If your solution to stated problem does not match the given solution, you should self-critique per instructions at

 

http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm.

 

Your solution, attempt at solution. 

 

If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it.  This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

 

014.  mean vs median

 

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Question:  `q001. Note that there are 10 questions in this assignment. 

 

{}{} What is the average, or mean value, of the numbers 5, 7, 9, 9, 10, 12, 13, and 15?  On the average how 'far' is each number from this mean value?

 

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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.

 

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Question:  `q002 What is the middle number among the numbers 13, 12, 5, 7, 9, 15, 9, 10, 8? 

 

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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.

 

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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'.

 

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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.

 

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Question:  `q004.  What is the median (the middle number) among the numbers 5, 7, 9, 9, 10, 12, 13, and 15?

 

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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.

 

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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?

 

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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.

 

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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?

 

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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.

 

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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?

 

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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.

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Question:  `q008.  In the preceding problem ten employees make $700 per pay period, while five make $800 per pay period and the other two make $1000 per pay period; we just found that the mean pay per period was $823.  On the average, how much to the individual salaries differ from the mean?

 

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Given Solution:  The mean was found in the preceding problem to be $765.  The deviation of $700 from the mean is therefore $65, the deviation of $800 from the mean is $35 and the deviation of $1000 from the mean is $135. 

 

Since $700 is paid to 10 employees, $800 to five and $1000 to two, the total deviation is 10 *$65 + 5 * $35 + 2 * $235 = $1295.  The mean deviation is therefore $1295 / 17 = $76.18 , approx..

 

 

Question: `q009.  What is the mean of the numbers 1.05, 1.03, 1.06, 1.08, 1.06?

 

On the average by how much do these numbers deviate from the mean?

 

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Question: `q010.  What is the mean of a set of numbers in which 1.05 occurs 4 times, 1.03 occurs 3 times, 1.06 occurs 10 times and 1.08 occurs 3 times?

 

On the average by how much do the numbers in this set deviate from their mean?

 

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