Exercises

1. For the following sets of data, find to one decimal place
   1.  
      1. the mean
      2. the median, and
      3. the mode
   1. 0 – 0 – 0 – 0 – 1 – 0 – 0 – 0 – 0 – 0 – 0 Answer 1a
   2. 2 – 1 – 2 – 3 – 1 – 3 – 0 – 2 – 4 – 2 – 2 Answer 1b
   3. 2.4 – 3.9 – 1.8 – 1.7 – 4.0 – 2.1 – 3.9 – 1.5 – 3.9 – 2.6 Answer 1c
   4. 153.8 – 154.7 – 156.9 – 154.3 – 152.3 – 156.1 – 152.3 Answer 1d
2. For the following sets of data, find
   1.  
      1. the mean
      2. the median
      3. the mode
   Briefly describe the positions of the mean, median and mode and their relation to one another for each data set.
   1.  
      

      Table 1

      x	frequency
      -2	3
      -1	7
      0	8
      1	5
      2	4

      Answer 2a
   2.  

      Table 2

      x	frequency
      6.3	2
      6.4	1
      6.5	6
      6.6	5
      6.7	13
      6.8	4

      Answer 2b
   3.  
      

      Table 3

      x	frequency
      1	15
      2	5
      3	3
      4	1
      5	2

      Answer 2c
3. For each of the following stem and leaf plots, find
   1. 1. the median, and
      2. the modal-class interval
   1.  
      

      Table 4

      Stem	Leaf
      2	2 3 8
      3	1 1 4 2
      4	2 2 3 5 8 9 9
      5	2 4 7 7 8
      6	0 3 2
      7	4

      Stem 4, Leaf 2 represents 42 Answer 3a
   2.  

      Table 5

      Stem	Leaf
      0(0)	2
      0(5)	5 6 8
      1(0)	0
      1(5)	5 5 6 6 7 8 8 9
      2(0)	0 0 0 1 1 2 3 3 3 4 4 4
      2(6)	6 6 7 8 8 9 9
      3(0)	0 4
      3(5)	5 6 7 7 8

      Stem 3, Leaf 5 represents 35 Answer 3b
4. Imagine that the annual population increases over a 10 year period are given in the table below:
   
   

   Table 6. Population increase

   Year	Increase from previous
   1	53,377
   2	52,170
   3	67,000
   4	90,332
   5	72,681
   6	65,226
   7	76,777
   8	83,657
   9	77,753
   10	82,892

   1. Calculate the mean annual population increase over a 10 year period. Answer 4a
   2. Calculate the median annual population increase over a 10 year period. Answer 4b
   3. Do you think the difference between these two measures is significant? Give reasons for your answer, and explain which result gives a better indication of the data's centre. Answer 4c
   4. For what purposes would one use measures such as these? Answer 4d
5. Forty students took a math test marked out of 10 points. Their results were as follows:
   
   9, 10, 7, 8, 9, 6, 5, 9, 4, 7, 1, 7, 2, 7, 8, 5, 4, 3, 10, 7,
   3, 7, 8, 6, 9, 7, 4, 2, 3, 9, 4, 3, 7, 5, 5, 2, 7, 9, 7, 1
   
   
   1. Prepare a frequency table of the scores. Answer 5a
   2. Using the frequency table, calculate the mean, median and mode. Answer 5b
   3. Interpret these results. Answer 5c
6. Imagine that the number of unemployed people is given in the table below
   
   

   Table 7.  Unemployment

   Age group	No. unemployed
   15 to 19	3,688
   20 to 24	4,031
   25 to 34	5,432
   35 to 44	4,360
   45 to 54	3,162
   55 to 64	1,702

   1. Copy the table into your notebook and find the midpoint of each interval. Calculate the average age of an unemployed person using the midpoint. Answer 6a
   2. What is the modal-class interval? Answer 6b
   3. In what age group does the median lie? Answer 6c
   4. Briefly discuss the comparison of these three results. Answer 6d
   5. Why do you think the number of unemployed people decreases after the age group 25 to 34? Answer 6e
   6. How might social welfare organizations use these figures? Answer 6f
7. A random survey of 100 married men gave the following distribution of hours spent per week doing unpaid household work:
   
   

   Table 8.  Hours spent per week doing unpaid household work

   Hours	No. of men
   0 to < 5	1
   5 to < 10	18
   10 to < 15	24
   15 to < 20	25
   20 to < 25	18
   25 to < 30	12
   30 to < 35	1
   35 to < 40	1

   1. Copy the table into your notebook and include columns to find the endpoint (upper value) for each interval. Figure out the cumulative frequency and cumulative percentages and insert them into your table. Answer 7a
   2. Draw the ogive (or distribution curve) with the cumulative frequency on the y axis. Answer 7b
   3. From the curve, find an approximate median value. What does this value indicate? Answer 7c
   4. What is the modal-class interval? Answer 7d
   5. Calculate the mean. What does this value indicate? Answer 7e
   6. Briefly describe the comparison between the mean, median and mode values. Answer 7f
   7. How would you find out whether women spent more hours doing unpaid household work per week than men? Answer 7g
8. The following is a hypothetical table of annual income of people aged 15 years or more:
   
   

   Table 9.  Annual income of people aged 15 years or more

   Income ($)	Persons
   0 to 2,079	114,195
   2,080 to 4,159	44,817
   4,160 to 6,239	45,862
   6,240 to 8,319	139,611
   8,320 to 10,399	114,192
   10,400 to 15,599	148,276
   15,600 to 20,799	123,638
   20,800 to 25,999	121,623
   26,000 to 31,199	103,402
   31,200 to 36,399	73,463
   36,400 to 41,599	59,126
   41,600 to 51,999	68,747
   52,000 to 77,999	56,710

   1. What is the modal-class interval? Answer 8a
   2. Copy the table into your notebook and include columns to find the upper endpoint of each interval. Calculate cumulative frequencies and cumulative percentages. Answer 8b
   3. Draw the ogive (or distribution curve). Answer 8c
   4. From the curve, give an approximate value for the median annual individual income. Answer 8d
   5. Calculate the mean annual income. (Hint: in the above table, the interval 2,080 to 4,159 actually represents 2,080 to < 4,160, so the midpoint is 3,120.) Answer 8e
   6. Briefly compare the mean, median and mode values. Answer 8f
   7. Which measure gives the most accurate picture of the data's centre? Answer 8g
   8. What types of organization would use information such as this? Answer 8h

Class activities

1. Measure the height of each student in your class to the nearest centimetre. Are there any outliers? Use an appropriate method to find the mean, median and mode. Compare all three measures. Which value gives the best measure of central tendency? Why? Which organizations or companies would find such statistics useful?
2. Find out what your grade or school's student population has been for the last 10 years. Are there any outliers? Use an appropriate method to find the mean, median and mode. Compare all three measures. Which value gives the best measure of central tendency? Why? How would your school or school board use such statistics?
3. Find the final scores of your favourite school sport from your school's records. Collect the scores, both wins and losses, for the last 10 years. (If the data are not available, use data for your favourite sporting team.)
   • What was the mean final score, including both wins and losses, for the past 10 years?
   • What was the median final score, including both wins and losses, for the past 10 years?
   • Are any of the mean final scores similar to the corresponding median final score?
   • Given these values, what can be said about the distributions?
   • What are some of the problems you might come across in trying to use statistics to compare school or other sports teams of the past with those of today?
     
4. For ordinal data, can you think of occasions where the mode would be of more use than the median or mean? Discuss as a class.