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Interpreting Statistics: A Case of Muddying the Waters
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  • Assessment Task: Unhappy Campers (15 minutes)
    • Ask students to do this task in the next lesson, or for homework.

      • Give each student a copy of the assessment task, Unhappy Campers.
      • Explain that this task used very similar math to the lessons on river pollution, but in a new context.
        • What is a wind turbine?
        • What are they used for?
        • What do decibels measure?
        • How loud is 50 decibels? 30 decibels? [0 decibels is the threshold for human hearing. A whisper in a quiet library is about 25 decibels. Normal conversation at about five feet is around 60 decibels.]
      • Ask students to work on their own on the post-assessment task, bearing in mind what they have learned during the previous lesson.
        • I want you to work on this task, using those same ideas about a fair mathematical critique.
        • Remember not to believe all the arguments someone gives using statistics.
      • After the assessment, you may find it useful to ask students to compare their first and second assessment tasks, so they can see the progress they have made.
  • Solutions: Muddying the Waters
    • Question 1.

      Interpreting the scatter chart:

      • The water was tested on a monthly basis for 10 months, and at the same time, the number of visitors to the center was recorded.
      • There is a negative correlation.
      • The number of visitors per month varies from 122 to 130. The range is 8 and the mean is 126.
      • Chemical concentration varies from 1 to 14 mg per m3. The range is 13 mg per m3 and the mean is 7.4 mg per m3.

      Interpreting the pie chart:

      • Eighteen people are involved in the survey. 13 replied yes, 1 replied no, and 4 were unsure.

      Question 2.

      The Riverside Manager's argument is misleading in various ways:

      • The scatter plot has a misleading scale. It gives the impression that, correlated with a rise in pollution, there has been a massive drop in visitor numbers. In fact, there is a fall of only 8. Overall the decrease is 6 percent.
      • There is a negative correlation on the scatter chart. This may not be causal, as there are many other reasons why the visitor numbers fell, such as change in season. If the dam was reducing the amount of water in the river, this might have made it less attractive to visitors. The survey was over 10 months, not a year.
      • The pie chart is based on a survey that uses a biased question: people may not have noticed a smell until they were asked about it.
      • The sample size for the pie chart is small. The results of the survey are unlikely to be a true representation of all the visitors to the center. Providing the number of people as well as the percentages in each response category would be helpful.
  • Solutions: Case Notes
    • The concentration of the chemical in the river has risen above the legal limit.

      The bar chart is appropriate, and is clearly shows that three distinct tests have been carried out. The concentration of the chemical in the river is now above the legal limit.

      The levels were within the limit in the previous two years. The factory is discharging the same amount of chemical, but the flow rate of the water has reduced, meaning that the concentration is now above the legal limit

      Students may have calculated the concentration of the chemical in the water:

      Last year:

      60 = 3mg/m3

      This year:

      60 = 15mg/m3

      Arguing that the mean concentration is within the legal limit is a misuse of statistics, because the low measurements in the first two years disguise the much higher figure in the last year:

      1 + 3 + 14 = 6mg/m3

      A more sympathetic judge might argue that there are not enough test sites to prove that the pollution was not caused by another source: it might have been useful to test the water upstream of the factory, to find out whether that water was polluted to start with.

      There has been an increase in the number of diseased fish due to the rise in chemical pollution.

      The survey is misleading because the sample sizes are different. Arguing that there are now 10 times more diseased fish is incorrect because it ignores the sample sizes. If students have calculated proportions or percentages, they will get a better sense of the data than if they rely on numbers:

      Two years ago:

      6 × 100 = 2%

      This year:

      64 × 100 = 4%

      Using this to argue that the number of diseased fish has doubled is still a misleading use of statistics: finding slightly more or fewer diseased fish in either survey (due to weather, the way the fish were caught, or just "the luck of the draw") would make a big difference to the percentages.

