Box plots

Overview of Box Plots

• Box plots are a method used for summarizing and comparing distributions.

• They are particularly useful for identifying extreme values known as outliers.

Data Background

• An experiment involving students in introductory statistics presented 30 colored rectangles for them to name as quickly as possible.

• Times taken to name the colors are recorded.

• Participants consisted of 16 men and 31 women.

Constructing Box Plots

Basic Structure of Box Plots

1. Draw the y-axis.

2. Identify and draw the 25th and 75th percentiles (quartiles) for each gender:

   • Women:

     • 25th percentile: 17

     • 75th percentile: 20

   • Men:

     • 25th percentile: 19

     • 75th percentile: 25.75

3. Connect the lines to form boxes representing each gender's distribution.

Identifying the Median

• Median computation: The median is located between the 25th and 75th percentiles:

  • Women: Median = 19.5

  • Men: Median = 22.5

• Observation: Men took longer than women to name the colors because their percentiles are higher.

Box Plot Terminology

Hinge Variables

• Upper hinge: 75th percentile.

• Lower hinge: 25th percentile.

• H spread: Difference between upper and lower hinges. Calculated as:

  • For women:
    $H ext{ spread} = ext{Upper hinge} - ext{Lower hinge} = 20 - 17 = 3$

Steps and Fences

• Step: Defined as 1.5 times the H spread.

  • For women, Step = $1.5 imes H ext{ spread} = 1.5 imes 3 = 4.5$

• Inner Fences:

  • Upper inner fence = Upper hinge + Step = $20 + 4.5 = 24.5$

  • Lower inner fence = Lower hinge - Step = $17 - 4.5 = 12.5$

• Outer Fences:

  • Upper outer fence = Upper hinge + 2 imes Step = $20 + 2 imes 4.5 = 29$

  • Lower outer fence = Lower hinge - 2 imes Step = $17 - 2 imes 4.5 = 8$

Adjacent Values and Whiskers

Adjacent Values

• Upper adjacent value: The largest value below the upper inner fence:

  • For women, the value is 24.

• Lower adjacent value: The lowest value above the lower inner fence:

  • For women, the value is 14.

Drawing Whiskers

• Whiskers: Vertical lines ending in a horizontal stroke:

  • For women, one whisker extends from 20 (upper hinge) to 24 (upper adjacent value) and another from 17 (lower hinge) to 14 (lower adjacent value).

Identifying Outliers

• Outside values: Values beyond the inner fence but within the outer fence:

  • For women: The upper inner fence is 24.5.

  • The value 29 meets this criterion and is considered an outside value. (Marked by small 'o')

• Far Out Values: Values beyond the outer fences:

  • No far-out values present in this data. (Marked by asterisks)

Including the Mean Score

• The mean score for the group is often indicated with a plus sign.

Summary of Observations

• The box plot effectively summarizes the data:

  • For women, half the times range between 17 and 20.

  • For men, half the times range between 19 and 25.

  • It is noted that women generally named the colors faster than men, despite one slower individual among women being slower than almost all men.

  • The median time for women (19.5) is close to the 25th percentile for men (19), indicating that a woman slower than half the women still performs better than about 25% of the men.

Conclusion

• Key Terms Recap:

  • Upper hinge, lower hinge, H spread, step, inner fences, outer fences, upper/lower adjacent values, whiskers, outside values, far out values, and the mean score.