Detailed Study Notes on Box-Whisker Plot

Lecture Overview

Course: W26 QMS 230
Topic: Box-Whisker Plot
Instructor: Dr. Boža Tasić
Date: January 18, 2026

Learning Objectives - Box-Whisker Plot

  • Introduction to a new graphical method for presenting data: Box-Whisker Plot.

  • Instruction on using the Casio FX-9750 calculator to graph a Box-Whisker Plot.

  • Establish a connection between the shape of the data distribution and the Box-Whisker Plot representation of a data set.

Box-Whisker Plot Definition

  • Context: Discussion builds on previous measures of central tendency and variability.

  • Functionality:

    • Box-Whisker Plot is a graphical representation summarizing a numerical data set.

    • It allows for a visualization of key statistical measures encapsulated in a five-number summary.

Five-Number Summary

  • The Five-Number Summary is essential for constructing a Box-Whisker Plot and consists of the following elements:

    1. Minimal data set value: $X_{minimum}$

    2. First quartile (Q1): The value below which 25% of the data fall.

    3. Median (Q2): The middle value that divides the data set into two equal halves.

    4. Third quartile (Q3): The value below which 75% of the data fall.

    5. Maximal data set value: $X_{maximum}$

  • Application: The five-number summary provides the foundation for constructing the Box-Whisker Plot.

Box-Whisker Plot Construction Steps

  • Requirements: No outliers or suspect data values.

  • Steps to Construct:

    1. Draw the scale: Create an evenly spaced scale that fully encompasses all data values.

    2. Construct the box:

    • Use the identified Q1 and Q3 as the left and right sides of the box.

    • Join the vertical sides with horizontal sides to form a rectangle.

    1. Add the median:

    • Draw a vertical line across the box to signify the median (Q2).

    1. Draw the left whisker:

    • Connect Q1 to the minimum value $X_{minimum}$ via a horizontal line.

    1. Draw the right whisker:

    • Connect Q3 to the maximum value $X_{maximum}$ via a horizontal line.

    1. Indicate the mean:

    • Mark the mean ($ar{x}$) with a '+' symbol on the diagram.

Identifying Outliers

  • Definitions:

    • Suspect Outliers: Data points that lie between the inner and outer fences, represented with a circle (◦).

    • Outliers: Data points that lie outside the outer fences, represented with an asterisk (∗).

Determination of Fences

  • Inner and Outer Fences Defined:

    • Right Inner Fence (RIF):
      RIF=Q3+(1.5imesIQR)RIF = Q3 + (1.5 imes IQR)

    • Right Outer Fence (ROF):
      ROF=RIF+(1.5imesIQR)ROF = RIF + (1.5 imes IQR)

    • Left Inner Fence (LIF):
      LIF=Q1−(1.5imesIQR)LIF = Q1 - (1.5 imes IQR)

    • Left Outer Fence (LOF):
      LOF=LIF−(1.5imesIQR)LOF = LIF - (1.5 imes IQR)

  • Interquartile Range (IQR):

    • Defined as:
      IQR=Q3−Q1IQR = Q3 - Q1

  • Note: RIF and ROF represent values on the right; LIF and LOF represent values on the left of the box, calculated but not plotted.

Whisker Length Determination

  • Whiskers are restricted by the following rules:

    1. They cannot extend beyond the inner fences.

    2. They must terminate at a data value.

  • Application of Rules:

    • Left Whisker: Ends at a minimum value greater than the left inner fence (LIF).

    • Right Whisker: Ends at a maximum value less than the right inner fence (RIF).

Box-Whisker Plot with Outliers - Example

  • Sample data representing days absent for 50 employees is analyzed.

  • Five-Number Summary for the Data:

    • Minimal value $X_{minimum} = 1$

    • First quartile $Q1 = 13$

    • Median $Q2 = 27.5$

    • Third quartile $Q3 = 52$

    • Maximal value $X_{maximum} = 173$

  • Additional Statistics:

    • Mean is $ar{x} = 40.7$

    • IQR calculated as $IQR = 39$.

Calculation of Fences (Continued)

  • Calculating Left Inner Fence (LIF):

    • LIF=Q1−(1.5imesIQR)=13−(1.5imes39)=−45.5LIF = Q1 - (1.5 imes IQR) = 13 - (1.5 imes 39) = -45.5

  • Calculating Left Outer Fence (LOF):

    • LOF=LIF−(1.5imesIQR)=−45.5−(1.5imes39)=−104LOF = LIF - (1.5 imes IQR) = -45.5 - (1.5 imes 39) = -104

  • Calculating Right Inner Fence (RIF):

    • RIF=Q3+(1.5imesIQR)=52+(1.5imes39)=110.5RIF = Q3 + (1.5 imes IQR) = 52 + (1.5 imes 39) = 110.5

  • Calculating Right Outer Fence (ROF):

    • ROF=RIF+(1.5imesIQR)=110.5+(1.5imes39)=169ROF = RIF + (1.5 imes IQR) = 110.5 + (1.5 imes 39) = 169

Visual Representation of the Box-Whisker Plot

  • The plot displays suspect outliers (◦) and true outliers (∗) based on calculations.

Using CASIO Calculator for Box-Whisker Plot

  • Steps to Graph Using CASIO FX-9750:

    1. Select the STAT mode from the main menu and press EXE.

    2. Input data into List 1.

    3. Press F1 to select GRPH.

    4. Press F6 to select SET and input the following:

    • Graph Type: F2 (Box)

    • XList: List 1

    • Frequency: F1 (1)

    • Outliers: F1 (On)

    1. Press EXE to confirm.

    2. View the plot: Press F1 to select GPH1 or F6 for DRAW to return to the plot.

Data Analysis with Box-Whisker Plot

  • The five-number summary assists in describing the center, spread, and shape of the data set.

  • Insights about distribution type:

    • Data analysis for the number of days absent shows relationships visualizing the skewness through computations:

    • Comparisons:

      • $Q2 - X{min} = 26.5$ and $X{max} - Q2 = 145.5$ indicates right-skewness.

      • $Q1 - X{min} = 12$ and $X{max} - Q3 = 121$ also indicates right skewness.

      • $Q2 - Q1 = 14.5$ and $Q3 - Q2 = 24.5$ confirm the distributions.

  • Conclusion: - The analysis indicates a right-skewed data distribution.

Shape Relationship

  • Visual representations illustrate the correspondence between distribution shape and Box-Whisker Plot characteristics.