Comprehensive Guide to Frequency Distributions, Ogives, and Relative Frequency Graphs

Class Intervals and Frequency Table Setup

• Class Width Calculation:

  • When calculating class width from a decimal result, always round up to the next whole integer.
  • Rounding up provides a consistent class width that dictates the vertical size of the table by defining the number of rows or classes to construct.

• Frequency Table Structure:

  • Table layout begins with setting up columns, starting with the class column on the far left.
  • Rows correspond directly to the given number of classes. For example, if a dataset specifies $6$ classes, the table must contain $6$ rows.
  • Class boundaries define continuous intervals for each class:
  • Lower class boundary is calculated by subtracting $0.5$ from the lower class limit.
  • Upper class boundary is calculated by adding $0.5$ to the upper class limit.

• Weight Dataset Specifications:

  • Total number of sample values (sample size): $N = 22$ students.
  • Total number of classes: $6$ classes.
  • Class boundary values in sequential order: $105.5$, $136.5$, $167.5$, $198.5$, $229.5$, $260.5$, and $292.5$.
  • Frequencies ($f$) recorded for each class:
  • Class 1 ($105.5$ to $136.5$): $f_1 = 6$
  • Class 2 ($136.5$ to $167.5$): $f_2 = 3$
  • Class 3 ($167.5$ to $198.5$): $f_3 = 10$
  • Class 4 ($198.5$ to $229.5$): $f_4 = 2$
  • Class 5 ($229.5$ to $260.5$): $f_5 = 0$
  • Class 6 ($260.5$ to $292.5$): $f_6 = 1$
  • Frequency summation verification:     $\sum f = 6 + 3 + 10 + 2 + 0 + 1 = 22$

Cumulative Frequency and Ogive Graph Construction

• Concept of Cumulative Frequency:

  • Cumulative frequency measures the accumulated total of frequencies across sequential classes, adding each row's frequency to the running sum of all previous frequencies.

• Calculation Step (Zigzag Addition Method):

  • Row 1: Equal to the first class frequency.     $\text{Cumulative Frequency}_1 = 6$
  • Row 2: Add Row 1 cumulative frequency ($6$) to Row 2 class frequency ($3$).     $6 + 3 = 9$
  • Row 3: Add Row 2 cumulative frequency ($9$) to Row 3 class frequency ($10$).     $9 + 10 = 19$
  • Row 4: Add Row 3 cumulative frequency ($19$) to Row 4 class frequency ($2$).     $19 + 2 = 21$
  • Row 5: Add Row 4 cumulative frequency ($21$) to Row 5 class frequency ($0$).     $21 + 0 = 21$
  • Row 6: Add Row 5 cumulative frequency ($21$) to Row 6 class frequency ($1$).     $21 + 1 = 22$

• Ogive (Cumulative Frequency Graph) Formatting and Plotting:

  • Definition: An Ogive is the graphical representation of a cumulative frequency distribution.
  • X-Axis Formatting:
  • Include a "thunderbolt" symbol (axis break) near the origin to signify that the x-axis has been shortened.
  • Segment the axis using the sequential class boundary values: $105.5$, $136.5$, $167.5$, $198.5$, $229.5$, $260.5$, and $292.5$.
  • Y-Axis Formatting:
  • Label the axis as Cumulative Frequency.
  • Axis range spans from the minimum cumulative value ($6$) up to the total sample size ($22$).
  • Axis scale increments must be uniform. Incrementing by $5$s ($5$, $10$, $15$, $20$, $25$) is standard, though uniform increments of $1$, $2$, $3$, or $4$ are also permissible.
  • Coordinate Plotting Steps:
  • Origin baseline start: Plot a baseline dot at the lower boundary of the first class ($105.5$) at $y = 0$.
  • Class 1 Upper Boundary ($136.5$): Plot dot at $y = 6$ ($6$ students weigh $136.5\,\text{lbs}$ or less).
  • Class 2 Upper Boundary ($167.5$): Plot dot at $y = 9$ ($9$ students weigh $167.5\,\text{lbs}$ or less).
  • Class 3 Upper Boundary ($198.5$): Plot dot at $y = 19$ ($19$ students weigh $198.5\,\text{lbs}$ or less).
  • Class 4 Upper Boundary ($229.5$): Plot dot at $y = 21$ ($21$ students weigh $229.5\,\text{lbs}$ or less).
  • Class 5 Upper Boundary ($260.5$): Plot dot at $y = 21$ ($21$ students weigh $260.5\,\text{lbs}$ or less; forms a horizontal segment relative to Class 4).
  • Class 6 Upper Boundary ($292.5$): Plot dot at $y = 22$ ($22$ students weigh $292.5\,\text{lbs}$ or less).
  • Connecting Points:
  • Points on an Ogive must be connected using straight line segments. Curved lines are not permitted.

