Measure of Central Tedency

The meaning of Measure of Central Tedency..

A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. As such, measures of central tendency are sometimes called measures of central location. 

They are also classed as summary statistics. The mean (often called the average) is most likely the measure of central tendency that you are most familiar with, but there are others, such as the median and the mode.

The mean, median and mode are all valid measures of central tendency, but under different conditions, some measures of central tendency become more appropriate to use than others. 

In the following sections, we will look at the mean, mode and median, and learn how to calculate them and under what conditions they are most appropriate to be used.

Mean is an Average which mean the sum of data divided by the number of data (Do not round your answer unless directed to do so.)

Median is 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 is the value (number) that appear 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:

Two sets o scores above is the same except for the first score. Set on the left shows the actual score. Set on the right shows what would happen if one of the score is WAY out of range with respect to the other score. Such a period is called outliers. With outliers, the mean change. With outliers, the median is unchanged.


How do I know which measure of central tedency 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.

This is the example to find the Mean, Median and Mode;

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, you have to 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.