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Quartiles

 

Quartiles divide data into four sets of equal size, or at least as close as you can get to four sets of equal size.

The first quartile, Q1, is the 25th percentile

The second quartile, Q2, is the 50th percentile, which is also the median

The third quartile, Q3, is the 75th percentile

 

The quick way to find the quartiles is to first put the data in order.  For example data is 3, 1, 6, 7, 4, 2, 12, 31, 22, 19, 16.

In order the data is 1, 2, 3, 4, 6, 7, 12, 16, 19, 22, 31

First find  Q2, which is the median: Q2 = 7

You then look at the data left of where the median is 1, 2, 3, 4, 6.

Then Q1 is the median of these values:  Q1= 3.

Finally look at the data to the right of where the median is: 12, 16, 19, 22, 31

Then Q3 is the median of these values:  Q3= 19.

Note that highlighting the quartiles is red shows that you are as close as you can be to four sets of equal size

1, 2, 3, 4, 6, 7, 12, 16, 19, 22, 31

 

Now suppose you add a value to get data 1, 2, 3, 4, 6, 7, 12, 16, 19, 22, 31, 64

First find  Q2, which is the median: Q2 = (7+12)/2 = 9.5

Now the position of the median is between the sixth anf seventh values, so data to the left of the position of the median is

1, 2, 3, 4, 6, 7

Then Q1 is the median of these values:  Q1= 3.5.

Finally look at the data to the right of where the median is: 12, 16, 19, 22, 31, 64

Then Q3 is the median of these values:  Q3= 20.5.

More examples.

 

The Five Number Summary

 

The five number summary is

Lowest value, Q1, Q2, Q3, Highest value

  Looking at what we did above the five number summary for 1, 2, 3, 4, 6, 7, 12, 16, 19, 22, 31 is

1, 3, 7, 19, 31

 

And the five number summary for 1, 2, 3, 4, 6, 7, 12, 16, 19, 22, 31, 64 is

1, 3.5, 9.5, 20.5, 64

 

Outliers   (Not Covered or Tested Fall 2016)

 

Sometimes a set of data contains a value that is suspiciously large or small, such a value may be an outlier.  More Explanation

 

To check for outliers you find the interquartile range, which is IQR = Q3  ̶  Q1.

Then values bigger than Q3 + 1.5 IQR or less than Q1  ̶  1.5 IQR are outliers

 

For the data 1, 2, 3, 4, 6, 7, 12, 16, 19, 22, 31, 64 given above

IQR = Q3  ̶  Q1 = 20.5 - 3.5 = 17

Then Q1  ̶  1.5 IQR = 3.5  ̶  1.5(17) =  ̶ 22  and  Q3 + 1.5 IQR == 20.5 +1.5(17) = 46

You conclude 64 is an outlier

 

Box Plots

 

A box plot shows the five number summary in graphical form.  You have dots above the smallest and highest values, lines above the quartiles and a box around the lines.

 

For 1, 2, 3, 4, 6, 7, 12, 16, 19, 22, 31 the box plot is:

Note that the box covers the range for the middle 50% of values