Organizing Quantitative Data: Histograms and Stem and Leaf Plots

Organizing Quantitative Data

Discrete vs. Continuous Data

• Discrete Data:
  • Countable, using numbers 0, 1, 2, …, n.
  • Examples:
    • Number of people entering a store (8 AM - 10 AM).
    • Number of patients in a hospital due to traffic accidents.
    • Number of calls made per day.
    • Number of times a pair of dice is rolled with a sum of 8.

Histograms

• Similar to bar graphs, used for organizing discrete and continuous data.
• Definition: A histogram is constructed by drawing rectangles for each class (category).
  • The height of each rectangle represents the frequency or relative frequency of the class.
  • The width of each rectangle is the the same.
  • All rectangles touch each other.
• Class Determination:
  • For qualitative and many discrete data sets, classes are determined directly from the data.
  • For continuous data, intervals of numbers are constructed (e.g., 20-29, 30-39, …).

Example: Histogram for Discrete Data

• Experiment: Two fair dice thrown 100 times; sum of pips recorded.
• Data:
  • Values of the Dice: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
  • Frequencies are listed for a sample dataset.
• Construction:
  • Variable (dice sum) on the x-axis.
  • Corresponding frequency on the y-axis.

Histogram for Continuous Data

• Divide the real number line into non-overlapping intervals.
  • Examples: [0, 1), [1, 2), [2, 3), … or [0, 0.99], [1, 1.99], [2, 2.99], …

Example: Earthquake Magnitudes in 2007 (Magnitude < 8.0)

• Data:
  • Magnitude Intervals and Corresponding Number of Earthquakes
    • [0, 0.9]: 1,600
    • [1, 1.9]: 33
    • [2, 2.9]: 2,983
    • [3, 3.9]: 8,028
    • [4, 4.9]: 10,465
    • [5, 5.9]: 1,466
    • [6, 6.9]: 151
    • [7, 7.9]: 11

Class Width Calculation

• To determine the width of each class, use the formula:
  • Class Width = \frac{Largest Data Value - Smallest Data Value}{Number of Classes}

Stem and Leaf Plots

Displaying Data

• Method for grouping and displaying data.

• Example Data Set: 8.2, 8.4, 8.9, 9.0, 9.5, 10.2, 10.3, 10.5, 10.7, 10.8

• Stem and Leaf Representation:

  • Stem: Represents the leading digit(s).
  • Leaf: Represents the trailing digit.

• Example:

  • 8 | 2, 4, 9
  • 9 | 0, 5
  • 10 | 2, 3, 5, 7, 8

Examples and Exercises

Internet Usage

• A random sample of 30 high school students is selected.
• Each student is asked how much time he or she spent on the Internet during the previous week.
• Times (in hours) recorded: (List of values given in transcript)
• Task: Construct a frequency histogram for the given data.

June Precipitation

• June precipitation amounts (in inches) for 40 cities.
• Data provided (List of values given in transcript).
• Task: Construct a frequency distribution and a relative frequency distribution using eight classes.

Almond Weights

• Table summarizing weights of almonds (in grams) in a one-pound bag.

• Data provided in terms of Weight (g) ranges and corresponding Frequencies.

• Task: Find the class width of the classes.

  • Weight (g) Frequency
  • 0. 7585-0.8184 1
  • 0. 8185-0.8784 1
  • 0. 8785-0.9384 1
  • 0. 9385-0.9984 3
  • 0. 9985-1.0584 157
  • 1. 0858-1.1184 171
  • 1. 1185-1.1784 8

Utility Bills

• March utility bills (in dollars) of 30 homeowners.
• Data: (List of values given in transcript).
• Task: Construct a stem-and-leaf plot for the data.
  • 44 38 41 50 36 36 43 42 49 48
  • 35 40 37 41 43 50 45 45 39 38
  • 50 41 47 36 35 40 42 43 48 33