Question

bell ringer - sep 18
find the first and third quartiles of the following data sets.

1. 25 56 78 45 39 80 70 65 90 29
2. 112 456 650 690 349 268 400 322

Explanation:

Step1: Sort the first data - set

First, sort the data set \(25,56,78,45,39,80,70,65,90,29\) in ascending order: \(25,29,39,45,56,65,70,78,80,90\).

Step2: Find the position of the first quartile (\(Q_1\))

The formula to find the position of \(Q_1\) for a data - set of size \(n\) is \(i=\frac{n + 1}{4}\). Here \(n = 10\), so \(i=\frac{10+1}{4}=2.75\). The first quartile is \(Q_1=39+(0.75)\times(45 - 39)=39 + 4.5=43.5\).

Step3: Find the position of the third quartile (\(Q_3\))

The formula to find the position of \(Q_3\) is \(i=\frac{3(n + 1)}{4}\). For \(n = 10\), \(i=\frac{3\times(10 + 1)}{4}=8.25\). The third quartile is \(Q_3=78+(0.25)\times(80 - 78)=78 + 0.5=78.5\).

Step4: Sort the second data - set

Sort the data set \(112,456,650,690,349,268,400,322\) in ascending order: \(112,268,322,349,400,456,650,690\).

Step5: Find the position of the first quartile (\(Q_1\)) for the second data - set

Here \(n = 8\), so \(i=\frac{n + 1}{4}=\frac{8 + 1}{4}=2.25\). The first quartile is \(Q_1=268+(0.25)\times(322 - 268)=268+13.5 = 281.5\).

Step6: Find the position of the third quartile (\(Q_3\)) for the second data - set

The formula for the position of \(Q_3\) gives \(i=\frac{3(n + 1)}{4}=\frac{3\times(8 + 1)}{4}=6.75\). The third quartile is \(Q_3=456+(0.75)\times(650 - 456)=456+145.5 = 601.5\).

Answer:

1. First quartile (\(Q_1\)): \(43.5\), Third quartile (\(Q_3\)): \(78.5\)
2. First quartile (\(Q_1\)): \(281.5\), Third quartile (\(Q_3\)): \(601.5\)