Question

suppose the data represent the inches of rainfall in april for a certain city over the course of 20 years.
determine the quartiles.
0.85 1.95 3.55 5.22
1.17 2.29 3.89 5.58
1.42 2.63 4.17 5.97
1.58 2.74 4.58 6.29
1.83 3.09 4.88 6.75
what are the quartiles?
$q_1 = \square$
(round to three decimal places as needed.)

Explanation:

Step1: Sort the data set

First, arrange all 20 data points in ascending order:
$0.85, 1.17, 1.42, 1.58, 1.83, 1.95, 2.29, 2.63, 2.74, 3.09, 3.55, 3.89, 4.17, 4.58, 4.88, 5.22, 5.58, 5.97, 6.29, 6.75$

Step2: Find median (Q2) to split data

Since $n=20$ (even), the median is the average of the 10th and 11th values. This splits the data into lower and upper halves (each with 10 values):
Lower half: $0.85, 1.17, 1.42, 1.58, 1.83, 1.95, 2.29, 2.63, 2.74, 3.09$
Upper half: $3.55, 3.89, 4.17, 4.58, 4.88, 5.22, 5.58, 5.97, 6.29, 6.75$

Step3: Calculate Q1 (lower quartile)

Q1 is the median of the lower half. For 10 values, average the 5th and 6th values:
$$Q_1 = \frac{1.83 + 1.95}{2}$$
$$Q_1 = \frac{3.78}{2} = 1.890$$

Step4: Calculate Q2 (median)

Q2 is the average of the 10th and 11th values of the full sorted set:
$$Q_2 = \frac{3.09 + 3.55}{2}$$
$$Q_2 = \frac{6.64}{2} = 3.320$$

Step5: Calculate Q3 (upper quartile)

Q3 is the median of the upper half. For 10 values, average the 5th and 6th values of the upper half:
$$Q_3 = \frac{4.88 + 5.22}{2}$$
$$Q_3 = \frac{10.1}{2} = 5.050$$

Answer:

$Q_1 = 1.890$, $Q_2 = 3.320$, $Q_3 = 5.050$