Statistics and Probability

Question

The Highway Loss Data Institute’s Injury and Collision Loss Experience Report rates car models on the basis of the number of insurance claims filed after accidents. Index ratings near 100 are considered average. Lower ratings are better, indicating a safer car model. Shown below are the ratings for 20 mid-size and 20 small cars.

Mid-size 91 93 127 68 81 60 51 58 75 100

Small 73 100 127 100 124 103 119 109 113 108

Mid-size 103 119 82 80 128 76 68 81 91 82

Small 118 103 120 102 122 96 100 133 80 140

(a) For each type of car calculate the median, lower quartile and upper quartile.

(b) Construct box plots for each car-type, and make two comments on how they compare.

(c) For each type of car calculate the quartile coefficient of skewness and comment on it.

(d) For each type of car identify the outliers, if there are any

Answer 1

Put the numbers in order.

To find the position number of median, compute (sample size+1)/2.

If position number is an integer, the median is the number in that position, e.g., if it is 5, then the median is the number in the 5th position

If position number is not an integer, the median is the average of the numbers in the positions below and above that position number, e.g., if it is 5.5., then the median is the average of the numbers in positions 5 and 6.

To find the position number of the 1st quartile, compute (sample size + 1)/4 and apply the same rules as for the median.

To find the position number of the 3rd quartile, compute (sample size + 1) * 3/4 and apply the same rules as for the median

The box plot contains the lower value, 1st quartile, median, 3rd quartile and upper value.

Quartile coefficient of skewness = Q3+Q1−2Q2Q3−Q1

Outliers are data below Q1 - 1.5 IQR and Q3 + 1.5 IQR, where IQR = Q3 - Q1.