Question

A survey was given to 18 students. One question asked about the one-way distance the student had to travel to attend college. The results, in miles, are shown in the following table. Use the median procedure for finding quartiles to find the first, second, and third quartiles for the data. Distance Traveled to Attend College

46 50 18 26 64 78 4 38 44 44 10 70 74 44 86 32 26 48

Q1 =

Q2 =

Q3 =

Answer 1

Given distances travelled to attend college are : 46, 50, 18, 26, 64, 78, 4, 38, 44, 44, 10, 70, 74, 44, 86, 32, 26, 48. We have to find the first quartile, second quartile and third quartile. First we have to arrange these values in ascending order. Thus, we get : 4, 10, 18, 26, 26, 32, 38, 44, 44, 44, 46, 48, 50, 64, 70, 74, 78, 86. Here, we have 18 values (i.e., even data set).

The median is the middle most value of any arranged data set. For any even data set, the median is the average of (n/2)th term and {(n/2)+1}th term. The numbers to the left side of the median is called the lower half and the numbers to the right side of the median is called the upper half.

We know, the first quartile, Q1 = Median of the lower half of the data. The second quartile, Q2 = Median of the given data set. The third quartile, Q3 = Median of the upper half of the data.

Now, first we will find the median. Here, n = 18. Thus, Median is the average of (18/2)th term and {(18/2)+1}th term. Thus, Median is the average of 9th term and 10th term. In our arranged data set, the 9th term is 44 and the 10th term is 44. Thus, Median = Average of 44 and 44. Thus, Median = (44+44)/2 = 44.

Thus, Median = Second Quartile, Q2 = 44.

As the median is the average value of 9th and 10th term, thus, we will include the 9th term in the lower half and the 10th term in the upper half.

Now, the numbers to the left side of the median = Lower half = {4,10,18,26,26,32,38,44,44}. Thus, we know, First Quartile = Median of lower half.

Now, we have to find the median of {4,10,18,26,26,32,38,44,44}. Here, there are 9 values (i.e., odd number). Thus, the median is the {(n+1)/2}th term. Here, n = 9. Thus, the median is {(9+1)/2}th term. Thus, the median is 5th term. Here, the 5th term is 26. Thus, Q1 = 26.

Thus, the first quartile, Q1 = 26.

Now, the numbers to the right side of the median = Upper half = {44,46,48,50,64,70,74,78,86}. Thus, we know, Third Quartile = Median of upper half.

Now, we have to find the median of {44,46,48,50,64,70,74,78,86}. Here, there are 9 values (i.e., odd number). Thus, the median is the {(n+1)/2}th term. Here, n = 9. Thus, the median is the {(9+1)/2}th term. Thus, the median is the 5th term. Here, the 5th term is 64. Thus, Q3 = 64.

Thus, the third quartile, Q3 = 64.

Thus, the First Quartile = 26, Second Quartile = 44 and the Third Quartile = 64.

Thus, Q1 = 26, Q2 = 44 and Q3 = 64.