Frequency Distributions and Graphs

• Frequency Distribution


  A group of data into categories showing the number of observations in each of the non-overlapping classes
• Tabular Form


  • Mutually Exclusive & Exhaustive
• Grouped Frequency Distribution


  Used when the range of the data set is large. The data are grouped into classes
• Grouped Frequency Distribution


  • Categorical
  • Interval or Ratio
• Constructing Frequency Distribution


  1. Grouped Frequency

  2. Categorical Frequency
• Categorical Frequency


  Used to organize nominal-level or ordinal-level type of data
• Examples of Categorical Frequency

  • Gender
  • Political affiliation
  • Business type
  • Year level
• Constructing Grouped Frequency Distribution

  1. Step 1: Construct a table

  2. Step 2: Tally the raw data

  3. Step 3: Convert the tallied data into numerical frequencies

  4. Step 4: Determine the percentage
• Percentage Formula


  Frequency of the class / Total number of values * 100
• Determining Class Interval


  1. Rule 1: 2k ≥ n, where k = number of classes, n = total number of values

  2. Rule 2: i = (Range) / (3.322 log N + 1), where i = class interval, N = total number of values

  3. Rule 3: i = (Range) / (Number of classes)
• Example 2: Constructing Grouped Frequency Distribution
• Steps in Example 2

  • Arrange raw data in ascending/descending order
  • Determine number of classes using 2k ≥ n rule
  • Calculate class interval
  • Select starting point for lowest class limit
  • Determine lower and upper class limits
  • Tally raw data
  • Convert tallied data to numerical frequencies
  • Determine relative frequency
  • Determine percentage
  • Determine cumulative frequencies
  • Determine midpoints
• Example 3: Constructing Grouped Frequency Distribution
• Steps in Example 3

  1. Arrange raw data in ascending/descending order

  2. Determine number of classes using 3.322 log N + 1 rule

  3. Calculate class interval

  4. Select starting point for lowest class limit

  5. Determine lower and upper class limits

  6. Tally raw data

  7. Convert tallied data to numerical frequencies

  8. Determine relative frequency
• Lower Limit


  Smallest value in a class interval
• Upper Limit


  Largest value in a class interval
• Stem-and-Leaf plot


  A method that overcomes the loss of actual observations brought about by the histogram. The stem is the leading digit or digits, and the leaf is the trailing digit.
• Stem-and-Leaf plot was introduced by John Tukey
• The advantage of the stem-and-leaf plot over the histogram is that we can see the actual observations
• Histogram


  A graph in which the classes are marked on the horizontal axis (x-axis) and the class frequencies on the vertical axis (y-axis)
• Frequency Polygon


  A graph that displays the data using points which are connected by lines
• Cumulative Frequency Polygon (Ogive)


  A graph that displays the cumulative frequencies for the classes in a frequency distribution
• Pareto Chart


  A frequency distribution for a categorical data (or nominal-level) where frequencies are displayed by the heights of vertical bars, which are arranged in order from highest to lowest
• Bar Chart (Bar Graph)

  The bases of the rectangles are arbitrary intervals whose centers are the codes. The height of each rectangle represents the frequency of that category.
• Pie Chart (Circle Graph)


  A circle divided into portions that represent the relative frequencies (or percentages) of the data belonging to different categories
• Time Series Graph


  Represents data that occur over specific period of time under observation, showing a trend or pattern on the increase or decrease over the period of time
• Pictograph


  Appropriate pictures arranged in a row (sometimes in a column) present the quantities for comparison
• Scatter Plot


  Used to examine possible relationships between two numerical variables, where the two variables are plotted on the x-axis and y-axis
• Guidelines for Developing Graphs/Charts

  • The graph or chart should include a title
  • The scales for all axes should be included
  • The scale on the y-axis should start at zero
  • The graph or chart should not disfigure the data
  • The x-axis and y-axis should be properly labeled
  • The graph or chart should not contain unnecessary decorations
  • The simplest possible graph or chart should be used for any data set