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Math 1 Lab Day 4 Statistics

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Math 1 Lab Day 4 Statistics Day 11.3 - Range & Measures of Central Tendency One of the topics to be covered in statistics is the analysis of the data. We will do that by looking at the : RANGE, MEAN, MEDIAN, AND MODE. RANGE The RANGE of a set of data is the difference between the greatest and the least values. DATA 6, 8, 3, 2, 1, 5, 9, 4 The range is MEAN The MEAN , or average, denoted by 푥̅, of a set of numbers is found by adding the numbers and then dividing that total by the number of data items in the set. Example 1: DATA 80, 92, 85, 91, 95, 88 The mean is Example 2: Mary got 80, 78, 85, and 82 on four math tests. What score must she get on the fifth test to get an 80 average? Example 3: The average grade for 26 students in Mr. Feheley’s math lab is 82. The average For the 24 students in Mr. Bill’s math lab is 86. What is the mean grade for all the students in both classes? Mr. Feheley : Mr. Bill: 퐹푒ℎ푒푙푒푦 푠푢푚+퐵푙푙 푠푢푚 Combined mean : 푡표푡푎푙 푠푡푢푑푒푛푡푠 Example 4: Find the mean of the data in this table: DATA 3.5 3.4 3.7 3.8 4.0 FREQUENCY 6 3 2 3 1 Example 5: Frank’s scores on his first 4 tests are 65, 72, 83, and 91. What does Frank need to earn on the fifth test to earn an average of 80? MEDIAN The MEDIAN is the middle value when a set of numbers is arranged in order from least to greatest or greatest to least. if there is an odd number of items the median is the middle number Example: 2, 5, 7, 19, 20 The median is: if there is an even number of items, the median is the average of the two middle numbers Example: 3, 4, 6, 7, 8, 9 The median is: **** it is possible that the median could be a number not in the set ( as in above example) **** the median means that 50% scored below and 50% scored above this number Example 1: Find the median of (3, 6, 7, 2, 4, 8, 3 ) Example 2: Find the median of (3, 7, 8, 1, 3, 8, 4, 6) Example 3: INTERVAL FREQUENCY CUMULATIVE Determine the median interval based FREQUENCY on the table to the left. 40 - 49 2 2 50 - 59 2 4 60 - 69 10 14 70 - 79 15 29 80 - 89 11 40 90 - 99 8 48 MODE The MODE of a data set is any data that has the greatest frequency. Example 1: { 3, 6, 7, 2, 4, 8, 3 } The mode is: Example 2: In the preceding table the mode interval is: **** If each data value has a frequency of 1 or all have the same frequency you can say that the set has no mode. **** Data can have 2 or 3 or more modes if the data values have the same high frequency. Two modes are called bimodal sets ************************************************************************************ SOME IMPORTANT FACTS The mean, median and mode are called measures of central tendencies because they are giving us an idea of where the "center" of the data lies. The mode is useful when dealing with data that is not numerical, such as the color of eyes, type of cars, etc. The mean is influenced by very large or very small numbers while the median is not. A "good" or "fair" test is one where the median and the mean are close in value. A value that is widely separated from the rest of the data in a data set is called and outlier. This value will influence the mean more than the median. It should be taken into account when you analyze the measures of central tendencies. Before you determine the range, mean, median & mode, put the data in numerical order. ************************************************************************************ Ex. Mr. Bill had the following cell phone bills over the course of a year: 45, 45, 45, 45, 45, 120, 90, 45, 45, 45, 45, 45. His cell phone company called and offered him a plan for $50 a month to save him money based on his average cell phone bill. He refused, based on the median and mode of his bills. What was the mean, median, and mode for his bills. Should he have accepted the offer? Why or why not? 2) Given the scores: 3, 3, 4, 4, 4, 5, 5. Which of the following statements is true? 1) mean > median 2) mean > mode 3) median < mode 4) mean = median 3) The prices of seven race cars sold last week are listed in the table below. What is the mean value of these race cars, in dollars? What is the median value of these race cars, in dollars? State which of these measures of central tendency best represents the value of the seven race cars. Justify your answer. Day 11.3 Homework 1. Which value of x will make the mean of the data equal to 6? _______ { 3, 3, 4, 5, 6, 7, 8, x } a) 8 b) 12 c) 16 d) 48 2. If the average ( mean) of 4, -8, 6 and x is 3 then x is equal to _______ a) -6 b) 1 c) 4 d) 10 3. Jane's test grades were 90, 75, 90, 83, 87. What is the median. _______ 4. Which data has 54 as its median. _______ a) 10, 30, 40, 45, 50, 58, 60, 65, 75, 90 b) 25, 25, 30, 35, 50, 56, 65, 70, 75, 80 c) 30, 40, 45, 47, 54, 55, 60, 65, 75, 85 d) 10, 20, 30, 40, 50, 54, 60, 70, 80, 90 5. Which statement is ALWAYS true ? _______ a) The median is a number in the data set. b) The median is also the maximum value. c) The median and the mean are the same. d) 50% of the data is at or below the median. 6. The accompanying graph shows the high temperatures in Elmira, New York, for a 5-day period in January. Which statement describes the data? (1) median = mode (3) mean < mode (2) median = mean (4) mean = mode 7. INTERVAL FREQUENCY Which interval contains the ________ 14 - 16 2 Median? 17 - 19 6 20 - 22 9 23 - 25 8 8. VOTER SURVEYS This bar graph shows the number of 700 600 people in favor of a proposition in 4 500 400 different surveys. What is the mean 300 200 number of favorable responses in the 4 100 NUMBER OF VOTERS OF NUMBER 0 surveys? #1 #2 #3 #4 SURVEY NUMBER 9. Alex earned scores of 60, 74, 82, 87, 87, and 94 on his first six algebra tests. What is the relationship between the measures of central tendency of these scores? (1) median < mode < mean (3) mode < median < mean (2) mean < mode < median (4) mean < median < mode 10. Jerry's quiz grade are 91, 83, 92, 85, 85, 90 , 94. His teacher says he may use either his mean or median for his report card grade. Jerry wants the highest grade. Which should he choose and what will the grade be? (Show work) 11. The average number of CD's owned by John, Sue, Juan and Kate is 114. John owns 200, Sue owns 121 and Kate owns 53. How many CD's does Juan own? 12. The average (mean) of Fred's 6 test scores is 75. His teacher will drop his lowest grade of 55. What is the mean of the remaining 5 test scores? .
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