Thursday, September 13, 2012

Interquartile Range a stepwise approach

Interquartile Range Statistics
To understand an Interquartile range first we need to learn what quartiles mean.  When a list of values is divided into quarters we get the value called the quartiles. Consider the list of values, 6, 12, 7, 9, 13, 5, 8. First we need to arrange the list of values in the numerical order which would be 5, 6, 7, 8,  9, 12, 13. There are seven values in the list. Let us now divide the values into quarters. The middle value in the list gives us the second quartile, which is 8 from the list given. Consider the values to the left of the second quartile, 5, 6, 7. The middle value in this list gives us the first quartile which is 6. Finally consider the values to the right of the second quartile, 9,12,13. The middle value in this list would be the third quartile which is 12. So, the quartiles from the list are Q1 = 6, Q2= 8 and Q3= 12. Now let us find Interquartile range which would be the value got by subtracting the first quartile from the third quartile. Here the Interquartile range, IQR is Q3 – Q1 = 12 – 6 = 6.

In calculating the Interquartile Range we subtract the first quartile from the third quartile of a given list of values. To find Interquartile Range we shall follow the following steps:
Step1: First the given list of values are arranged in the numerical order
Step2: Count the number of values in the list
Step3: If the count is odd, then the middle value would be Q2, the middle value of the first half and
            second half of the list will be Q1 and Q3 respectively
            If the count is even, then the mean of the two middle values would be Q2, the mean of the two
            middle values of the first half and the second half of the list give us Q1 and Q3 respectively
Step4: Once we are ready with Q3 and Q1 the Interquartile Range=IQR = Q3 – Q1

Example of Interquartile Range of a given list 7, 15, 18, 3, 7, 9, 10, 11.
First we arrange the given list in the numerical order, 3, 7, 7, 9, 10, 11, 15, 18.
There are even number of values in the given list so the median or the second quartile would be the mean of the two middle values which would be (9+10)/2 = 19/2 = 9.5 = Q2.
To find the first quartile we consider the values to the left of the median which are 3,7,7,9. So, the first quartile would be the mean of the two middle values which would be (7+7)/2 = 7 and
finally the third quartile would be the mean of the two middle values which are to the right of the median, that would be (11+15)/2 = 13. So, IQR = Q3 – Q1 = 13 – 7 = 6

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