How to find the 5-number summary

In the last lab (#4), question 04 asked to find the 5-number summary of the data “Exam 1”. The 5-number summary are: Minimum, First Quartile, Median, Third Quartile and Maximum. In order to find those values, the first thing we have to do is sort the data.

##  [1] 25 38 45 50 52 53 57 59 60 61 63 65 67 67 69 69 69 69 70 70 70 70 71
## [24] 72 73 74 74 75 75 75 75 76 76 77 77 77 78 78 78 78 78 79 79 79 79 79
## [47] 80 80 80 80 81 81 81 81 81 82 82 82 82 82 82 83 84 84 85 85 85 85 85
## [70] 85 86 87 87 87 87 87 89 89 89 89 89 89 89 89 90 91 92 92 93 93 94 94
## [93] 94 94 95

Just by ordering the data we can see that the Minimum value is 25 and the Maximum value is 95.

The median is the midpoint of a distribution. Half of the observations are smaller than the median and half are larger than the median. Because we have 95 observations (odd number), the median is the center observation in the ordered list. If the number of observations were even, then the median would be the mean of the two center observations in the ordered list. The position of the median in the ordered list is given by

\[\dfrac{(n+1)}{2}\]

Therefore, in our case, the position of the median is

\[\dfrac{(n+1)}{2} = \dfrac{95 + 1}{2} = 48\]

The 48th value in the ordered list is our median . Hence, the median is 80.

The first quartile is the median of the observations whose position in the ordered list are to the left of the location of the median. That means that the first quartile is the median of the new dataset the consists of all observations in “Exam 1” that are lower than the median. The new dataset is:

##  [1] 25 38 45 50 52 53 57 59 60 61 63 65 67 67 69 69 69 69 70 70 70 70 71
## [24] 72 73 74 74 75 75 75 75 76 76 77 77 77 78 78 78 78 78 79 79 79 79 79
## [47] 80

In this new dataset we have 47 observations. We do the same procedure as before to find the median. It’s position is going to be \(\dfrac{47 + 1}{2} = 24\). Therefore, the median of the new dataset (and consequently the first quartile of “Exam 1”) is 72.

We know that the median splits the dataset in two subset with the same number of observations. Therefore, the position of the third quartile of “Exam 1” is also 24 in the new dataset: all observations in “Exam 1” that are greater than the median. Or we can see it as \(48+24 = 72\) th value in “Exam 1”. Therefore, third quartile is 87.

Then, the 5-number summary of the data “Exam 1” is:

Minimum First quartile Median Third quartile Maximum
25 72 80 87 95