Frequency Distributions, Graphical Representations, and Distribution Shapes

Fundamentals of Frequency Distributions and Histograms

• Frequency Distribution Concepts:

  • Vertical bar heights in a histogram represent the frequency of each class.

  • Example class frequencies from initial data:

    • Class 1: Frequency = $2$

    • Class 2: Frequency = $8$ (for range $5$ to less than $9$)

    • Class 3: Frequency = $18$

    • Class 4: Frequency = $13$

    • Class 5: Frequency = $12$ or $15$

    • Class 6: Frequency = $15$ or $12$

    • Class 7: Frequency = $7$

• Class Boundaries vs. Class Limits:

  • Class boundaries must be used to construct histograms so that there are no gaps between adjacent vertical bars.

  • Class limits leave visual gaps on a graph because adjacent class limits do not overlap.

  • The base width of each class bar on the x-axis must be uniform and equal.

  • The x-axis marks represent class boundaries (e.g., from $99.5$ to $134.5$).

• Determining Axis Marks and Scales:

  • X-axis marking: Calculated using boundary values. For a distribution with $7$ classes, $8$ boundary marks are plotted along the axis starting from the second position, or $9$ marks overall.

  • Y-axis scaling: Set based on the range from the lowest frequency ($1$) to the highest frequency ($18$).

  • To avoid an excessively tall Y-axis, scale tick marks by grouping units. Dividing maximum frequency by the desired number of tick marks yields the scale per mark:         $\text{Scale per mark} = \frac{18}{6} = 3\,\text{units/mark}$         This maps an $18$-unit range onto $6$ vertical grid marks.

Frequency Polygons and Ogives

• Frequency Polygons:

  • A frequency polygon uses connected line segments between plotted points instead of vertical rectangular bars.

  • Plots class midpoints (middle points) on the x-axis rather than class boundaries.

  • Class Midpoint Calculation (algebraic average):         $\text{Midpoint} = \frac{\text{Lower Limit} + \text{Upper Limit}}{2}$

  • Midpoints can also be calculated sequentially by adding the class width ($w = 5$) to the preceding midpoint.

  • Sample set of midpoints: $105$, $110$, $115$, etc.

  • Sample plotted points corresponding to midpoints: $2$, $8$, $18$, $13$, $7$, $11$.

• Anchoring Frequency Polygons:

  • Frequency polygons must start and end at a frequency of zero on the x-axis.

  • Lower anchor midpoint: Subtract class width ($w = 5$) from the first class midpoint ($102$ or $107$):         $\text{Lower Anchor} = 102 - 5 = 97$

  • Upper anchor midpoint: Add class width ($w = 5$) to the last class midpoint ($132$):         $\text{Upper Anchor} = 132 + 5 = 137$

  • Connecting the graph to $\text{Frequency} = 0$ at these lower and upper anchor points completely closes the polygon.

• Ogives (Cumulative Frequency Graphs):

  • An ogive represents cumulative frequencies plotted against upper class boundaries (right boundaries).

  • Cumulative frequency is calculated by taking the sum of frequencies for a given class and all preceding classes:

    • Class 1 cumulative frequency = $2$

    • Class 2 cumulative frequency = $2 + 8 = 10$

    • Class 3 cumulative frequency = $10 + 18 = 28$

  • The cumulative frequency graph begins at zero for values strictly less than the lower boundary of the first class ($< 99.5$).

  • The final value on an ogive equals the total sample size ($N$).

  • Ogives are strictly non-decreasing (always increasing or flat) curves.

Summary of Primary Graphical Representations

• Histogram: Uses class boundaries on the x-axis and vertical bars with no gaps between classes.

• Frequency Polygon: Uses class midpoints on the x-axis with connected line segments, anchored to frequency zero at both ends by extending one class width below the first midpoint and above the last midpoint.

• Ogive: Uses upper class boundaries on the x-axis and cumulative frequencies on the y-axis, forming a non-decreasing curve starting at frequency zero at the lowest boundary.

Relative Frequency Distributions

• Relative Frequency Formula:

  • Relative frequency represents the proportion of total data values falling within a specific class:         $\text{Relative Frequency} = \frac{f}{N}$         where $f$ is class frequency and $N$ is total sample size (total number of data values).

• Example Application (50 Small Businesses Net Worth):

  • Dataset context: Net worth in thousands of dollars for $50$ small businesses ($N = 50$).

  • For a class with frequency $f = 8$:         $\text{Relative Frequency} = \frac{8}{50} = 0.16$

• Graphical Substitution:

  • Relative frequency histograms, relative frequency polygons, and relative frequency ogives follow the exact same construction rules as raw frequency graphs, replacing raw frequencies with relative frequencies on the vertical axis.

Distribution Shapes and Classification

• Bell-Shaped Distribution (Symmetrical):

  • Features a single central peak tapering off equally on both left and right sides.

