The results for each topic test are shown below:

a

Which topic did the class know more about?

Worked Solution

Create a strategy

Compare the scores for each histogram.

Apply the idea

The scores for the dinosaur topic test range from 4 to 7, while the scores for the nanotechnology topic test range from 7 to 10. So, the class knows more about nanotechnology topic.

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b

Which statistical piece of evidence supports your answer?

A

The positive skew of the graph

B

The mean

C

The range

Worked Solution

Create a strategy

Compare the shape and range of each graph.

Apply the idea

The shape of the nanotechnology topic has a negative skew, so option A is not correct. Both topics have the same value of the range, which is 3, so option C is not correct.

The correct answer is option B.

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c

Which statistic is the same for each topic?

A

The mode

B

The range

Worked Solution

Create a strategy

Evaluate the mode and range for each topic.

Apply the idea

The mode is the score with the highest frequency. The mode for the dinosaur topic test is 5, and the mode for the nanotechnology topic test is 9.

The range for the dinosaur topic test is 7-4=3, and the range for the nanotechnology topic test is 10-7=3.

The range is the same for both topics. So the correct answer is option B.

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d

Calculate the mean for the dinosaur topic test. Give your answer correct to one decimal place.

Worked Solution

Create a strategy

Use the formula: \text{Mean}=\dfrac{\text{Sum of scores}}{\text{Number of scores}}

Apply the idea

\displaystyle \text{Sum}	\displaystyle =	\displaystyle 5\times 4+9\times 5+4\times 6+2\times 7	Multiply each score by its frequency
	\displaystyle =	\displaystyle 103	Evaluate
\displaystyle \text{Mean}	\displaystyle =	\displaystyle \dfrac{103}{20}	Use the formula
	\displaystyle \approx	\displaystyle 5.2	Evaluate the division

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e

Calculate the mean for the nanotechnology topic test. Give your answer correct to one decimal place.

Worked Solution

Create a strategy

Use the formula: \text{Mean}=\dfrac{\text{Sum of scores}}{\text{Number of scores}}

Apply the idea

\displaystyle \text{Sum}	\displaystyle =	\displaystyle 2\times 7+4\times 8+8\times 9+6\times 10	Multiply each score by its frequency
	\displaystyle =	\displaystyle 178	Evaluate
\displaystyle \text{Mean}	\displaystyle =	\displaystyle \dfrac{178}{20}	Use the formula
	\displaystyle \approx	\displaystyle 8.9	Evaluate the division

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