Answer:
The first quartile (Q1) is 19, and the third quartile (Q3) is 28.5 for the given data set.
Explanation:
To find the first and third quartiles of the given data set, you first need to arrange the data in ascending order. Here's the sorted data set:
6, 15, 18, 20, 21, 22, 24, 25, 26, 28, 29, 30
The first quartile (Q1) represents the 25th percentile, and the third quartile (Q3) represents the 75th percentile of the data.
To find Q1:
Calculate the index (k) corresponding to the 25th percentile: k = (25/100) * (n + 1), where n is the number of data points.
If k is an integer, Q1 is the value at the kth position in the sorted data set.
If k is not an integer, round it up to the nearest whole number to get the position of the value (k_ceiling). Q1 will be the average of the values at positions k_ceiling and k_ceiling + 1.
To find Q3:
Calculate the index (k) corresponding to the 75th percentile: k = (75/100) * (n + 1).
If k is an integer, Q3 is the value at the kth position in the sorted data set.
If k is not an integer, round it up to the nearest whole number to get the position of the value (k_ceiling). Q3 will be the average of the values at positions k_ceiling and k_ceiling + 1.
Let's calculate Q1 and Q3:
Q1:
k = (25/100) * (12 + 1) = 3.25
k_ceiling = ceil(3.25) = 4
Q1 = (18 + 20) / 2 = 38 / 2 = 19
Q3:
k = (75/100) * (12 + 1) = 9.75
k_ceiling = ceil(9.75) = 10
Q3 = (28 + 29) / 2 = 57 / 2 = 28.5
So, the first quartile (Q1) is 19, and the third quartile (Q3) is 28.5 for the given data set.