#$&* course MTH 152 3/27 2 014. mean vs median Question: `q001. Note that there are 8 questions in this assignment.
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Given Solution: To get the mean value of the numbers, we first note that there are eight numbers. Then we had the numbers and divide by eight. We obtain 5 + 7 + 9 + 9 + 10 + 12 + 13 + 15 = 80. Dividing by 8 we obtain mean = 80 / 8 = 10. The difference between 5 and the mean 10 is 5; the difference between 7 and the mean 10 is 3; the difference between 9 and 10 is 1; the differences between 12, 13 and 15 and the mean 10 are 2, 3 and 5. So we have differences 5, 3, 1, 1, 0, 2, 3 and 5 between the mean and the numbers in the list. The average difference between the mean and the numbers in the list is therefore ave difference = ( 5 + 3 + 1 + 1 + 0 + 2 + 3 + 5 ) / 8 = 20 / 8 = 2.5. Self-critique: OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: `q002 What is the middle number among the numbers 13, 12, 5, 7, 9, 15, 9, 10, 8? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: In order to determine the middle number among our set of numbers, we must first put the numbers in order. Our set of numbers then becomes [5, 7, 8, 9, 9, 10, 12, 13, and 15]. There are nine numbers in our set of numbers, so the middle number will be the fifth number from the left of the set. Therefore, our middle number is 9. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: It is easier to answer this question if we place the numbers in ascending order. Listed in ascending order the numbers are 5, 7, 8, 9, 9, 10, 12, 13, and 15. We see that there are 9 numbers in the list. If we remove the first 4 and the last 4 we are left with the middle number. So we remove the numbers 5, 7, 8, 9 and the numbers 10, 12, 13, and 15, which leaves the second '9' as the middle number. Self-critique: OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: `q003. On a list of 9 numbers, which number will be the one in the middle? Note that the middle number is called the 'median'. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: We discovered in the previous problem that a set of nine numbers has a median number that is located 5 numbers from the left. This is because there are exactly 4 numbers located on each side of the fifth number. In a set of numbers where the amount of numbers is odd, the middle number is always the single number left between two even sets of numbers. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: If the 9 numbers are put in order, then we can find the middle number by throwing out the first four and the last four numbers on the list. We are left with the fifth number on the list. In general if we have an odd number n of number in an ordered list, we throw out the first (n-1) / 2 and the last (n-1) / 2 numbers, leaving us with the middle number, which is number (n-1)/2 + 1 on the list. So for example if we had 179 numbers on the list, we would throw out the first (179 - 1) / 2 = 178/2 = 89 numbers on the list and the last 89 numbers on the list, leaving us with the 90th number on the list. Note that 90 = (179 - 1) / 2 + 1, illustrating y the middle number in number (n-1)/2 + 1 on the list. Self-critique: OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: `q004. What is the median (the middle number) among the numbers 5, 7, 9, 9, 10, 12, 13, and 15? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: There are 8 numbers in our new set of numbers. In order to find the median of a set in which there is an even amount of numbers, we must take the average of the two numbers found in the middle. The numbers in our new set are already ordered, so we do not have to change the order to find the two middle numbers. In this instance the middle numbers are 9 and 10; they are set between two sets of three numbers at the beginning and end of the set. The sum of 9 and 10 is 19. When we divide by two, since we are finding the average of only two numbers, we discover that the average of 9 and 10 and the middle number of our set is 9.5. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: There are 8 numbers on this list. If we remove the smallest then the largest our list becomes 7, 9, 9, 10, 12, 13. If we remove the smallest and the largest from this list we obtain 9, 9, 10, 12. Removing the smallest and the largest from this list we are left with 9 and 10. We are left with two numbers in the middle; we don't have a single 'middle number'. So we do the next-most-sensible thing and average the two numbers to get 9.5. We say that 9.5 is the middle, or median, number. Self-critique: OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: `q005. We saw that for the numbers 5, 7, 9, 9, 10, 12, 13, and 15, on the average each number is 2.5 units from the average. Are the numbers in the list 48, 48, 49, 50, 51, 53, 54, 55 closer or further that this, on the average, from their mean? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The mean of the set [48, 48, 49, 50, 51, 53, 54, and 55] is 408 (48+48+49+50+51+53+54+55)/8 (the number of numbers in our set), or 51. We can find the average distance of the numbers from the mean by creating a new set of numbers with each respective distance. The distance from 48 to 51 is 3, the distance from 49 to 51 is 2, and so on and so forth. Our new set of numbers is therefore [3, 3, 2, 1, 0, 2, 3, and 4]. The mean of this set of numbers is 18/8, or 2.25. Therefore, this set of numbers is closer to the mean than the set of numbers from the previous problem. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: The mean of the numbers 48, 48, 49, 50, 51, 53, 54, and 55 is (48 + 48 + 49 + 50 + 51 + 53 + 54 + 55) / 8 = 408 / 8 = 51. 48 is 3 units away from the mean 51, 49 is 2 units away from the mean 51, 50 is 1 unit away from the mean 51, and the remaining numbers are 2, 3 and 4 units away from the mean of 51. So on the average the distance of the numbers from the mean is (3 + 3 + 2 + 1 + 0 + 2 + 3 + 4) / 8 = 18 / 8 = 2.25. This list of numbers is a bit closer, on the average, then the first list. Self-critique: OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: `q006. On a 1-10 rating of a movie, one group gave the ratings 1, 8, 8, 9, 9, 10 while another gave the ratings 7, 7, 8, 8, 9, 10. Find the mean (average) and the median (middle value) of each group's ratings. Which group would you say liked the movie better? