Question

1. b) given the following data, create a box plot. show all calculations and clearly label all parts. 20, 28, 35, 40, 45, 52, 60, 60, 65, 72, 80, 90, 92, 95, 100

Explanation:

Step1: Confirm sorted data

Data (already sorted): $20, 28, 35, 40, 45, 52, 60, 60, 65, 72, 80, 88, 92, 95, 100$
Count: $n=15$

Step2: Find minimum and maximum

Minimum ($Min$) = smallest value: $20$
Maximum ($Max$) = largest value: $100$

Step3: Calculate median (Q2)

Median = value at position $\frac{n+1}{2} = \frac{15+1}{2}=8$
8th value: $60$

Step4: Calculate Q1 (lower quartile)

Q1 = median of lower half (first 7 values: $20,28,35,40,45,52,60$)
Position: $\frac{7+1}{2}=4$
4th value: $40$

Step5: Calculate Q3 (upper quartile)

Q3 = median of upper half (last 7 values: $65,72,80,88,92,95,100$)
Position: $\frac{7+1}{2}=4$
4th value: $88$

Step6: Calculate IQR

$IQR = Q3 - Q1 = 88 - 40 = 48$

Step7: Identify outliers (none here)

Lower fence: $Q1 - 1.5\times IQR = 40 - 1.5\times48 = -32$
Upper fence: $Q3 + 1.5\times IQR = 88 + 1.5\times48 = 160$
All data points are within $-32$ to $160$, so no outliers.

Step8: Define box plot components

• Whiskers: Extend from $20$ (min) to $40$ (Q1), and $88$ (Q3) to $100$ (max)
• Box: Spans from $40$ (Q1) to $88$ (Q3), with a line at $60$ (median)

Answer:

Box plot key values:

• Minimum: $20$
• First Quartile (Q1): $40$
• Median (Q2): $60$
• Third Quartile (Q3): $88$
• Maximum: $100$
• Interquartile Range (IQR): $48$

The box plot has:

1. A horizontal box from 40 to 88, with a vertical line at 60 (median)
2. Left whisker from 20 to 40
3. Right whisker from 88 to 100
4. No outliers, as all data falls within the range $-32$ to $160$