Measure of Central Tendency
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The term "measures of central tendency" refers to
finding the mean, median and mode

Mean: Average.
The sum of a set of data divided by the number of data.
(Do not round your answer unless directed to do so.)
Median: The middle value, or the mean of the middle two values, when the data is arranged in numerical order.   Think of a "median" being in the middle of a highway.
 
Mode:  The value ( number) that appears the most.
It is possible to have more than one mode, and it is possible to have no mode.  If there is no mode-write "no mode", do not write zero (0) . 

 

Consider this set of test score values:


Normal listing of scores.

Scores with the lowest score replaced with outlier.

The two sets of scores above are identical except for the first score.  The set on the left shows the actual scores.  The set on the right shows what would happen if one of the scores was WAY out of range in regard to the other scores.  Such a term is called an outlier.
                                          With the outlier, the
mean changed.
                                 With the outlier, the
median did NOT change.
 

How do I know which measure of central tendency to use?

MEAN
Use the mean to describe the middle of a set of data that does not have an outlier.

Advantages:
    
Most popular measure in fields such as business, engineering and computer science.
     It is unique - there is only one answer.
     Useful when comparing sets of data.

Disadvantages:
    
Affected by extreme values (outliers)

MEDIAN
Use the median to describe the middle of a set of data that does have an outlier.

Advantages:
    
Extreme values (outliers) do not affect the median as strongly as they do the mean.
     Useful when comparing sets of data.
     It is unique - there is only one answer.


Disadvantages:
    
Not as popular as mean.

MODE
Use the mode when the data is non-numeric or when asked to choose the most popular item.

Advantages:
    
Extreme values (outliers) do not affect the mode.

Disadvantages:
  
 Not as popular as mean and median.
     Not necessarily unique - may be more than one answer
     When no values repeat in the data set, the mode is every value and is useless.
     When there is more than one mode, it is difficult to interpret and/or compare.

 

What will happen to the measures of central tendency if we add the same amount to all data values, or multiply each data value by the same amount?

  Data Mean Mode Median
Original Data Set:  6, 7, 8, 10, 12, 14, 14, 15, 16, 20 12.2 14 13
Add 3 to each data value 9, 10, 11, 13, 15, 17, 17, 18, 19, 23 15.2 17 16

Multiply 2 times each data value

12, 14, 16, 20, 24, 28, 28, 30, 32, 40

24.4 28 26

When added:  Since all values are shifted the same amount, the measures of central tendency all shifted by the same amount.  If you add 3 to each data value, you will add 3 to the mean, mode and median.  

When multiplied:  Since all values are affected by the same multiplicative values, the measures of central tendency will feel the same affect.  If you multiply each data value by 2, you will multiply the mean, mode and median by 2.

 


 


Example #1

Find the mean, median and mode for the following data:  5, 15, 10, 15, 5, 10, 10, 20, 25, 15. 
        
    
                                           (You will need to organize the data.)
                                            5, 5, 10, 10, 10, 15, 15, 15, 20, 25

Mean:       

Median:     5, 5, 10, 10, 10, 15, 15, 15, 20, 25

          Listing the data in order is the easiest way to find the median.
         The numbers 10 and 15 both fall in the middle.
 Average these two numbers to get the median.      10 + 15 = 12.5
                                                                              2

Mode:    Two numbers appear most often:  10 and 15.
                  There are three 10's and three 15's.
                  In this example there are two answers for the mode.



Example #2
     

For what value of  x  will  8 and x have the same mean (average) as 27 and 5?

First, find the mean of 27 and 5:

27 + 5 = 16
     2

Now, find the x value, knowing that the average of x and 8 must be 16:

x + 8 = 16              
   2                          

32 = x + 8     cross multiply
-8         - 8
24 = x             and solve

 


Example #3 :  
                            
                                                       

On his first 5 biology tests, Bob received the following scores:  72, 86, 92, 63, and 77.  What test score must Bob earn on his sixth test so that his average (mean score) for all six tests will be 80?  Show how you arrived at your answer.

Possible solution:
        
Set up an equation to represent the situation.  Remember to use all 6 test scores:  
                                             72 + 86 + 92 + 63 + 77 + x   =  80
                                                                6

 cross multiply and solve:                 (80)(6) = 390 + x
                                                          480 = 390 + x
                                                        - 390   -390
                                                            90 =          x
                                         Bob must get a 90 on the sixth test.


Example #4    


The mean (average) weight of three dogs is 38 pounds.  One of the dogs, Sparky, weighs 46 pounds.  The other two dogs, Eddie and Sandy, have the same weight.  Find Eddie's weight.

Let x = Eddie's weight        ( they weigh the same, so they are both represented by "x".)
Let x = Sandy's weight                        

Average:   sum of the data divided by the number of data.

                  x + x + 46 = 38                 cross multiply and solve
                      3(dogs)                                 

                 (38)(3) = 2x + 46
                      114 = 2x + 46
                       -46          -46 
                        68 = 2x 
                         2      2

                        34 = x     Eddie weighs 34 pounds.


You can always check your work with a calculator!!

See how to use your
TI-83+/TI-84+ graphing calculator  with mean, mode, median.
Click calculator.

See how to use your
TI-83+/TI-84+ graphing calculator  with mean, mode, median and grouped data.
Click calculator.