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The mean of the first set of numbers [1, 8, 8, 9, 9, and 10] is 45 (1+8+8+9+9+10)/ 6 (the number of numbers in the set), or 7.5. The median of the first set of numbers is 17 (8+9)/ 2, or 8.5 The mean of the second set of numbers [7, 7, 8, 8, 9, and 10] is 49 (7+7+8+8+9+10)/ 6 (the number of numbers in the set), or 8.16. The median of the second set of numbers is 16 (8+8)/ 2, or 8. The results of the problem indicate that the first group liked the movie better. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: The mean of the first list is (1 + 8 + 8 + 9 + 9 + 10) / 6 = 45 / 6 = 7.5. The median is obtained a throwing out the first 2 numbers on the list and the last 2 numbers. This leaves the middle two, which are 8 and 9. The median is therefore 8.5. The mean of the numbers on the second list is (7 + 7 + 8 + 8 + 9 + 10) / 6 = 49 / 6 = 8 .16. The median of this list is found by removing the first 210 the last 2 numbers on the list, leaving the middle two numbers 8 and 8. The median is therefore 8. The first group had the higher median and the lower mean, while the second group had the lower median but the higher mean. Since everyone except one person in the first group scored the movie as 8 or higher, and since everyone in both groups except this one individual scored the movie 7 or higher, it might be reasonable to think that the one anomalous score of 1 is likely the result of something besides the quality of the movie. We might also note that this score is much further from the mean that any of the other scores, giving it significantly more effect on the mean than any other score. We might therefore choose to use the median, which limits the otherwise excessive weight given to this unusually low score when we calculate the mean. In this case we would say that the first group liked the movie better. Self-critique: OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: `q007. Suppose that in a certain office that ten employees make $700 per pay period, while five make $800 per pay period and the other two make $1000 per pay period. What is the mean pay per period in the office? What is the median? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: In order to determine the mean and median of the pay period in the office, we must create a set of numbers that includes each instance of pay in the office. When we do this we end up with a set of [700, 700, 700, 700, 700, 700, 700, 700, 700, 700, 800, 800, 800, 800, 800, 1000, and 1000]. The mean of this set is 13,000 [700+700+700+700+700+700+700+700+700+700+800+800+800+800+800+1000+1000]/ 17 (the number of numbers in the set), or 764.7. The median value of this set of numbers is 700, as it is the odd number located in the middle of the set of numbers. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: There are a total of 10 + 5 + 2 = 17 employees in the office. The total pay per pay period is 10 * $700 + 5 * $800 + 2 * $1000 = $13,000. The mean pay per period is therefore $13,000 / 17 = $823 approx.. The median pay is obtained by 'throwing out' the lowest 8 and the highest 8 in an ordered list, leaving the ninth salary. Since 10 people make $700 per period, this leaves $700 as the median. STUDENT QUESTION: Is it typical to use the median value if there are ‘oddball’ scores in a group? INSTRUCTOR RESPONSE A few 'oddball' scores have little effect on the median, but can have a great effect on the mean. Other factors can also be important depending on the situation, but if a lot of 'oddball' scores, or 'outliers', are expected the median is often the better indication of average behavior than the mean. Self-critique: I’m not sure why you got $823 as the average. Whenever I divide 13,000 by 17 I always get 764.7. I tried it several times and always came up with the same answer. Am I doing something wrong? ------------------------------------------------ Self-critique rating: OK ********************************************* Question: `q007. Suppose that in a certain office that ten employees make $700 per pay period, while five make $800 per pay period and the other two make $1000 per pay period. What is the mean pay per period in the office? What is the median? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: In order to determine the mean and median of the pay period in the office, we must create a set of numbers that includes each instance of pay in the office. When we do this we end up with a set of [700, 700, 700, 700, 700, 700, 700, 700, 700, 700, 800, 800, 800, 800, 800, 1000, and 1000]. The mean of this set is 13,000 [700+700+700+700+700+700+700+700+700+700+800+800+800+800+800+1000+1000]/ 17 (the number of numbers in the set), or 764.7. The median value of this set of numbers is 700, as it is the odd number located in the middle of the set of numbers. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: There are a total of 10 + 5 + 2 = 17 employees in the office. The total pay per pay period is 10 * $700 + 5 * $800 + 2 * $1000 = $13,000. The mean pay per period is therefore $13,000 / 17 = $823 approx.. The median pay is obtained by 'throwing out' the lowest 8 and the highest 8 in an ordered list, leaving the ninth salary. Since 10 people make $700 per period, this leaves $700 as the median. STUDENT QUESTION: Is it typical to use the median value if there are ‘oddball’ scores in a group? INSTRUCTOR RESPONSE A few 'oddball' scores have little effect on the median, but can have a great effect on the mean. Other factors can also be important depending on the situation, but if a lot of 'oddball' scores, or 'outliers', are expected the median is often the better indication of average behavior than the mean. Self-critique: I’m not sure why you got $823 as the average. Whenever I divide 13,000 by 17 I always get 764.7. I tried it several times and always came up with the same answer. Am I doing something wrong? ------------------------------------------------ Self-critique rating: OK #*&!