12.2 Bias, error and misuse  (Page 2/2)

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

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.

   

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.
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.

   

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?

   

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.

   

   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

End of chapter summary

• Data types can be divided into primary and secondary data. Primary data may be further divided into qualitative and quantitative data.
• We use the following as measures of central tendency:
  • Mean The mean of a data set, $x$ , denoted by $\overline{x}$ , is the average of the data values, and is calculated as:
    $$\overline{x}=\frac{\mathrm{sum\; of\; values}}{\mathrm{number\; of\; values}}$$
  • 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 are measures of dispersion:
  • Range The range of a data set is the difference between the lowest value and the highest value in the set.
  • Quartiles Quartiles are the three data values that divide an ordered data set into four groups containing equal numbers of data values. The median is the second quartile.
  • Percentiles Percentiles are the 99 data values that divide a data set into 100 groups.
  • Inter quartile range The inter quartile range is a measure which provides information about the spread of a data set, and is calculated by subtracting the first quartile from the third quartile, giving the range of the middle half of the data set, trimming off the lowest and highest quarters, i.e. $Q_3-Q_1$ . Half of this value is the semi-interquartile range.
• The five number summary is a way to summarise data. A box and whisker plot is a graphical representation of the five number summary.
• Random errors are found in all sets of data and arise from estimating data values. Bias or systematic error occurs when you consistently under or over estimate data values.
• You must always consider the data and the statistics that summarise the data

Exercises

1. Calculate the mean, median, and mode of Data Set 3.
   
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.
   
3. The students in Bjorn's class have the following ages: 5, 6, 7, 5, 4, 6, 6, 6, 7, 4. Find the mode of their ages.
   
4. 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.

   	Test 1	Test 2	Test 3	Test 4	Test 5	Test 6	Test 7	Test 8	Test 9	Test 10	Average
   Bike 1	1.55	1.00	0.92	0.80	1.49	0.71	1.06	0.68	0.87	1.09
   Bike 2	0.9	1.0	1.1	1.0	1.0	0.9	0.9	1.0	0.9	1.1

   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.
5. The heights of 40 learners are given below.

   154	140	145	159	150	132	149	150	138	152
   141	132	169	173	139	161	163	156	157	171
   168	166	151	152	132	142	170	162	146	152
   142	150	161	138	170	131	145	146	147	160

   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)?
6. 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.

   Distance in km	1-5	6-10	11-15	16-20	21-25	26-30	31-35	36-40	41-45
   Frequency	4	5	9	10	7	8	3	2	2

   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?
7. 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.

   Trained	121	137	131	135	130
   	128	130	126	132	127
   	129	120	118	125	134
   Untrained	135	142	126	148	145
   	156	152	153	149	145
   	144	134	139	140	142

   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.
8. A small firm employs nine people. The annual salaries of the employers are:

   R600 000	R250 000	R200 000
   R120 000	R100 000	R100 000
   R100 000	R90 000	R80 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?
9. The marks for a particular class test are listed here:

   67	58	91	67	58	82	71	51	60	84
   31	67	96	64	78	71	87	78	89	38
   69	62	60	73	60	87	71	49

   Complete the frequency table using the given class intervals.

   Class	Tally	Frequency	Mid-point	Freq $\times$ Midpt
   30-39		34,5
   40-49		44,5
   50-59
   60-69
   70-79
   80-89
   90-99
   		Sum =		Sum =