Section 3.3 Notes: Dotplots, Stemplots, and Time-Series Plots

Numerical Data and Types

  • Numerical data is obtained when the answer to a question is a number.
  • Discrete Data:
    • Data with gaps between possible values.
    • Examples:
      • Number of people in a room.
      • Number of states in America.
      • Amount of money someone has (finite decimals, typically 2).
  • Continuous Data:
    • Data with no gaps between possible values.
    • Examples:
      • Someone's height to an EXACT measurement.
      • Amount of water in the ocean to an EXACT measurement (possibility of infinite decimals).

Dotplots

  • Dotplots are used to visualize the frequency of data points.
  • Example: Ages of students in a class.
    • Given data: 18, 18, 24, 20, 19, 19, 17, 22, 18, 18, 20, 28, 20, 20, 18, 19, 19, 21, 25, 20, 24, 21, 21, 20, 22

Dotplot Analysis

  1. Frequency of 21-year-olds: 3
  2. Age with the largest frequency: 20 years old
  3. Proportion of observations between 20 and 22 (inclusive): 11/25
  4. Proportion of observations at least 24 years old: 4/25
  5. Outlier: 28 years old
  6. Percentile of a 21-year-old:
    • 16 students are younger than 21.
    • Adding one 21-year-old gives 17 students.
    • Percentile = 17/25 = 0.68 = 68%

Percentiles

  • Percentile: A measure of the percentage of data below a given point.
  • Example: A student in the 80th percentile performed better than 80% of other students.
  • How to find the percentile:
    • Add up the number of data points below the target spot.
    • Add 1 for the data point at the target spot.
    • Divide this sum by the total number of data points.

Stemplots

  • Stemplots are used to display quantitative data in a format similar to a histogram.
  • Example: Highway gas mileages of Kia car models. Data is considered discrete due to gaps between possible values.
    • Data: 28, 39, 29, 37, 24, 35, 26, 36, 31, 24, 34, 34, 30, 30, 31, 34, 24, 30, 29, 29, 37, 28, 31, 34, 26, 36, 24, 40

Stemplot Construction

  • Order the data from smallest to largest. Separate each observation into a stem and a leaf. The leaf contains the last digit.
  • Stem Leaf: Represents the first digit(s) followed by the last.
  • Split Stem: A stemplot where each stem is divided into two, one for leaves 0-4 and the other for 5-9.

Example Stemplot and Split Stemplot:

  • Stemplot:

    • Stem | Leaf
    • 2 | 4 4 4 4 6 6 8 8 9 9 9
    • 3 | 0 0 0 1 1 1 4 4 4 4 5 6 6 7 7 9
    • 4 | 0

  • Split Stemplot:

    • Stem | Leaf
    • 2 | 4 4 4 4
    • 2 | 6 6 8 8 9 9 9
    • 3 | 0 0 0 1 1 1 4 4 4 4
    • 3 | 5 6 6 7 7 9
    • 4 | 0

Stemplot Analysis

  1. Best gas mileage (Optima Hybrid): 40 mpg
  2. Percentile of 26 mpg:
    • 4 cars have less than 26 mpg.
    • Adding one for the 26 mpg car gives 5 cars.
    • Percentile = 5/28 = 0.179 = 17.9
  3. 50th percentile:
    • (Total amount) * (percentile) = 28 * 0.5 = 14th data point
    • The 14th smallest data point is 30 mpg.

Time-Series Plots

  • Time-series plots display data points in chronological order. They are used to analyze trends over time.
  • Example: Percentages of U.S. households with only a cell phone (no landline) for various years.

Time-Series Plot Construction

  • Plot the data points with time on the x-axis and the variable of interest on the y-axis.

Example Data:

YearPercent
20032
20058
200713
200922
201131
201339

Time-Series Plot Analysis

  1. Trend: The percentage of households with only a cell phone is increasing over time.
    • Graphs are read from left to right. An upward trend signifies an increase and a downward trend a decrease.
  2. Greatest change over a two-year period: 9% (between 2007-2009 and 2009-2011).
  3. Change in percent over the entire time period (10 years): 37%.
  4. Prediction for 2023:
    • Based on the trend, the predicted percentage would be 39% (in 2013) + 37% (change over 10 years) = 76%.