Frequency Distributions, Histograms, and Frequency Polygons

Grouped Frequency Distributions

• Construction Steps:
  • Select Number of Intervals: Preliminary choice of intervals (e.g., $14$ intervals).
  • Determine Interval Width: Calculated as range divided by the number of intervals (e.g., $\frac{62}{14} = 4.43$\,lb, rounded to $5$\,lb for convenience).
  • Establish Limits: Set the lower limit of the lowest interval to include the minimum value (e.g., $135$ for a lowest point of $135$). An interval of $135-139$\,lb contains $5$ whole values: $135$, $136$, $137$, $138$, and $139$
  • Create Contiguous Intervals: Construct equal-width adjacent intervals up to the maximum data point (e.g., ending at $195-199$\,lb for a maximum of $197$\,lb).
  • Tally Frequencies: Count observations per interval. The sum of all frequencies ($f$) must equal the total sample size ($N = 25$).
• Key Characteristics:
  • Zero Frequency Intervals: Grouped frequency distributions explicitly retain class intervals with $0$ frequency (e.g., $165-169$\,lb), unlike ungrouped frequency distributions which omit zero-frequency values.
  • Trade-Offs: Grouping improves visual clarity and readability but sacrifices individual data point precision.

Histograms

• Definition: A graphical representation of grouped interval/ratio data where frequency ($f$) is plotted as contiguous vertical bars over class intervals.
• Construction Rules:
  • Axes: The horizontal axis ($X$-axis or abscissa) lists equal-width class intervals with the variable name and units (e.g., "Weight (lb)"). The vertical axis ($Y$-axis) is labeled $f$ and formatted to approximately two-thirds to three-quarters the length of the horizontal axis.
  • Axis Break: An axis break symbol is placed on the horizontal axis if it intersects the vertical axis at a value other than $0$
  • Bar Structure: Vertical lines represent interval boundaries. Bars share common borders without spaces between them because class intervals are contiguous.
  • Bar Height: Equal to the frequency of values falling within each specific class interval.

Frequency Polygons

• Definition: A graphical distribution method equivalent to a histogram, created by plotting points at class interval midpoints and connecting them with straight line segments.
• Midpoint Calculation: Midpoints are determined by averaging the upper and lower limits of an interval: $\text{Midpoint} = \frac{\text{Lower Limit} + \text{Upper Limit}}{2}$ (e.g., $\frac{160 + 169}{2} = 164.5$\,lb).
• Closure: The polygon is completed by connecting the first and last data points diagonally to the horizontal axis with dashed lines.

Relative Frequency and Large Samples

• Relative Frequency: Calculated as frequency divided by total sample size, expressed as a proportion or percentage (e.g., $\frac{1}{25} = 4\%$\,per individual). The vertical axis can be labeled as percentage rather than raw frequency ($f$) without altering graph shape.
• Continuous Distributions: As sample size increases and class interval width decreases, the histogram or frequency polygon becomes progressively smoother, approaching a smooth continuous curve as sample size approaches infinity.