The following scores on the midterm exam in Chemistry 102 were recorded:
93 81 59 69 82 73 61 77 95 84 88 71
86 97 63 72 89 80 60 98 91 62 78 83
76 81 94 66 83 96
Find the interquartile range (IQR).
I came up with 15 as the answer. 96-81=15. Is this answer correct?
To find the interquartile range (IQR), we first need to determine the values of the first quartile (Q1) and the third quartile (Q3).
Step 1: Sort the scores in ascending order:
59, 60, 61, 62, 63, 66, 69, 71, 72, 73, 76, 77, 78, 80, 81, 81, 82, 83, 83, 84, 86, 88, 89, 91, 93, 94, 95, 96, 97, 98
Step 2: Determine the median (Q2) of the entire data set:
The median is the middle value of the sorted list. In this case, since we have an odd number of data points, the median is the score at the (n + 1)/2 position, where n is the number of data points.
In this case, n = 30, so the median is the score at the (30 + 1)/2 = 15.5th position, which corresponds to the 15th value in the sorted list.
Therefore, Q2 = 81.
Step 3: Determine the first quartile (Q1):
The first quartile is the median of the lower half of the data set. It represents the value below which 25% of the data falls.
The lower half of the data set consists of the 14 scores before the median. Since we have an even number of data points in this subset, we need to take the average of the two middle values to find Q1.
The two middle values are the scores at the 7th and 8th positions, which are 69 and 71, respectively.
Therefore, Q1 = (69 + 71) / 2 = 70.
Step 4: Determine the third quartile (Q3):
The third quartile is the median of the upper half of the data set. It represents the value below which 75% of the data falls.
The upper half of the data set consists of the 15 scores after the median. Since we have an odd number of data points in this subset, the median is the score at the (n + 1)/2 position, where n is the number of data points.
In this case, n = 15, so the median is the score at the (15 + 1)/2 = 8th position, which corresponds to the 8th value in the sorted list.
Therefore, Q3 = 82.
Step 5: Calculate the interquartile range (IQR):
The interquartile range is the difference between the third quartile (Q3) and the first quartile (Q1).
IQR = Q3 - Q1 = 82 - 70 = 12.
Therefore, the interquartile range (IQR) of the given scores is 12.