Question

a poll asks shoppers at the mall how many cups of water they typically drink in a day. the results of the poll are listed below.
3, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 11
what is value of the first quartile?
3
6
8
11

Explanation:

Step1: Count the number of data points

There are \(n = 21\) data points.

Step2: Calculate the position of the first quartile

The formula for the position of the first quartile \(Q_1\) is \(i=\frac{n + 1}{4}\). Substituting \(n = 21\), we get \(i=\frac{21+ 1}{4}=\frac{22}{4}=5.5\).

Step3: Find the value of the first quartile

When \(i\) is not an integer, we use linear interpolation. The \(5^{th}\) value is \(6\) and the \(6^{th}\) value is \(6\). Using the formula \(Q_1=x_{j}+(x_{j + 1}-x_{j})\times (i - j)\) where \(j = 5\) (the integer part of \(i = 5.5\)), \(x_{j}=6\), \(x_{j + 1}=6\) and \(i-j=0.5\). Then \(Q_1=6+(6 - 6)\times0.5=6\)

Answer:

6