Question

1.66
2.86
2.37
2.09
1.91
2.18
3.79
1.69
3.89
2.74
2.48
3.83
1.74
3.18
(a) find the first quartile.
(b) find the third quartile.
(c) find the mean.

Explanation:

Step1: Sort the data

First, we sort the given data set: \(1.66,1.74,1.91,2.09,2.18,2.37,2.48,2.74,2.86,3.18,3.79,3.83,3.89\)

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

The formula for the position of \(Q_1\) is \(i=\frac{n + 1}{4}\), where \(n = 13\) (number of data points).
\(i=\frac{13+1}{4}=3.5\)
\(Q_1=x_3+0.5(x_4 - x_3)\)
\(x_3 = 1.91\), \(x_4=2.09\)
\(Q_1=1.91+0.5(2.09 - 1.91)=1.91 + 0.09=2.0\)

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

The formula for the position of \(Q_3\) is \(i=\frac{3(n + 1)}{4}\)
\(i=\frac{3\times(13 + 1)}{4}=10.5\)
\(Q_3=x_{10}+0.5(x_{11}-x_{10})\)
\(x_{10}=3.18\), \(x_{11}=3.79\)
\(Q_3=3.18+0.5(3.79 - 3.18)=3.18+0.305 = 3.485\)

Step4: Find the mean (\(\bar{x}\))

The formula for the mean is \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}\)
\(\sum_{i=1}^{13}x_i=1.66+1.74+1.91+2.09+2.18+2.37+2.48+2.74+2.86+3.18+3.79+3.83+3.89 = 34.62\)
\(\bar{x}=\frac{34.62}{13}\approx2.663\)

Answer:

(a) \(2.0\)
(b) \(3.485\)
(c) \(2.663\)