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3.3 Median Try These You will need Order from least to greatest. i) 2298, 2196, 2279, 2275 2196 , 2275 , 2279 , 2298 29 , 28 , 26 , 0 , 1 , 10 ii) 1, 0, 29, 10, 26, 28 iii) 9.8, 9.05, 9.75, 9.5, 9.25 9.05 , 9.25 , 9.5 , 9.75 , 9.8 • paper for tracing • a millimetre ruler median the middle term in a set of data arranged in order; if there is an even number of numbers, the median is halfway between the middle values Keira is a repair technician. She uses text messages to communicate with clients and co-workers. What is the median number of daily text messages that Keira sent last week? Day Monday Tuesday Wednesday Thursday Friday 101 117 108 95 104 Number of text messages 1 Order the numbers from least to greatest. Circle the middle number. , 95 2 101 , 104 , 108 , 117 What is the median number of daily text messages that Keira sent last week? The median number of daily text messages is Reflecting Does the median change if the numbers are ordered from greatest to least? Hint When determining the median for a data set, use the number of times they are in the data set. 104 . Example 1 A school basketball team scored these numbers of points in their first 8 games: 86, 61, 72, 86, 74, 59, 91, and 80 What is the median number of points the team scored? Solution A. Order the numbers of points from least to greatest. Circle the two middle numbers. 59 , 61 , 72 , 74 , 80 , 86 , 86 , 91 B. What is the median number of points for these games? Median: mean of two middle numbers 5 ( 77 ) 4 2, or 77 The median number of points is 68 Apprenticeship and Workplace 12 03_AW12_Ch03.indd 68 74 1 80 . NEL 02/03/12 11:39 AM Example 2 Liam is a real-estate agent in Winnipeg. Last month, he sold nine homes. The prices are in the chart at the right. Selling Prices of Homes $264 000 $223 000 $272 000 $319 000 $248 000 $288 000 $257 000 $188 000 $784 500 Is the median or the mean a better measure of central tendency for Liam’s typical selling price? Solution A. What is the mean selling price for these homes? Mean price ($): 2 843 500 4 9 5 315 944.44 measure of central tendency a way to summarize the values in a data set with a single value The mean selling price is $315 944.44 . B. Order the selling prices from least to greatest. Circle the median selling price. $ 188 000 , $ 223 000 , $ 248 000 , $ 257 000 , $ 264 000 , $ 272 000 , $ 288 000 , $ 319 000 , $ 784 500 The median selling price is $ 264 000 . C. Which selling price is an outlier? $ 784 500 D. Is the median or the mean a better measure of central tendency for Liam’s typical selling price? Explain. e.g., The median is a better measure of central tendency for Reflecting How would removing the outlier affect the mean and the median? Liam’s typical selling price. Seven of the nine homes that Liam sold cost less than the mean. Practice 1. Jacob uses a spring caliper to measure the width of a metal part on his bicycle. To reduce measurement error, he measures seven times. The measurements, in centimetres, are 7.2, 7.0, 7.1, 7.1, 6.9, 7.3, and 7.2. Order the measurements from least to greatest. What is the median width? 6.9, 7.0, 7.1, 7.1, 7.2, 7.2, 7.3 The median width is 7.1 cm. NEL 03_AW12_Ch03.indd 69 Chapter 3 Statistics 69 02/03/12 11:39 AM 2. Erin wants to buy a new car. She has compared the fuel economy of several vehicles. This list gives the number of litres of fuel that is used by these vehicles for 100 km of city driving: 10.3, 14.1, 11.9, 4.6, 11.6, 10.7, 10.8, 11.1 a) What is the median fuel economy for these cars? For 100 km (L): 4.6, 10.3, 10.7, 10.8, 11.1, 11.6, 11.9, 14.1 Median: (10.8 L 1 11.1 L) 4 2 5 10.95 L The median fuel economy is 10.95 L/100 km. b) One car that Erin likes is a hybrid. It uses electricity for some city driving, so it uses less fuel. How does the median change if this outlier is removed? Outlier: 4.6 L Median without outlier (L): 10.3, 10.7, 10.8, 11.1, 11.6, 11.9, 14.1 The median increases from 10.95 L/100 km to 11.1 L/100 km. c) How does removing the outlier affect the mean? Original mean: 85.1 L 4 8 5 10.6375 L Mean without outlier: 80.5 L 4 7 5 11.5 L The mean increases from about 10.6 L/100 km to 11.5 L/100 km. 3. Hector completed his carpentry apprenticeship. He applied for six jobs in construction. This list is the hourly wages for these jobs: $27.48, $29.30, $28.96, $28.80, $33.85, and $28.80. a) What is the median hourly wage? $27.48, $28.80, $28.80, $28.96, $29.30, $33.85 ($28.80 1 $28.96) 4 2 5 $28.88 The median hourly wage is $28.88. b) What is the mean hourly wage? $177.19 4 6 5 $29.53 c) Suppose that the lowest wage increased from $27.48 to $28.48. How would this affect the median and the mean? e.g., The median would stay the same. The mean would increase. 70 Apprenticeship and Workplace 12 03_AW12_Ch03.indd 70 NEL 02/03/12 11:40 AM 4. Whitney received these donations for a charity run. Donations Received ($) a) What error did Whitney make when she determined the median? Describe and correct her error. 10 5 20 30 Donations($): 5, 10, 20, 30, 40, 50, 100, 200 10 50 50 20 40 40 20 200 20 100 5 ($30 1 $40) 4 2 5 $35 The median donation was $35. Whitney’s error: She left out the repeated values. Correct list of donations ($): 5, 5, 10, 10, 20, 20, 20, 20, 30, 40, 40, 50, 50, 100, 200 The median donation was $20. b) What was the mean donation? $620 4 15 5 $41.33 The mean donation was $41.33. C03-F03-AW12.ai c) Which donations could be considered outliers? e.g., $100, $200 d) Do you think the median or the mean is a better measure of a typical donation in this situation? Explain. e.g., The median is a better measure because four people gave this much and it would not change if you removed the outliers. 5. Environment Canada keeps track of weather data for Canadian cities. What is the median daily maximum temperature in Calgary? Typical Daily Maximum Temperatures in Calgary (8C) Jan Feb Mar 22.8 20.1 4.0 Apr AW12 May Jun 0176519637 11.3 Figure 16.4 20.2 Number Company Jul Aug Sep Oct Nov Dec 22.9 22.5 17.6 C03-F03-AW12.ai 12.1 2.8 21.3 MPS Technical Temperatures (8C): 22.8, 21.3, 20.1, 2.8, 4.0, 11.3, 12.1, 16.4, 17.6, 20.2, 22.5, 22.9 Pass (11.3 8C 1 12.1 8C) 4 2 5 11.7 8C 1st pass Approved The median daily maximum Not temperature is 11.7 8C. Approved 6. Trace your fingers. Then measure and record the widths of your fingers. What is each measure of central tendency? Your median finger width (cm): e.g., 1.2, 1.6, 1.6, 1.8, 2.1 Your mean finger width (cm): e.g., 1.66 Your trimmed mean finger width (cm): e.g., 1.666… NEL 03_AW12_Ch03.indd 71 Chapter 3 Statistics 71 02/03/12 11:40 AM