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CHAPTER 3 3-1 MEAN, MEDIAN, MODE, AND RANGE IWBAT compute the mean, median, mode, and range from a data set VOCABULARY Mean – Average using division Median – Average using the central number Mode – Average using the number that occurs most frequently Range – How far is the data spread out Measures of Central Tendency – The mean, median, and mode. They indicate what is typical of a set of data. MEAN Find the mean of the number of people at a play during 8 showings. 100, 100, 89, 90, 95, 100, 90, 104 1) Add all numbers in data set 100 + 100 + 89 + 90 + 95 + 100 + 90 + 104 768 2) Divide the sum by the number of data in the set 768 ÷ 8 96 The Mean is 96 MEDIAN Find the median of the number of people at a play during 8 showings. 100, 100, 89, 90, 95, 100, 90, 104 1) Write data in order from least 89 90 90 95 100 100 100 104 to greatest 2) Starting from opposite ends, find the number(s) in the middle *If there are two numbers left in the middle: find the mean of the two numbers MODE AND RANGE Find the mode of the number of people at a play during 8 showings. 100, 100, 89, 90, 95, 100, 90, 104 1) Mode: Which number(s) occur the most frequently 100 Mode: 100 2) Range: Subtract lowest number and the greatest number 104 – 89 = 15 Range is 15 EXAMPLES Find the mean, median, mode and range of the following data set. 31, 25, 37, 25, 42 1) Mean 31 + 25 + 37 + 25 + 42 160 160 ÷ 5 32 2) Median 25 25 31 37 42 31 3) Mode 4) Range 25 42 – 25 = 17 INDEPENDENT PRACTICE Find the mean, median, mode, and range of the following data set. 35, 20, 7, 12, 35, 12, 9, 6 3-2 DATA WITH OUTLIERS IWBAT compute the mean, median, mode, and range of data sets with and without outliers to determine their effect on central tendencies VOCABULARY Outlier – A number in a data set that is very different from the rest of the numbers. Data sets can have more than one outlier. IDENTIFY THE OUTLIERS. THEN FIND THE MEAN, MEDIAN, AND MODE WITH AND WITHOUT THE OUTLIERS. ROUND MEAN TO THE NEAREST TENTH. 15, 18, 12, 9, 46 1) Identify the outlier(s) 2) Find mean, median, mode with outlier 46 Mean: 15+18+12+9+46 5 100 5 = 20 Median: 9 12 15 18 46 15 Mode: None 3) Find mean, median, and mode without the outlier Mean: 15+18+12+9 4 54 4 = 13.5 Median: 9 12 15 18 12 + 15 2 27 2 = 13.5 Mode: None *How did the outlier effect the mean? (Increase or decrease) *How did the outlier effect the median? The mean was increased by the outlier The median was increased by the outlier EXAMPLE Identify the outliers. Then find the mean, median, and mode of the data with and without the outlier. Round the mean to the nearest tenth. 25, 33, 28, 14, 35, 60 Outliers 14 and 60 With outliers Mean: 25 + 33 + 28 + 14 + 35 + 60 = 195 195 = 32.5 6 Median: 14 25 28 33 35 60 28 + 33 = 61 Mode: None 61 = 30.5 2 Without outliers Mean: 25 + 33 + 28 + 35 = 121 121 = 30.25 ≈ 30.3 4 Median: 25 28 33 35 28 + 33 = 61 61 = 30.5 2 Mode: None *How did the outliers effect the mean? (Increase or decrease) The mean increased with the outliers *How did the outliers effect the median? The median was not effected by the outliers *How did the outliers effect the mode? The mode was not effected by the outliers 3-3 STEM-AND-LEAF PLOTS IWBAT use stem-and-leaf plots to analyze central tendencies of data with 2 or 3 digit numbers MAKING A STEM-AND-LEAF PLOT Average Olympic Basketball Scores Angola Argentina Australia Brazil China Croatia 56 70 96 92 72 85 Greece Lithuania Puerto Rico South Korea USA Yugoslavi a 80 85 89 81 104 96 1) Make a stem-and-leaf plot of the data 2) Use the stem-and-leaf plot to find the range, median, and mode. EXAMPLE Average June Temperature (°F) of Some Western US Cities Albuquerque 75 Flagstaff 60 Missoula 59 Salt Lake City 67 Bismarck 65 Fort Worth 82 Portland 64 San Francisco 61 Boise 66 Lincoln 73 Rapid City 65 Seattle 60 1) Make a stem-and-leaf plot of the data. 2) Use the stem-and-leaf plot to find the range, median, and mode. USING A STEM-AND-LEAF PLOT Average Yearly Precipitation in Some US Cities (in Inches) Stem Leaf 4 1 1 2 3 5 7 5 1 4 4 1) Find the Range 2) Find the Median 3) Find the Mode EXAMPLE Heights of Buildings Downtown (in feet) Stem Leaf 2 3 4 5 6 7 8 9 10 11 12 5 2 0 2 2 3 5 0 3 5 6 1) Why is the leaf of stem 6 blank? 2) Find the range 3) Find the mode 5 8 8 9 5 5 9 2 3 8 4 7 8 5 1 4 4 7 8 5 9 8 3-4 CHOOSING THE BEST GRAPH IWBAT choose the type of graph that best describes a set of numerical data and relationships. TYPES OF GRAPHS Bar Graph – shows specific number of multiple groups (ex. Number of students in each class) can also use a pictograph Double Bar Graph – compares and contrasts two groups side-by-side (ex. Sports that girls play vs. sports that boys play) Line Graph – shows changes over time or other kinds of interrelationships (Ex. Temperatures over a month, Population over several years) Circle Graph – Describes parts of a whole (Ex. The percentage of students that like specific types of music) EXAMPLES Community Theater Group Year # of Dramas # of Comedies Attendance Cost per Ticket 1996 10 8 6,500 $10.50 1997 15 7 4,000 $12.00 1998 12 9 5,500 $12.50 1999 12 10 8,000 $14.00 1) Which sets of data show totals for each year? Number