Thursday 22 December 2016
- For the following sets of data, find to one decimal place
-
- the mean
- the median, and
- the mode
- 0 – 0 – 0 – 0 – 1 – 0 – 0 – 0 – 0 – 0 – 0
- 2 – 1 – 2 – 3 – 1 – 3 – 0 – 2 – 4 – 2 – 2
- 2.4 – 3.9 – 1.8 – 1.7 – 4.0 – 2.1 – 3.9 – 1.5 – 3.9 – 2.6
- 153.8 – 154.7 – 156.9 – 154.3 – 152.3 – 156.1 – 152.3
-
- For the following sets of data, find
-
- the mean
- the median
- the mode
-
- the median, and
- the modal-class interval
- 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 - Calculate the mean annual population increase over a 10 year period.
- Calculate the median annual population increase over a 10 year period.
- 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.
- For what purposes would one use measures such as these?
- 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- Prepare a frequency table of the scores.
- Using the frequency table, calculate the mean, median and mode.
- Interpret these results.
- 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 - Copy the table into your notebook and find the midpoint of each interval. Calculate the average age of an unemployed person using the midpoint.
- What is the modal-class interval?
- In what age group does the median lie?
- Briefly discuss the comparison of these three results.
- Why do you think the number of unemployed people decreases after the age group 25 to 34?
- How might social welfare organizations use these figures?
- 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 - 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.
- Draw the ogive (or distribution curve) with the cumulative frequency on the y axis.
- From the curve, find an approximate median value. What does this value indicate?
- What is the modal-class interval?
- Calculate the mean. What does this value indicate?
- Briefly describe the comparison between the mean, median and mode values.
- How would you find out whether women spent more hours doing unpaid household work per week than men?
- 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 - What is the modal-class interval?
- Copy the table into your notebook and include columns to find the upper endpoint of each interval. Calculate cumulative frequencies and cumulative percentages.
- Draw the ogive (or distribution curve).
- From the curve, give an approximate value for the median annual individual income.
- 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.)
- Briefly compare the mean, median and mode values.
- Which measure gives the most accurate picture of the data's centre?
- What types of organization would use information such as this?
Answer:
1.a)
- 0.1
- 0
- 0
1.b)
- 2
- 2
- 2
1.c)
- 2.8
- 2.5
- 3.9
1.d)
- 154.3
- 154.3
- 152.3
2.a)
- 0
- 0
- 0
- The mean, median and mode are equal. This distribution is almost symmetrical.
2.b)
- 6.6
- 6.7
- 6.7
- Distribution is skewed left, so the mean is less than the median. The mode and median are the same. In skewed distributions, the median is the better measure of central tendency.
2.c)
- 1.85
- 1
- 1
- The median and mode are the same. The distribution is skewed right, so the mean is more than the median. In skewed distributions, the median is the better measure of central tendency. In b) and c), the mean has been influenced by a few low and high values.
3.a)
- 48
- 40 to 49
3.b)
- 23
- 20 to 24
4.a) 72,186.5
4.b) 74,729
4.c) The measures are quite close together, given the size of each observation. The median probably gives the best indication of the data's centre, as there is a large diversity of observation values. The median would not be affected by the very large or very small values.
4.d) A government could use these measures when planning for building schools, hospitals and road construction. The government could also use them to help predict revenue intake from taxation.
5.a)
Table 1. Math test results, marked out of 10 points Score (x) Tally Frequency (f) 0 1 ll 2 2 lll 3 3 llll 4 4 llll 4 5 llll 4 6 ll 2 7 lllll lllll 10 8 lll 3 9 lllll l 6 10 ll 2 Total 40
5.c) The median is higher than the mean because most of the observations have high values. The mean is influenced by the lower scores. The mode is equal to the median.
6.a) 36.2 to 34 (Note: interval sizes are not the same. If they were, the 15 to 24 interval would be the modal-class interval.)
6.b) 25 to 34
6.c) All three results lie within the same interval, but distribution is skewed (or slanted) to the right.
6.d) The younger age groups, 15 to 19 and 20 to 24, are filled with students who are still in school or graduates who have not yet been able to get a job. The age groups after 25 to 34 contain a smaller proportion of unemployed people because these people have joined the work force full time and are no longer attending school.
6.e) Social welfare organizations might use these figures to plan employment programs catering to younger people.
7.a)
Table 2. Hours spent per week doing unpaid household work Hours No. of men (x) Endpoint Cumulative frequency Cumulative percentage 0 to < 5 1 5 1 1 5 to < 10 18 10 19 19 10 to < 15 24 15 43 43 15 to < 20 25 20 68 68 20 to < 25 18 25 86 86 25 to < 30 12 30 98 98 30 to < 35 1 35 99 99 35 to < 40 1 40 100 100
7.b)
7.c) The approximate median value is 17 hours. This indicates that the middle of the distribution is 17 hours.
7.d) The modal-class interval is 15 to < 20 hours.
7.e) The mean value is 16.8 hours. This indicates that the average number of hours that a married man spends doing unpaid household work is 16.8 hours.
7.f) The mean and median are very similar, and all measures lie in the modal-class interval. The distribution is almost symmetrical.
7.g) A survey could be conducted and analysed in a similar fashion. Then, the results of both surveys could be compared.
8.a) The modal-class interval is $10,400 to $15,599. (Note: interval sizes are not the same.)
8.b)
| Income ($) | Persons | Endpoint | Cumulative frequency | Cumulative percentage |
|---|---|---|---|---|
| 0 | 0 | 0.0 | ||
| 0 to < 2,080 | 114,195 | 2,080 | 114,195 | 9.4 |
| 2,080 to < 4,160 | 44,817 | 4,160 | 159,012 | 13.1 |
| 4,160 to < 6,240 | 45,862 | 6,240 | 204,874 | 16.9 |
| 6,240 to < 8,320 | 139,611 | 8,320 | 344,485 | 28.4 |
| 8,320 to < 10,400 | 114,192 | 10,400 | 458,677 | 37.8 |
| 10,400 to < 15,600 | 148,276 | 15,600 | 606,953 | 50.0 |
| 15,600 to < 20,800 | 123,638 | 20,800 | 730,591 | 60.2 |
| 20,800 to < 26,000 | 121,623 | 26,000 | 852,214 | 70.2 |
| 26,000 to < 31,200 | 103,402 | 31,200 | 955,616 | 78.7 |
| 31,200 to < 36,400 | 73,463 | 36,400 | 1,029,079 | 84.8 |
| 36,400 to < 41,600 | 59,126 | 41,600 | 1,088,205 | 89.7 |
| 41,600 to < 52,000 | 68,747 | 52,000 | 1,156,952 | 95.3 |
| 52,000 to < 78,000 | 56,710 | 78,000 | 1,213,662 | 100.0 |
8.c)
8.d) The median annual individual income is approximately $15,500.
8.e) The mean annual income is $19,986.
8.f) It is difficult to compare the mode with the mean and median because of the difference between the sizes of the intervals. The mean is higher than the median because it is affected by the higher incomes. This means that the distribution is skewed or slanted to the right.
8.g) The median gives the most accurate picture of the data's centre because it is not influenced by extreme values.
8.h) Some possible answers include the following:
- social welfare organisations interested in the number of low-income earners;
- businesses interested in the number of high-income earners; and
- governments and other service providers interested in data, broken down by such characteristics as age, sex and geographic area, in order to locate services appropriately.