Calculating Relative Frequency

• Concept of Relative Frequency:

  • Relative frequency computes the proportion or percentage of total observations that fall within a given class interval relative to the total dataset size ($N$).

• Mathematical Formula and Computations ($N = 22$):   $\text{Relative Frequency} = \frac{\text{Class Frequency}}{N}$

  • Class 1:     $\frac{6}{22} \approx 0.27$
  • Class 2:     $\frac{3}{22} \approx 0.14$
  • Class 3:     $\frac{10}{22} \approx 0.45$
  • Class 4:     $\frac{2}{22} \approx 0.09$
  • Class 5:     $\frac{0}{22} = 0.00$
  • Class 6:     $\frac{1}{22} \approx 0.05$

• Relative Frequency Summation Check:

  • Summing all values across the relative frequency column gives:     $0.27 + 0.14 + 0.45 + 0.09 + 0.00 + 0.05 = 1.00$
  • A total of $1.00$ represents $100\%$ of the sample population.
  • Due to intermediate rounding, acceptable column totals range from $0.98$ to $1.01$ (e.g., $0.98$, $0.99$, $1.00$, $1.01$, or $1.1$).
  • Sums significantly outside this range (such as $0.75$ or $2.5$) signal an error in calculation.

Cumulative Relative Frequency and Graph Construction

• Definition and Calculation Steps:

  • Cumulative relative frequency sums the relative frequency values sequentially down the table.
  • Row 1: Equals the first class relative frequency.     $\text{Cumulative Relative Frequency}_1 = 0.27$
  • Row 2: Add Row 1 cumulative relative frequency ($0.27$) to Row 2 relative frequency ($0.14$).     $0.27 + 0.14 = 0.41$
  • Row 3: Add Row 2 cumulative relative frequency ($0.41$) to Row 3 relative frequency ($0.45$).     $0.41 + 0.45 = 0.86$
  • Row 4: Add Row 3 cumulative relative frequency ($0.86$) to Row 4 relative frequency ($0.09$).     $0.86 + 0.09 = 0.95$
  • Row 5: Add Row 4 cumulative relative frequency ($0.95$) to Row 5 relative frequency ($0.00$).     $0.95 + 0.00 = 0.95$
  • Row 6: Add Row 5 cumulative relative frequency ($0.95$) to Row 6 relative frequency ($0.05$).     $0.95 + 0.05 = 1.00$

• Graphing Cumulative Relative Frequency:

  • Axis Configuration:
  • Graph Title: Relative Frequency Graph (or Cumulative Relative Frequency Graph).
  • X-Axis: Labeled with continuous class boundary values ($105.5$, $136.5$, $167.5$, $198.5$, $229.5$, $260.5$, $292.5$).
  • Y-Axis: Scaled from $0.00$ to $1.00$ ($100\%$). Axis segments must use consistent increments, such as steps of $0.2$ ($0.2$, $0.4$, $0.6$, $0.8$, $1.0$), $0.1$, or $0.3$.
  • Plotting Points and Interpretation:
  • Baseline start ($105.5$): Plot dot at $y = 0.00$.
  • Upper Boundary $136.5$: Plot dot at $y = 0.27$ ($27\%$ of students weigh $136.5\,\text{lbs}$ or less).
  • Upper Boundary $167.5$: Plot dot at $y = 0.41$ ($41\%$ of students weigh $167.5\,\text{lbs}$ or less).
  • Upper Boundary $198.5$: Plot dot at $y = 0.86$ ($86\%$ of students weigh $198.5\,\text{lbs}$ or less).
  • Upper Boundary $229.5$: Plot dot at $y = 0.95$ ($95\%$ of students weigh $229.5\,\text{lbs}$ or less).
  • Upper Boundary $260.5$: Plot dot at $y = 0.95$ ($95\%$ of students weigh $260.5\,\text{lbs}$ or less).
  • Upper Boundary $292.5$: Plot dot at $y = 1.00$ ($100\%$ of students weigh $292.5\,\text{lbs}$ or less).
  • Point Connection: Connect points sequentially using straight lines.

Summary of Verification Checks and Exam Guidelines

• Table Verification Checks:

  • Last entry in Cumulative Frequency column must equal total sample size $N$ ($22$).
  • Sum of Relative Frequency column must equal $1.00$ (or $0.98\text{--}1.01$ due to rounding).
  • Last entry in Cumulative Relative Frequency column must equal $1.00$ (or $0.99\text{--}1.1$ due to rounding).

• Practical Exam Guidelines:

  • Examination Requirements: Bring a simple standard calculator to exams.
  • Restricted Items: Cell phones and mobile devices are strictly forbidden during exams.