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10 Statistics
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Australia
Year 8

10.07 Making statistical comparisons

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Statistical comparisons

While calculating the  mean, median, mode, and range  can tell us a lot about a data set, these calculations can also be very powerful in comparing and contrasting two different data sets.

When comparing two data sets, the data set with the higher mean is considered to have generally higher scores. The data set with the lower range is considered to have more consistent scores, because the scores are less spread out.

Examples

Example 1

The runs scored by each player in a cricket match are displayed below.

\text{Team A}70073253152120278
\text{Team B}283312605124419157
a

Find the mean runs for Team A to one decimal place.

Worked Solution
Create a strategy

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

Apply the idea
\displaystyle \text{Mean}\displaystyle =\displaystyle \dfrac{70+0+73+25+31+52+1+20+2+7+8}{11}Substitute the values
\displaystyle =\displaystyle 26.3Evaluate
b

Find the mean runs for Team B to one decimal place.

Worked Solution
Apply the idea
\displaystyle \text{Mean}\displaystyle =\displaystyle \dfrac{28+33+12+60+51+24+14+19+1+5+7}{11}Substitute the values
\displaystyle =\displaystyle 22.2Evaluate the division
c

Which team scored more runs?

Worked Solution
Create a strategy

Since each team has the same number of players, choose the team with the higher mean.

Apply the idea

Team A had a higher mean than Team B, so Team A scored more runs.

Example 2

An online shopping website records the user ratings for two similar products in the histograms below.

The image shows a histogram with the data of Product A. Ask your teacher for more information.
The image shows a histogram with the data of Product B. Ask your teacher for more information.
a

Find the median user rating for product A.

Worked Solution
Create a strategy

To find the median using a histogram, first find the total number of scores by adding the heights of each column.

Apply the idea
\displaystyle \text{Total}\displaystyle =\displaystyle 3+7+10+13+14+20+22+30+20+17+16Add the heights
\displaystyle =\displaystyle 172Evaluate

The middle score will be between the 86th and 87th scores. Both of these scores lie in the column for the rating of 3.0.

\text{Median}=3

b

Find the median user rating for product B.

Worked Solution
Apply the idea
\displaystyle \text{Total}\displaystyle =\displaystyle 28+19+21+17+13+10+14+17+17+24+20Add the heights
\displaystyle =\displaystyle 200Evaluate

The middle score will be between the 100th and 101st scores. Both of these scores lie in the column for the rating of 2.5.

\text{Median}=2.5

c

According to the website's rating system, a perfect rating is 5.0. Which product got more perfect ratings?

Worked Solution
Create a strategy

Compare the which product has a higher column for a user rating of 5.0.

Apply the idea

Based on both histograms:

  • Product A got 16 ratings of 5.0.
  • Product B got 20 ratings of 5.0.

So product B got more perfect scores.

Idea summary

We can use the mean, median, and mode to compare data sets to determine which set had higher results on average.

We can use the range to compare data sets to determine which set had more consistent results.

Outcomes

ACMSP206

Explore the practicalities and implications of obtaining data through sampling using a variety of investigative processes

ACMSP293

Explore the variation of means and proportions of random samples drawn from the same population

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