COMPARING THE MEAN AND THE MEDIAN

About "Comparing the mean and the median"

Comparing the mean and the median :

The mean and median of a data set may be equal, very close to each other, or very different from each other. For data sets where the mean and median differ greatly, one likely describes the data set better than the other.

Comparing the mean and the median - Examples

Example 1 :

The monthly earnings of several teenagers are $200, $320, $275, $250, $750, $350, and $310.

Questions (i) : 

Find the mean. Round to the nearest tenth.

Answer : 

Mean  =  (200 + 320 + 275 + 250 + 750 + 350 + 310) / 7

Mean  =  2455 / 7

Mean  =  350.7142........

Rounding to the nearest tenth 

Mean  =  350.7

Questions (ii) : 

Write the data values in order from least to greatest and find the median.

Answer : 

200, 250, 275, 310, 320, 350, 750

Number of observations (n)  =  7

Median  =  [(n+1)/2] th value

Plug n  =  7

Median  =  [(7+1)/2] th value

Median  =  4 th value

Median  =  310

Questions (iii) : 

What is the difference between the mean and the median ? Why ?

Answer : 

Difference between mean and median  =  |350.7 - 310|

Difference between mean and median  =  40.7

Reason : 

The mean is affected by the data value 750, which is much greater than the other data values. The median is not affected by this value.

Questions (iv) : 

Which measure of center better describes the typical monthly earnings for this group of teenagers—the mean or the median? Explain.

Answer : 

Median; The mean is greater than all but one of the data values. The median is a more central value.

Example 2 :

Luka’s final exam scores for this semester are 70, 72, 99, 72, and 69. 

Questions (i) : 

Find the mean.

Answer : 

Mean  =  (70 + 72 + 99 + 72 + 69) / 5

Mean  =  382 / 5

Mean  =  76.4

Questions (ii) : 

Write the data values in order from least to greatest and find the median.

Answer : 

69, 70, 72, 72, 99

Number of observations (n)  =  5

Median  =  [(n+1)/2] th value

Plug n  =  5

Median  =  [(5+1)/2] th value

Median  =  3 rd value

Median  =  72

Questions (iii) : 

What is the difference between the mean and the median ? Why ?

Answer : 

Difference between mean and median  =  |76.4 - 72|

Difference between mean and median  =  4.4

Reason : 

The mean is affected by the data value 99, which is much greater than the other data values. The median is not affected by this value.

Questions (iv) : 

Which is a better description of Luka’s typical exam score? Explain your thinking.

Answer : 

Median; the median is closer in value to most of the data values than the mean.

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