of dramas, number of comedies, and attendance 2) Which set of data shows a steady increase over time? Cost per ticket 3) Which type of graph would be best for the data on the number of dramas and number of comedies? Why? the specific numbers of two groups Double bar graph, compares 4) Which type of graph would be best for the data on cost per ticket? Why? Line graph, shows change of prices over several years WHICH GRAPH SHOULD YOU USE? 1) You want to show that the music store sells about 4 times as many country music CDs as jazz CDs. Bar Graph or Pictograph 2) You want to show that sales at the music store have been falling steadily for the last three years. Line Graph 3) You want to show that current sales of Elvis’ recordings are greater than sales of the Beatles’ recordings. Bar Graph or Pictograph 4) You want to show what part Oldies and Classic Rock CDs are of the entire collection of music in the store. Circle Graph 3-5 UNDERSTANDING SAMPLING IWBAT understand the use of sampling in describing group tendencies. VOCABULARY Population – The entire group of people or things being considered (Ex. All students at Holy Angels, The entire bag of marbles) Sample – Part of the population (Ex. Girls at Holy Angels, Only the 6th grade class, A handful of marbles from the bag) Samples are used when studying or surveying when the population is too difficult or impractical EXAMPLE #1 There is a barrel of marbles. You want to know how many green marbles are in the barrel. You scoop out a bucket of marbles. 1) What is the population? The whole barrel of marbles. 2) What is the sample? The bucketful of marbles. You want to take two samples to estimate a central tendency. Sample 1: There are 210 marbles in the bucket. Of those marbles, 51 are green. 1) Find the percentage 51 = 210 51 ÷ 210 ≈ 0.24 24% Sample 2: You scoop another bucketful. This time there are 188 marbles, 61 of those are green. 2) Find the percentage 61 188 61 ÷ 188 ≈ 0.32 32% 3) Compare both samples by finding the mean 24% and 32% 24 + 32 56 ≈ 28% ≈ 2 2 4) Round (make the number end with a zero or a 5) About 30% of the barrel is green EXAMPLE #2 In each situation, would it make more sense to study the entire population or a sample? David got a new shipment of dog treats. Each small bag is supposed to contain pieces of real beef jerky. David wants to know the mean number of pieces of beef jerky per bag. the or It would take too long to count jerky in all bags, so it’s best to take a sample from one two bags. EXAMPLE #3 Kristen wants to know the mean height of her cat’s new kittens. A cat can only have so many kittens. It won’t be terribly difficult measuring a few cats that live in the house, so you can study the entire population. EXAMPLE #4 Dog Owners Breed Percent Beagle 35% German Shepherd 30% Retriever 25% Other 10% The table shows the results of a poll in 1999 of 500 dog owners in a large city. Were the statistics drawn from a sample or from the entire population? The population is all the dog owners in the city. Is it possible to poll every single owner? No, it even tells you that only 500 dog owners were polled. 3-6 SAMPLING METHODS IWBAT understand how the method of sampling determines how representative the sample is of the population. VOCABULARY Biased – A sample that does not mirror the population. Random Sampling – A sample where each person or thing has an equal chance of being chosen. Representative – A sample that mirrors the population. Convenience Sampling – Any convenient method to use to choose the sample. Responses to a Survey – A form of sampling where you ask people to take a survey. Tends to be biased since those with strong opinions are more likely to take a survey. EXAMPLE #1 Cassie wants to determine which brand of athletic shoes is most popular with U.S. middle school students. She decides to stand near the entrance of the school and keep a count of what brands of shoes students are wearing when entering the school. 1. What sampling method is Cassie using? How do you know? Convenience Sampling because she’s going to her own school. 2. Is it likely to be representative or biased? Explain. Biased because she is only studying the students at her school and not at any other schools. MORE EXAMPLES Tell whether each sample is likely to be representative or biased. Explain your answers. Identify each as random sampling, convenience sampling, or responses to a survey. 1. A state legislator mails questionnaires about literacy to all the eligible voters in her district. Only 12% of the people return the questionnaire. Biased because people who are illiterate won’t be able to read the questionnaire. This is responses to a survey. 2. A school principal questions every student whose locker number ends in a 9 to gather student input about plans for the new library. Representative because students of different grade levels and interests would be studied. This is random sampling. 3-7 MAKE A GRAPH IWBAT make graphs to illustrate data and solve problems. 3-8 ANALYZING STATISTICAL RESULTS IWBAT evaluate the effect of questions on statistical results. 3-9 REPRESENTING A POINT OF VIEW IWBAT analyze data displays and evaluate the influence of the display on the conclusions.