Question

find the quartiles for the following set of data. data 3 7 7 11 13 14 16 19 21 22 q1 = q2 = q3 =

Explanation:

Step1: Arrange data in order

The data is already in ascending - order: 3, 7, 7, 11, 13, 14, 16, 19, 21, 22. The number of data points $n = 10$.

Step2: Calculate the position of $Q_1$

The formula for the position of $Q_1$ is $i_1=\frac{1}{4}(n + 1)$. Substitute $n = 10$ into the formula: $i_1=\frac{1}{4}(10 + 1)=\frac{11}{4}=2.75$. So, $Q_1$ is the value at the 2.75 - th position. Interpolate between the 2nd and 3rd values. The 2nd value is 7 and the 3rd value is 7. $Q_1=7+(7 - 7)\times0.75 = 7$.

Step3: Calculate the position of $Q_2$

The formula for the position of $Q_2$ (the median) is $i_2=\frac{1}{2}(n + 1)$. Substitute $n = 10$ into the formula: $i_2=\frac{1}{2}(10 + 1)=5.5$. Interpolate between the 5th and 6th values. The 5th value is 13 and the 6th value is 14. $Q_2=13+(14 - 13)\times0.5=13.5$.

Step4: Calculate the position of $Q_3$

The formula for the position of $Q_3$ is $i_3=\frac{3}{4}(n + 1)$. Substitute $n = 10$ into the formula: $i_3=\frac{3}{4}(10 + 1)=\frac{33}{4}=8.25$. Interpolate between the 8th and 9th values. The 8th value is 19 and the 9th value is 21. $Q_3=19+(21 - 19)\times0.25=19.5$.

Answer:

$Q_1 = 7$
$Q_2 = 13.5$
$Q_3 = 19.5$