Frequency Distribution and Graphical Methods Study Guide

Overview of Frequency Distribution and Grouped Data

• Frequency Distribution Defined: A frequency distribution is a systematic arrangement of data designed to show the frequency of different values or groups of values of a specific variable.
• Grouped Data: Data that has been organized into a frequency distribution format (classes or categories) is referred to as grouped data.
• Array: When data is arranged in either ascending or descending order, it is called an array.

Procedure for Constructing a Frequency Distribution

To construct an exhaustive frequency distribution, one must follow these eight sequential steps:

• Step 1: Determine the Range: Calculate the range ($R$) by finding the difference between the highest value ($H$) and the lowest value ($L$) in the data set.
  • Formula: $R = H - L$
• Step 2: Determine number of categories and Class Interval: Divide the range by an appropriate number to determine the number of classes or the size of the interval.
  • The most recommended number of categories or classes is between $10$ and $20$.
  • The divisor used becomes the class interval. It is possible to derive two or more potential intervals.
• Step 3: Preference for Odd Intervals: An odd number for the class interval is usually preferred. This facilitates the determination of the class midpoint (the middle value within a class).
• Step 4: Determine the Starting Point: Divide the highest value in the data set by the chosen interval and take note of the remainder.
• Step 5: Set the Topmost Lower Limit: Subtract the remainder found in Step 4 from the highest value. This difference serves as the starting point (the lower limit) of the topmost class.
  • Exceptions: If there is no remainder, the highest value itself becomes the starting point/lower limit of the topmost class.
• Step 6: Determine Class Limits: The upper limit of a class is determined by the size of the interval. Going downward from the top class, determine the lower and upper limits of each subsequent class by decreasing each limit by the value of the interval. Stop once you reach the class that contains the lowest value in the data set.
• Step 7: Tallying: Perform a tally by marking each individual data value within its appropriate class interval. This is traditionally recorded in a second column.
• Step 8: Calculate Frequencies: Count the number of tally marks in each class to find the frequency ($f$). List these counts in a third column.
• Step 9: Summation: Calculate the sum of the frequencies using the formula $\sum f$.
  • The capital Greek letter sigma ($\sum$) signifies "the sum of."
  • The total number of values is represented by $n$.
  • Verification Rule: $\sum f$ should always be equal to $n$.

Case Illustration: Student Typing Speeds

• Data Set Context: The data represents the average words per minute ($wpm$) typed by $50$ randomly assigned students after receiving a specific amount of instruction.
• Raw Data ($wpm$):
  • $48, 45, 39, 76, 44, 43$
  • $52, 72, 47, 42, 52, 47, 69, 61, 43, 36, 59, 44$
  • $26, 45, 40, 75, 46, 60, 52, 39, 67, 65, 35, 49, 46$
  • $62, 55, 49, 49, 37, 68, 71, 34, 46, 48, 55, 46, 40, 50, 56, 48, 45, 41, 72$
• Application of Procedure:
  • 1. The range: $76\,wpm - 26\,wpm = 50\,wpm$.
  • 2. Possible intervals: Divisors of $3$, $4$, or $5$ could be used. Divisors of $3, 4, 5$ yield $10, 13,$ and $17$ classes respectively.
  • 3. If utilizing an interval of $5$: $\frac{76}{5} = 15$ with a remainder of $1$.
  • 4. Starting point: $76 - 1 = 75$.
  • 5. Class Limits: The highest class is $75-79$. The lowest class is $25-29$.

Table 1: Frequency Distribution for the Typing Speed of 50 Students

Graphical Representation of Frequency Distributions

Three primary methods are used to represent frequency distributions visually:

• 1. Frequency Polygon: A graphical representation in the form of lines. The data points are plotted at the midpoint of the classes.
• 2. Histogram: A graphical representation in the form of vertical bars. The data points are plotted at the exact lower limits (or class boundaries) of the classes.
• 3. Ogive Curve: A graph representing the cumulative frequency distribution.
  • Less Than Ogive: Represents the "less than cumulative frequency."
  • More Than Ogive: Represents the "more than cumulative frequency."

Detailed Mechanics of the Frequency Polygon

• Axis Orientation:
  • Y-axis (Vertical): Plotted with frequency ($f$) values. The axis must accommodate units up to the highest frequency in the set (e.g., $15$ in the typing speed example).
  • X-axis (Horizontal): Plotted with the class midpoints (e.g., $22, 27, 32, 37, 42, 47, 52, 57, 62, 67, 72, 77, 82$).
• Plotting: Frequencies are plotted directly above the corresponding midpoints.
• Connecting Points: Once all points are logged, they are connected with straight lines.
• Anchoring: To anchor the graph at both ends to the x-axis, the creator assumes the existence of an additional class above the highest and below the lowest class, assigning these theoretical classes a frequency of $0$.

Detailed Mechanics of the Histogram

• Comparison to Polygon: Similar to the frequency polygon, but uses bars instead of lines and boundaries instead of midpoints.
• Plotting Points: Frequencies are plotted against the exact limits (also known as class boundaries) of the classes.
• Initial Step: Start with the exact lower limit of the lowest class. For the typing data, the first boundary is $24.5$.
• Bar Construction: For each class, a bar is marked with a height corresponding to the class frequency.
• Class Boundary Example: For the first class ($25-29$), the boundaries are $24.5 - 29.5$.

Cumulative Frequency Distribution

• Definition: A cumulative frequency distribution displays the accumulated frequency up to a certain limit.
• Procedure for "Less Than" Cumulative Frequency ($cf$):
  • 1. Start at the bottom (lowest class interval).
  • 2. The initial cumulative frequency for the lowest class is equal to its own frequency ($f$).
  • 3. Add this count to the frequency of the next interval above it.
  • 4. Continue the process of adding the current total to the next class frequency.
  • 5. Verification: The cumulative frequency of the topmost interval must be equal to the total number of cases ($N$ or $n$).

Table 2: Cumulative Frequency Distribution for Weekly Savings

Target Population: $90$ Senior Working Students

Note on Table 2 derivation: The transcript identifies the process of starting at the bottom. For class $80-86$, the description notes a calculation starting at $2$, though the table data values may vary slightly in presentation (e.g., the bottom entry in the description is $2$, but the final table shows $9$ for the bottom interval).

The Ogive Curve Plotting Mechanics

• Visual Form: The Ogive presents the cumulative frequency distribution in a visual, flowing curve.
• Axis Orientation:
  • X-axis: Plotted using the exact upper limits of each class interval.
  • Y-axis: Plotted using the cumulative frequencies ($cf$).
• Specific Intervals (Table 2/Figure 9 Example):
  • Exact upper limits plotted on x-axis: $86.5, 93.5, 100.5, 107.5, 114.5, 121.5, 128.5, 135.5, 142.5$.
  • Cumulative frequency peaks at $90$.