  • Symmetrical shape where the middle class contains the highest frequency.

• Uniform Distribution:

  • Rectangular shape where all classes have approximately equal frequencies across the entire horizontal span.

• J-Shaped Distribution:

  • Frequencies start low at the first class and gradually increase to reach maximum height at the final class.

• Reverse J-Shaped Distribution:

  • Highest frequency occurs in the first class, tapering down rapidly across subsequent classes.

• Right-Skewed Distribution (Positively Skewed):

  • Peak frequency is positioned toward the left/lower values.

  • The left tail is short and the right tail (wing) is elongated.

• Left-Skewed Distribution (Negatively Skewed):

  • Peak frequency is positioned toward the right/higher values.

  • The right tail is short and the left tail (wing) is elongated.

• Unimodal Distribution:

  • A distribution containing a single peak or single class with the highest frequency.

• Bimodal Distribution:

  • A distribution containing two distinct peaks of approximately equal maximum frequency.

• U-Shaped Distribution:

  • Frequency is low in the center classes and high on both the far left and far right sides.

Comprehensive Step-by-Step Exercise (Entrance Exam Scores)

• Problem Statement & Dataset Setup:

  • Data context: Entrance exam scores for $180$ selected applicants.

  • Given Class Limits:

    • Class 1: $90 - 98$

    • Class 2: $99 - 107$

    • Class 3: $108 - 116$

    • Class 4: $117 - 125$

    • Class 5: $126 - 134$

  • Class Frequencies: Class 1 = $6$, Class 2 = $22$, Class 3 = $43$, Class 4 = $28$, Class 5 = $9$.

• Determining Class Width ($w$):

  • Subtract lower limit of Class 1 from lower limit of Class 2:         $w = 99 - 90 = 9$

  • The class width is $9$, not $8$.

• Detailed Class Computations:

  • Class 1 ($90 - 98$):

    • Lower Boundary: $89.5$

    • Upper Boundary: $98.5$

    • Midpoint: $\frac{90 + 98}{2} = 94$

    • Frequency: $6$

    • Cumulative Frequency: $6$

  • Class 2 ($99 - 107$):

    • Lower Boundary: $98.5$

    • Upper Boundary: $107.5$

    • Midpoint: $94 + 9 = 103$

    • Frequency: $22$

    • Cumulative Frequency: $6 + 22 = 28$

  • Class 3 ($108 - 116$):

    • Lower Boundary: $107.5$

    • Upper Boundary: $116.5$

    • Midpoint: $103 + 9 = 112$

    • Frequency: $43$

    • Cumulative Frequency: $28 + 43 = 71$

  • Class 4 ($117 - 125$):

    • Lower Boundary: $116.5$

    • Upper Boundary: $125.5$

    • Midpoint: $112 + 9 = 121$

    • Frequency: $28$

    • Cumulative Frequency: $71 + 28 = 99$

  • Class 5 ($126 - 134$):

    • Lower Boundary: $125.5$

    • Upper Boundary: $134.5$

    • Midpoint: $121 + 9 = 130$

    • Frequency: $9$

    • Cumulative Frequency: $99 + 9 = 108$

• Histogram Construction Steps:

  • X-axis points: Plot boundaries $89.5, 98.5, 107.5, 116.5, 125.5, 134.5$.

  • Y-axis scale: Maximum frequency is $43$. Mark Y-axis in increments of $6$ units ($6, 12, 18, 24, 30, 36, 42$).

  • Draw vertical bars between boundaries with heights equal to respective frequencies ($6, 22, 43, 28, 9$) with zero gap between bars.

• Frequency Polygon Construction Steps:

  • X-axis points: Plot midpoints $94, 103, 112, 121, 130$.

  • Plotted coordinates: $(94, 6)$, $(103, 22)$, $(112, 43)$, $(121, 28)$, $(130, 9)$.

  • Anchor points at $\text{Frequency} = 0$:

    • Left anchor midpoint: $94 - 9 = 85$ at coordinate $(85, 0)$.

    • Right anchor midpoint: $130 + 9 = 139$ at coordinate $(139, 0)$.

  • Connect coordinates sequentially with straight lines starting at $(85, 0)$ and ending at $(139, 0)$.

• Ogive Construction Steps:

  • X-axis points: Plot upper boundaries $89.5, 98.5, 107.5, 116.5, 125.5, 134.5$.

  • Plotted cumulative coordinates: $(89.5, 0)$, $(98.5, 6)$, $(107.5, 28)$, $(116.5, 71)$, $(125.5, 99)$, $(134.5, 108)$.

  • Connect points sequentially to display a continuously non-decreasing curve.

• Assignment Submission Details:

  • Section 2.1 / Exercise 2.2 assignment requires constructing histograms, frequency polygons, and ogives.

  • Submission requirement: Must be submitted in two weeks on Friday with name and date ('7) clearly written on the paper.