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Measures of Center: Mean, Median, Mode Please view this tutorial and answer the follow-up questions on loose leaf to turn in to your teacher.

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Measures of Center Basics Mean: the average, add up all the values and divide by the number of values Median: the middle number in an ordered set of data Mode: the most frequent number in a set of data; the number that occurs the most

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Mean The following set of data is Greg’s Test scores from his Math Class: 95, 88, 84, 47, 72, 68, 98, 100, 78, 72, 96, 84, 55, 72, 90, 72, 100 1) First you should add up all the test scores together. What do you get? You should have gotten 1,371. 2) Next, figure out how many tests Greg took altogether? 17 tests To compute the mean, take the sum of all the test scores (1,371) and divide by the total number of tests he took. You should have gotten 80.6. This means that Greg’s average test score was an 80.6.

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Finding the mean on the TI-83 Plus First enter all the data into a list by hitting the STAT button and then 1: Edit… 95, 88, 84, 47, 72, 68, 98, 100, 78, 72, 96, 84, 55, 72, 90, 72, 100 **Be sure to take your time when typing in the numbers, one small mistake can make a big difference. It might be a good idea to double check that you entered everything correctly.

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Finding the mean on the TI-83 Plus Once everything is typed into a list, hit the 2 nd button and then the STAT button Press right until you get to the MATH screen and then choose 3: mean( You should still get the same mean of 80.6. Once that is up on your home screen, hit the 2 nd button and then whatever number list you put your data into and hit enter. For example, 2 nd ….1, will give you L1.

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Median To find the median, the first step we must do is to get all the data in ascending order (from least to greatest) 95, 88, 84, 47, 72, 68, 98, 100, 78, 72, 96, 84, 55, 72, 90, 72, 100 You should get… 47, 55, 68, 72, 72, 72, 72, 78, 84, 84, 88, 90, 95, 96, 98, 100, 100

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Median (odd amount of data) In data sets that have an odd amount of data: To find the median (or middle number) cross out the lowest number and the largest number 47, 55, 68, 72, 72, 72, 72, 78, 84, 84, 88, 90, 95, 96, 98, 100, 100 84 is the median since there are exactly the same amount of numbers above it and below it. Notice that there are eight numbers smaller than 84 and eight numbers larger.

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Median (even amount of data) If there is an even amount of data the process is similar, but you will have no exact middle number. Let’s say we use the same data except that we get rid of one of the 100’s 47, 55, 68, 72, 72, 72, 72, 78, 84, 84, 88, 90, 95, 96, 98, 100 As you can see the median falls between two values. When this happens take the average of those two numbers. For this problem the average of 78 and 84 is 81. So the median of this data set is 81

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Mode The mode is the number that happens the most (the most frequent value) What is the mode of the following set of data? 47, 55, 68, 72, 72, 72, 72, 78, 84, 84, 88, 90, 95, 96, 98, 100, 100 If you said 72, then you were correct! 72 occurs four times. The next most frequent values would be 84 and 100 which both occur twice. ** It is possible to have two modes, this happens when two numbers both occur the most.

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Outliers and Measures of Center Outliers affect one of the three measures of center. Let’s see which one… For our data set, we got a mean of 80.6, a median of 84, and a mode of 72. Let’s say Greg completed an extra credit assignment worth 50 points. This was added onto one of the tests that Greg scored a 100 on. So there is a new test with a score of 150. 47, 55, 68, 72, 72, 72, 72, 78, 84, 84, 88, 90, 95, 96, 98, 100, 100, 150 Compute the mean, median and mode of this new data set.

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Outliers and Measures of Center As you can see the mean is the measure of center that is most affected by outliers OldNew Mean80.684.5 Median84 Mode72

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Outliers and Measures of Center In conclusion, 1)The mean is the measure of center that is most affected by outliers. 2)If a data set has a small outlier, the mean becomes significantly lower, but the median stays around the same number. 3)If a data set has a large outlier, the mean becomes significantly higher, but the median stays around the same number.

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Follow-Up Questions 1)Consider the following list of high temperatures ( o F) for the first two weeks of October in a city in Maryland. 65, 72, 60, 64, 75, 59, 71, 63, 60, 67, 72, 85, 86, 60 a)Make a stem and leaf plot of the data (submit a hard copy to your teacher) b)Find the mean high temperature for this time period. c)Find the median high temperature for this time period. d)Find the mode high temperature for this time period. e)If a 100 o day was added to this data sheet, what would happen to the mean, median, and mode without computing the actual values?

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Follow-Up Questions 2.In the following shapes for histograms, determine the median would be greater than, less than, or equal to the mean. A)B) C)

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Follow-Up Questions 3)The median of the test scores displayed in the stem and leaf plot below is: a)68 b)72 c)75 d)76 e)78 23 3 44 7 52 2 3 6 61 4 5 5 8 72 5 6 6 6 8 8 81 3 4 4 92 6

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