Box Plots Notes

5.3 Box Plots

Introduction

• A box plot visually represents the distribution of a dataset along a number line, dividing the data into four parts using the median and quartiles.
• The term "box plot" often refers to an outlier box plot, also known as a box-and-whisker plot or a Tukey box plot.
• John Tukey introduced the box-and-whisker plot in 1969 to visually represent the "Five Number Summary" of a dataset.
• Box plots can be oriented vertically or horizontally.

Parts of Box Plots

• A box plot consists of a box and two whiskers.

Box

• The ends of the box represent the upper (Q3) and lower (Q1) quartiles, spanning the interquartile range (IQR).
• A vertical line inside the box indicates the median (Q2).

Whiskers

• The whiskers are the lines extending from the box to the highest and lowest observations within a certain range.

Anatomy

• A box-and-whisker plot visually represents the five-number summary of a dataset:
  • Minimum Value
  • First Quartile (Q1)
  • Median (Second Quartile, Q2)
  • Third Quartile (Q3)
  • Maximum Value

Five-Number Summary

• Minimum: The smallest value in the dataset.
• First Quartile (Q1): The median of the lower half of the dataset.
• Median (Q2): The middle value of the dataset, dividing it into two equal parts. It is also the second quartile.
• Third Quartile (Q3): The median of the upper half of the dataset.
• Maximum: The largest value in the dataset.

Other Terms

• Interquartile Range (IQR): The difference between the third and first quartiles. $IQR = Q3 - Q1$
• Outlier: Data points that lie far away from the rest of the data. Outliers are tested to be data points that fall outside a specified distance from the first and third quartiles.
• Outlier Rule: Outliers are values greater than $Q3 + (1.5 \, IQR)$ or less than $Q1 - (1.5 \, IQR)$.

Box Plot Representation

• Visual representation of key statistics:
  • Outliers are marked individually outside the whiskers.
  • The box spans from Q1 to Q3.
  • The median is marked within the box.
  • Whiskers extend to the minimum and maximum values within the outlier range.

Applications of Box Plots

1. Spread of Data: Box plots provide a good indication of how the values in the data are spread out.
2. Skewness: Box plots are useful for showing the skewness of a dataset.
3. Dispersion: Box plots are useful for showing the dispersion of a dataset.
4. Outliers: Box plots are useful for identifying outliers within a dataset.

Box Plot Distribution

• Positively Skewed: If the distance from the median to the maximum value is greater than the distance from the median to the minimum value, the box plot is positively skewed.
• Negatively Skewed: If the distance from the median to the minimum value is greater than the distance from the median to the maximum value, the box plot is negatively skewed.
• Symmetric: If the median is equidistant from the maximum and minimum values, the box plot is symmetric.

Steps in Making a Box-and-Whisker Plot

• Example Data: 21, 25, 15, 13, 17, 19, 19, 21

Step 1: Order Data and Find Key Values

1. Order the data from least to greatest: 13, 15, 17, 19, 19, 21, 21, 25
2. Identify the minimum and maximum values:
   • Minimum: 13
   • Maximum: 25
3. Calculate the quartiles and median:
   • Lower Quartile (Q1): $\frac{15 + 17}{2} = 16$
   • Upper Quartile (Q3): $\frac{21 + 21}{2} = 21$
   • Median (Q2): $\frac{19 + 19}{2} = 19$

Step 2: Draw a Number Line and Plot Points

1. Draw a number line and plot a point above each of the key values (minimum, Q1, median, Q3, maximum).

Step 3: Draw the Box and Whiskers

1. Draw the box extending from Q1 to Q3, with a line indicating the median.
2. Draw the whiskers extending from the box to the minimum and maximum values.