Question

Consider the following two sample data sets.

Sample A	Sample B
121, 171, 158, 173, 184, 163, 157, 85, 145, 165, 172, 196, 170, 159, 172, 161, 187, 100, 142, 166, 171	171, 152, 170, 168, 169, 171, 190, 183, 185, 140, 173, 206, 172, 174, 169, 199, 151, 180, 167, 170, 188

a.Construct a box plot for each data set.

b.Identify any outliers that may exist in the two data sets.

Short answer

(a) Sample A

Sample B

(b) Sample A = 85, 100, 121

Sample B = 140

Step-by-step solution

Calculating the values needed to construct a box plot for Sample A

Arranging the data in ascending order,

(85,100,121,142,145,157,158,159,161,163,165,166,170,171,171,172,172,173,184,187,196)

$$\begin{gathered} Q1=\frac{N+1}{4}=\frac{21+1}{4}=\frac{22}{4}=5.5^{th}term=151 \\ Q2=\frac{N+1}{2}=\frac{21+1}{2}=\frac{22}{2}={11}^{th}term=165 \\ Q3=\frac{3\left(N+1\right)}{4}=\frac{3\left(21+1\right)}{4}=\frac{66}{4}=16.5^{th}term=172 \end{gathered}$$

IQR = Q_U – Q_L

= 172 – 151

= 21

Lower Inner Fence = Q_L - 1.5(IQR)

= 151 – 1.5(21)

= 151 – 25.2

= 125.8

Upper Inner Fence = Q_U + 1.5(IQR)

= 172 + 1.5(21)

= 172 + 25.2

= 197.2

Median = 165, Q_U = 172, Q_L = 151, LIF = 125.8, UIF = 197.2

Creating the box plot for Sample A

Calculating the values needed to construct a box plot for Sample A

We will first arrange the data in ascending order,

(140,151,152,167,168,169,169,170,170,171,171,172,173,174,180,183,185,188,190,199,206)$$\begin{gathered} Q1=\frac{N+1}{4}=\frac{21+1}{4}=\frac{22}{4}=5.5^{th}term=168.5 \\ Q2=\frac{N+1}{2}=\frac{21+1}{2}=\frac{22}{2}={11}^{th}term=171 \\ Q3=\frac{3\left(N+1\right)}{4}=\frac{3\left(21+1\right)}{4}=\frac{66}{4}=16.5^{th}term=184 \end{gathered}$$

IQR = Q_U – Q_L

= 184 – 168.5

= 15.5

Lower Inner Fence = Q_L - 1.5(IQR)

= 168.5 – 1.5(15.5)

= 168.5 – 23.25

= 145.25

Upper Inner Fence = Q_U + 1.5(IQR)

= 184 + 1.5(15.5)

= 184 + 23.25

= 207.25

Median = 171, Q_U = 184, Q_L = 168.5, LIF = 145.25, UIF = 207.25

Constructing the box plot for Sample B

Identifying the outliers

Sample A = 85, 100, 121

Sample B = 140