      The judge could argue that the survey is poor evidence because the sample sizes are too small to detect a difference in such a small percentage of diseased fish. Or the judge might argue that there are not enough survey sites to show whether being downstream of the factory makes a difference. The reduced flow rate of the river might have affected the health of the fish regardless of the pollution. Why did the second survey look at so many more fish than the first: were the fish harder to find the first time?

      The number of invertebrates has not changed.

      There has been hardly any change in the mean number of invertebrates. Two years ago, the mean across four sites was 21, and now it is 19.

      However, two years ago the range was 4. Now the range is 20. This is a big increase.

      The sites most likely to have been affected by the pollution are A and B, downstream from the factory. Two years ago, the mean number of invertebrates at these sites was 21; now it is 12. This is quite a large decrease. In contrast, the mean at sites C and D has increased.

      Arguing that the mean number of invertebrates has hardly changed is a misuse of statistics: taking the mean of all four sites (including two which would not have been affected by pollution from the factory) hides the possibly significant reduction at the polluted sites.

      The number of birds has increased.

      Using a line graph to represent this data is inappropriate because it gives the impression that the birds were continually monitored. A bar chart with two bars would be more appropriate.

      The scale on the line graph is misleading because it gives the impression that there has been a dramatic increase in the number of birds. The data show that there has only been an increase of 6 birds (about 7 percent). This is insignificant, especially without more details of how the birds were counted or at what time of year.

      Arguing that the chart shows that the number of birds has increased dramatically in the last two years is a misleading use of statistics.

  • Solutions: Unhappy Campers
    • Question 1.

      Interpreting the scatter chart:

      • There are fourteen data points on the scatter chart; the survey took place over a two-week period. There is a negative correlation.
      • The number of visitors ranges from 70 to 78, a range of 8 with a mean average of 75. The noise level ranges from 10 to 60 decibels, with a range of 50 decibels and a mean average of 35 decibels1.

      Interpreting the pie chart:

      • The number of campers surveyed showed 50.
      • 80% of the sample responded yes, 16% were unsure, and 4% said no.
      • The numbers of respondents are 40, 8, and 2 respectively.

      Question 2.

      The camp manager's argument is biased in several ways.

      Her choice of math introduces bias:

      • The scatter plot has a misleading scale. The scale on the "number of campers" axis starts at 40 rather than 0. It gives the impression that, correlated with the rise in noise level, there has been a large drop in visitor numbers. The number of campers only varies by eight across the fourteen-day period, decreasing by about 10% between the quietest and noisiest day.
      • The survey statement and question is biased. Stating that the noise is "loud" and assuming that the respondent can hear the noise pushes the respondent towards a positive response; the use of "spoiling enjoyment" in the question also introduces potential bias.
      • The pie chart is based on a relatively small sample (50 campers). The survey took place on one day. On only 2 days on the scatter chart were there 50 campers. Both days were particularly noisy. Surveying only on a noisy day produces potential bias in the survey responses. It would have been helpful to show the number of respondents, not just percentages, on the pie chart, to aid interpretation of the results.

      Her interpretations of her data and statistics are incorrect.

      • Peggy claims that the noise from the turbines has caused a drop in camper numbers. The correlation between the noise level in decibels and the number of campers does not show there is a causal relationship between the two variables. There may be other explanations of why the number of campers and the noise level correlate. For example, the turbine noise increases with the wind level, so you would expect fewer people to want to camp at noisy times, because it is windier then.

      Although most of the people surveyed did state that the wind turbines spoiled their enjoyment, the questionnaire was biased, the sample was small, and the survey took place on a noisy day. As the results of the survey are dubious, there is no evidence to support Peggy's interpretation that most people coming to the camp would agree with the results of the survey. She can't generalize from a small, biased sample, and she can't rely on responses to a biased question.

      1 From the US Environmental Protection Website:

      The document identifies a 24-hour exposure level of 70 decibels as the level of environmental noise which will prevent any measurable hearing loss over a lifetime. Likewise, levels of 55 decibels outdoors and 45 decibels indoors are identified as preventing activity interference and annoyance. These levels of noise are considered those which will permit spoken conversation and other activities such as sleeping, working and recreation, which are part of the daily human condition.

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