Topic: Describing the Centre and Dispersion of a Variable

ONLY NUMBERS: 3,4 ,6-9, 11

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3 | Describing the Centre and Dispersion of a Variable 91 Practice What You Have Learned Check to see if you can apply the key concepts in this Table 3.7 Frequency Distribution of Students' chapter by answering the following questions. Keep two Monthly Housing Costs (Hypothetical Data) decimal places in any calculations. Approximately how much do you spend on housing costs each month (including utilities)? 1. A local agency has received funding to build new sen- Cumulative iors' residences. To find out what type of housing is Answer Frequency Percentage Percentage most in demand, the agency hires you to survey people $0 20 1.6 1.6 participating in seniors programs at a local commun- $1 to $99 43 3.4 5.0 ity centre. The survey results show that 36 per cent of re- $100 to $199 83 6.6 11.6 spondents want the agency to build independent-living $200 to $299 123 9.7 21.3 apartments, 32 per cent want assisted-living apart- $300 to $399 125 9.9 31.2 ments, 14 per cent want small townhouses, and 18 per $400 to $499 187 14.8 16.0 cent want housing for multi-generational families. $500 to $599 256 20.3 66.3 a. When this survey question becomes a variable, $600 to $699 166 13.2 79.5 what level of measurement will it have? $700 to $799 120 9.5 89.0 b. How can you describe the centre of a variable $800 to $899 97 7.7 96.7 with this level of measurement? C. What is the centre of this variable? $900 to $999 42 3.3 100.0 $1,000 or more 0.0 2. As a volunteer in a local election campaign, you go Total 1.262 100.0 door to door to find out which issues are most im- portant to potential voters. You ask people whether having affordable housing is "Very important," a. What level of measurement does this variable "Somewhat important," "Not very important," or have? "Not at all important" to them. By the end of the b. How can you describe the centre of a variable campaign, you collected information about afford- with this level of measurement? able housing from 86 potential voters: 36 said that C. What is the centre of this variable? it is "Very important," 24 said that it is "Somewhat 4. Using the information in Table 3.7: important," 14 said that it is "Not very important." a. What are two strategies you can use to describe and 12 said that it is "Not at all important." the dispersion of a variable with this level of a. When this information becomes a variable, measurement? what level of measurement will it have? b. Describe the dispersion of this variable using b. How can you describe the centre of a variable each of the two strategies you identified in ques- with this level of measurement? tion 4(a). c. What is the centre of this variable? 5. By hand, draw a box plot that shows the distribution 3. Your student union is concerned about whether local of the variable shown in Table 3.7. housing is affordable for students who are living away 6. Using the information in Table 3.7: from home, especially given rising tuition costs. The a. Find the tenth percentile of the variable. Explain union distributes a questionnaire to a random sample what it shows. of students who are not living at home and not living in b. Find the ninetieth percentile of the variable. Ex- residence, asking them to estimate their monthly hous- plain what it shows. ing costs, including utilities. Table 3.7 shows the results.92 PART I | Describing the Social World 7. Using the information in Table 3.7: b. Find the eightieth percentile of the variable. a. Find the cut-offs that divide the cases into quin- Explain what it shows. tile groups (the twentieth, fortieth, sixtieth, and 12. In the follow-up survey of seniors in described in eightieth percentiles). question 8, you also collect information about how b. Use the cut-offs you found in (a) to create a new many years each person has lived in their current frequency distribution that groups the cases into residence. Their answers are: quintile groups, based on housing costs. Your frequency distribution will be similar to that 2 years 7 years 10 years 5 years shown in Table 3.4. 4 years 5 years 4 years 12 years 8. As a follow-up to the survey of seniors described in 6 years 20 years 2 years 18 years question 1, you collect some additional information 10 years 3 years 5 years 15 years about how high housing costs influence people's 15 years 15 years 17 years 14 years lives. You ask 20 seniors about how often they worry about having enough money to pay for their home a. Find the mode of this variable. and utilities: less than once a month, once a month, a b. Find the median of this variable. few times a month, once a week, a few times a week, 13. Using the answers listed in question 12: or every day. Their answers are: a. Find the range of the variable. less than once a every once a b. Find the interquartile range of the variable. once a month day week month 14. Using the answers listed in question 12: a few less than a. Find the fifteenth percentile of the variable. Ex- once a times times once a month plain what it shows. a month a week month b. Find the sixty-fifth percentile of the variable. less than Explain what it shows. every once a once a once a day week month 15. The CMHC regularly publishes information about month housing and income in Canada. The graph in Figure less than a few once a every 3.6 uses CMHC data to compare the median incomes once a times week day of homeowners and renters across time. month a week a few less than a few a. What type of graph is it? every times once a times What three variables are included in the day a month month a month graph? C. List two claims that you can make based on the a. Find the mode of this variable. information in this graph. b. Find the median of this variable. 9. Using the answers listed in question 8: 16. Access to affordable housing is a nationwide prob- lem. But some provinces and municipalities are a. Find the range of the variable. doing better than others. Use the online interactive b. Find the interquartile range of the variable. map and database at www.rentalhousingindex.ca to 10. By hand, draw a box plot that shows the distribution investigate: of the answers listed in question 8. a. What percentage of renter households in your 11. Using the answers listed in question 8: municipality spend more than 30 per cent of a. Find the fortieth percentile of the variable. their before-tax income on rent and utilities? Explain what it shows