Topics for Today. Measures of Central Tendency. The Mean. The Median. The Mode. The Weighted Mean. Properties of the Measures

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1 Topics for Today Measures of Central Tendency The Mean The Median The Mode The Weighted Mean Properties of the Measures Stat203 Page 1 of 1

2 Central Tendency, and and a couple of variations What are they good for? Which is best? What do they mean? Do they really get used? Stat203 Page 2 of 2

3 Yes they really get used: Median From the Vancouver Sun (2009) ml?id=7750bb27-49e4-430e-9032-c4daa38d2f5e&k=96959 Key Points: Increased longevity, coupled with steadily declining fertility rates, mean British Columbia's population is aging. In 1976, the median age was just 29; today, it is close to 41; and by 2030, it is projected to reach The emergence of a population skewed toward older generations will have wide-ranging economic and social implications. Stat203 Page 3 of 3

4 Taking the existing population and recent trends in population growth (driven largely by immigration), we can estimate the size and growth of the future labour force. Using status quo assumptions about the proportion of the population that participates in the workforce, the outlook is somewhat troubling. Annual growth of B.C.'s labour force will drop below one per cent within five to seven years, and then fall further to perhaps 0.5 per cent by 2020 as the baby boom retirement wave gathers force. This compares to labour force growth rates averaging two per cent or more in the past. Under this scenario, many more employers would face hiring challenges and the rate of economic growth would be noticeably lower. Stat203 Page 4 of 4

5 What does this have to do with Central Tendency? The authors used the to summarize the of the entire population, and showed that it was increasing.. In 1976, the median age was just ; today, it is close to ; and by 2030, it is projected to reach How does the median help us to understand the authorʼs point? The median is simply the. Half of people are older than this point, and half are younger. Stat203 Page 5 of 5

6 Yes they really get used: Mode The is the value of a variable which is most common. Stat203 Page 6 of 6

7 look a little closer here Why does it appear that there are two modes here? Stat203 Page 7 of 7

8 There is only 1 mode by definition But if 2 modes are close to the max height we say the distribution is. Stat203 Page 8 of 8

9 Yes theyʼre used: The Mean I got a 65/90 on the midterm what was the? The mean is the average. You just up all the individual values and divide by the. 5 people have ages: 20, 24, 25, 21, 25 The average is ( ) 5 = but there are some sticky points. The mean is influenced by individuals with a large or small value of the variable. Stat203 Page 9 of 9

10 if one person was much older than the others.. 5 people have ages: 22, 22, 24, 21, 46 The average is ( ) 5 = The average gets pulled around if one value is much bigger than the others Stat203 Page 10 of 10

11 The median doesnʼt have this problem 5 people have ages: 20, 24, 25, 21, 25 To get the median: sort the data: 21, 22, 22, 24, 25 then pick the middle number: median = 5 people have ages: 22, 22, 24, 21, 46 To get the median: first sort: 21, 22, 22, 24, 46 The middle number (median) is The median doesnʼt get pulled around if one value is much bigger than the others Stat203 Page 11 of 11

12 5 people have ages: 22, 22, 24, 21, 46 To get the mode, find frequencies: 21 (1 person) 22 ( 2 people) 24 ( 1 person) 46 ( 1 person) The most common value (mode) is 22 Stat203 Page 12 of 12

13 Another example of extreme observations influencing the mean: Housing Prices The following is the sale price of 1500 homes sold through the first 9 months of 2011 in the Seattle area. Where do you think the mean is? Stat203 Page 13 of 13

14 The mean is What about the median? Stat203 Page 14 of 14

15 The median is only.. more than than the mean. Stat203 Page 15 of 15

16 Weighted Averages GPA is a measure of the average grade you have received while studying. But courses are worth 3, 4 or 5 credits, so we let an A in a 5 credit course be worth more than an A in a 3 credit course. To do this we use a Stat203 Page 16 of 16

17 Course grades and credits: credits credits credits The weighted average looks at the grades as though a 5 credit course is than a 3 credit course or as if it is based on a sample of 5 compared to a sample of 3 total credits = = 12 Multiply each grade by the number of credits for that course, add them together, then divide by the total number of credits (3.33*4 + 4* *3) 12 = Stat203 Page 17 of 17

18 To get the GPA, we weigh each grade by the number of the course was worth Same idea when combining information from different. We weigh by the number of people in a group and divide by the total number of everyone. Stat203 Page 18 of 18

19 Stat203 Page 19 of 19

20 Do you think they would have used a weighted average to adjust for the different sizes of the community? What if they had not? Stat203 Page 20 of 20

21 Maybe this will be helpful maybe not: Stat203 Page 21 of 21

22 Some properties If the distribution of data is symmetric, with a single mode, the mean, median and mode will all coincide. The Mean: It is the most measure. It is the most used measure. Its value may be distorted by a few observations. Stat203 Page 22 of 22

23 The Median: It is easy to calculate. It is easy to. Its value is not affected by outlying observations. The Mode: It is only rarely used as a measure. It need not be a ' ' value. There may be more than one mode in a set of data. Its value is not unduly affected by outlying observations. Stat203 Page 23 of 23

24 Reiterating (again): Main disadvantage of the mean It is strongly by observations. e.g. If the annual incomes of five students are $8,000, $10,800, $19,720, $12,760, $185,000 then the mean is $47,256 picture but the median is $12,760 picture Stat203 Page 24 of 24

25 Note: Always at the data set Check for and investigate their validity. Are outlying observations really ' ' of the population studied? If there is some reason for their being then we may wish to report them separately. Stat203 Page 25 of 25

26 Outlying Observations are, and they may be the observations in a data set because of information provided during their investigation. Stat203 Page 26 of 26

27 Circle the best measure of Central Tendency for each: Midterm Grade Mean Median Mode House Price Mean Median Mode Income Mean Median Mode M&M Colors Mean Median Mode Hours worked per week (MTF study) Mean Median Mode Stat203 Page 27 of 27

28 Todayʼs Topics Measures of Central Tendency - The Mean: average The Median: the middle - The Mode: the most common The Weighted Mean - when some individuals are more or less important than others Properties of the Measures - The Mean is influenced by outliers, but easy to understand and compare populations - The Median is robust to outliers, and easy to understand but difficult for comparing - The Mode robust to outliers, easy to understand, doesnʼt always convey the message Stat203 Page 28 of 28

29 Reading for next lecture Chapter 4 Stat203 Page 29 of 29