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Statistics: misuse of statistics (Grade 10) [NCS]

Module by: Free High School Science Texts Project. E-mail the author

Misuse of Statistics

In many cases groups can gain an advantage by misleading people with the misuse of statistics.

Common techniques used include:

  • Three dimensional graphs.
  • Axes that do not start at zero.
  • Axes without scales.
  • Graphic images that convey a negative or positive mood.
  • Assumption that a correlation shows a necessary causality.
  • Using statistics that are not truly representative of the entire population.
  • Using misconceptions of mathematical concepts

For example, the following pairs of graphs show identical information but look very different. Explain why.

Figure 1
Figure 1 (MG10C16_010.png)

Exercises - Misuse of Statistics

  1. A company has tried to give a visual representation of the increase in their earnings from one year to the next. Does the graph below convince you? Critically analyse the graph.
    Figure 2
    Figure 2 (MG10C16_011.png)
    Click here for the solution
  2. In a study conducted on a busy highway, data was collected about drivers breaking the speed limit and the colour of the car they were driving. The data were collected during a 20 minute time interval during the middle of the day, and are presented in a table and pie chart below.
    • Conclusions made by a novice based on the data are summarised as follows:
    • “People driving white cars are more likely to break the speed limit.”
    • “Drivers in blue and red cars are more likely to stick to the speed limit.”
    • Do you agree with these conclusions? Explain.
    Click here for the solution
  3. A record label produces a graphic, showing their advantage in sales over their competitors. Identify at least three devices they have used to influence and mislead the readers impression.
    Figure 3
    Figure 3 (MG10C16_013.png)
    Click here for the solution
  4. In an effort to discredit their competition, a tour bus company prints the graph shown below. Their claim is that the competitor is losing business. Can you think of a better explanation?
    Figure 4
    Figure 4 (MG10C16_014.png)
    Click here for the solution
  5. To test a theory, 8 different offices were monitored for noise levels and productivity of the employees in the office. The results are graphed below.
    Figure 5
    Figure 5 (MG10C16_015.png)
    The following statement was then made: “If an office environment is noisy, this leads to poor productivity.” Explain the flaws in this thinking.
    Click here for the solution

Summary of Definitions

  • mean: The mean of a data set, xx, denoted by x¯x¯, is the average of the data values, and is calculated as:
    x¯=sum of values number of values x¯=sum of values number of values
    (1)
  • median: The median is the centre data value in a data set that has been ordered from lowest to highest
  • mode: The mode is the data value that occurs most often in a data set.

The following presentation summarises what you have learnt in this chapter. Ignore the chapter number and any exercise numbers in the presentation.

Figure 6

Summary

  • Data types
  • Collecting data
  • Samples and populations
  • Grouping data Tally Frequency bins
  • Graphing data Bar and compound bar graphs Histograms and frequency polygons Pie charts Line and broken line graphs
  • Summarising data
  • Central tendency Mean Median Mode Dispersion Range Quartiles Inter-quartile range Percentiles
  • Misuse of stats

Exercises

  1. Calculate the mean, median, and mode of Data Set 3.
    Click here for the solution
  2. The tallest 7 trees in a park have heights in metres of 41, 60, 47, 42, 44, 42, and 47. Find the median of their heights.
    Click here for the solution
  3. The students in Bjorn's class have the following ages: 5, 9, 1, 3, 4, 6, 6, 6, 7, 3. Find the mode of their ages.
    Click here for the solution
  4. The masses (in kg, correct to the nearest 0,1 kg) of thirty people were measured as follows:
    Table 1
    45,157,967,957,450,761,163,967,569,771,7
    68,063,258,756,978,559,754,466,451,647,7
    70,954,859,160,360,152,674,972,149,549,8
    1. Copy the frequency table below, and complete it.
      Table 2
      Mass (in kg)TallyNumber of people
      45,0≤m<50,045,0≤m<50,0  
      50,0≤m<55,050,0≤m<55,0  
      55,0≤m<60,055,0≤m<60,0  
      60,0≤m<65,060,0≤m<65,0  
      65,0≤m<70,065,0≤m<70,0  
      70,0≤m<75,070,0≤m<75,0  
      75,0≤m<80,075,0≤m<80,0  
    2. Draw a frequency polygon for this information.
    3. What can you conclude from looking at the graph?
    Click here for the solution
  5. An engineering company has designed two different types of engines for motorbikes. The two different motorbikes are tested for the time it takes (in seconds) for them to accelerate from 0 km/h to 60 km/h.
    Table 3
     Test 1Test 2Test 3Test 4Test 5Test 6Test 7Test 8Test 9Test 10Average
    Bike 11.551.000.920.801.490.711.060.680.871.09 
    Bike 20.91.01.11.01.00.90.91.00.91.1 
    1. What measure of central tendency should be used for this information?
    2. Calculate the average you chose in the previous question for each motorbike.
    3. Which motorbike would you choose based on this information? Take note of accuracy of the numbers from each set of tests.
    Click here for the solution
  6. The heights of 40 learners are given below.
    Table 4
    154140145159150132149150138152
    141132169173139161163156157171
    168166151152132142170162146152
    142150161138170131145146147160
    1. Set up a frequency table using 6 intervals.
    2. Calculate the approximate mean.
    3. Determine the mode.
    4. How many learners are taller than your approximate average in (b)?
    Click here for the solution
  7. In a traffic survey, a random sample of 50 motorists were asked the distance they drove to work daily. This information is shown in the table below.
    Table 5
    Distance in km1-56-1011-1516-2021-2526-3031-3536-4041-45
    Frequency4591078322
    1. Find the approximate mean.
    2. What percentage of samples drove
      1. less than 16 km?
      2. more than 30 km?
      3. between 16 km and 30 km daily?
    Click here for the solution
  8. A company wanted to evaluate the training programme in its factory. They gave the same task to trained and untrained employees and timed each one in seconds.
    Table 6
    Trained121137131135130
     128130126132127
     129120118125134
    Untrained135142126148145
     156152153149145
     144134139140142
    1. Find the medians and quartiles for both sets of data.
    2. Find the Interquartile Range for both sets of data.
    3. Comment on the results.
    Click here for the solution
  9. A small firm employs nine people. The annual salaries of the employers are:
    Table 7
    R600 000R250 000R200 000
    R120 000R100 000R100 000
    R100 000R90 000R80 000
    1. Find the mean of these salaries.
    2. Find the mode.
    3. Find the median.
    4. Of these three figures, which would you use for negotiating salary increases if you were a trade union official? Why?
    Click here for the solution
  10. The marks for a particular class test are listed here:
    Table 8
    67589167588271516084
    31679664787187788938
    6962607360877149  

    Complete the frequency table using the given class intervals.

    Table 9
    ClassTallyFrequencyMid-pointFreq ×× Midpt
    30-39 34,5  
    40-49 44,5  
    50-59    
    60-69    
    70-79    
    80-89    
    90-99    
      Sum = Sum =

    Click here for the solution

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