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11.07 Making statistical comparisons

Lesson

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.

 

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 and range for Person A

Mode $=65$=65

Median $=57.5$=57.5

Mean $=57$=57

Range $=20$=20

 

(b) Calculate the mean, median, mode and range for Person B

Mode $=70$=70

Median $=60$=60

Mean $=54.5$=54.5

Range $=50$=50

 

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

Person A

 

(d) Which statistical measure supports your answer to part (c)?

The range. The smaller range for Person A indicates that the number of minutes they exercise each day is more consistent than that of Person B.

The range for Person B is more than double that of Person A, indicating more inconsistency in their exercise routine.

 

(e) Which person seems to train more per day?

There is not enough data to conclude who trains more.

 

(f) Which statistical measure(s) supports your answer to (e)?

The mean is higher for Person A, however, the median is higher for Person B. Both values are very close.

 

Reflect: There are two comparisons we made with the data here. For the first we were concerned with the consistency of the data, so the range was an appropriate value to test.

For the second question we were concerned with the centre of the data. To compare this, we could compare means, medians or modes. However, the modes for both people are also the maximum value, so the mode is not an appropriate value to use.

This leaves the median and the mean. An argument could be made for either of these. However, there are two things to be careful of. The differences of both the means and the medians was $2.5$2.5, which is much smaller than the ranges ($20$20 and $50$50). This implies that there is not much difference between the centres of the data sets, so neither of these values is really conclusive.

The other reason that we said the result was inconclusive was that the mean was higher for A but the median was higher for B. Depending on the exact question, one of these might be more appropriate than the other. In this case, A trained more in the $10$10 days, but because of the outliers B might train more in another $10$10 days. So either of these could be an appropriate answer.

In general, there is no objective way to answer these sorts of questions, and different people might come up with different answers. The important thing is that any answer you choose must be supported by the data.

 

Practice Questions

Question 1

Classes 7M and 9C were surveyed about whether they should receive more maths homework. The results are displayed in the table below.

  Agreed Disagreed
7M $6$6 $24$24
9C $8$8 $24$24
  1. Are these results likely to be representative of the student population?

    Yes

    A

    No

    B

    Yes

    A

    No

    B
  2. What can we conclude from this data?

    Most students agreed that they should do more homework.

    A

    Most students disagreed that they should do more homework.

    B

    The opinion of the students was evenly split.

    C

    Most students agreed that they should do more homework.

    A

    Most students disagreed that they should do more homework.

    B

    The opinion of the students was evenly split.

    C

question 2

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

Team $A$A $70$70 $0$0 $73$73 $25$25 $31$31 $52$52 $1$1 $20$20 $2$2 $7$7 $8$8
Team $B$B $28$28 $33$33 $12$12 $60$60 $51$51 $24$24 $4$4 $19$19 $1$1 $5$5 $7$7
  1. Find the mean runs for team $A$A to one decimal place.

  2. Find the mean runs for team $B$B to one decimal place.

  3. Which team scored more runs?

    Team $A$A

    A

    Team $B$B

    B

    Team $A$A

    A

    Team $B$B

    B

Question 3

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

  1. Find the median user rating for product A.

  2. Find the median user rating for product B.

  3. According to the website's rating system, a perfect rating is $5.0$5.0.

    Which product got more perfect ratings?

    Product A

    A

    Product B

    B

    Product A

    A

    Product B

    B
  4. According to the website's rating system, a positive rating is greater than $3$3.

    Which product got more positive ratings?

    Product A

    A

    Product B

    B

    Product A

    A

    Product B

    B

Outcomes

MA4-20SP

analyses single sets of data using measures of location, and range

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