Statistical Reasoning: Visual Displays of Data and Frequency Tables

Section 3.1: Frequency Tables and Learning Goals

• The primary learning goal of this section is to develop the technical ability to create and interpret frequency tables.

• This topic is situated within Chapter 3: Visual Displays of Data, of the Sixth Edition of Statistical Reasoning for Everyday Life.

Definitions and Fundamental Components

• Frequency Table: A basic frequency table is a structured data display consisting of two primary columns:

  • Categories: One column lists all the distinct categories or classifications of the data.

  • Frequency: The other column lists the frequency of each specific category.

• Frequency: Defined as the number of data values that fall within a specific category.

• Relative Frequency: The proportion or percentage of the total data set that falls within a particular category. It is calculated using the formula:

  • $\text{Relative Frequency} = \frac{\text{Frequency of Category}}{\text{Total Frequency}}$

• Cumulative Frequency: The total number of data values in a specific category combined with the values in all preceding categories.

Example 1: Taste Test Study

• Background: The Rocky Mountain Beverage Company conducted a taste test for its new product, Coral Cola.

• Data Set: The study included $20$ individuals who each rated the cola on a $5$-point scale.

• Variable of Interest: The taste rating, which is a qualitative variable at the ordinal level of measurement.

• Frequency Table (Table 3.2):

  • Taste Scale 1: Frequency = $2$

  • Taste Scale 2: Frequency = $3$

  • Taste Scale 3: Frequency = $9$

  • Taste Scale 4: Frequency = $4$

  • Taste Scale 5: Frequency = $2$

  • Total Frequency: $20$

Example 2: Relative and Cumulative Frequency Interpretation

• This example expands on the Taste Test data by adding relative and cumulative frequency metrics to Table 3.4.

• Calculating Relative Frequency: Each category's frequency was divided by the total sum ($20$).

  • Example: For the highest rating ($5$), the frequency is $2$. The relative frequency is $\frac{2}{20} = 0.10$, or $10\%$.

• Calculating Cumulative Frequency: The sum of values for a category and its predecessors.

  • Cumulative Frequency for rating $3$: $2 + 3 + 9 = 14$.

  • Interpretation: $14$ out of $20$ people ($70\%$) gave the cola a rating of $3$ or lower.

• Full Data (Table 3.4):

  • Taste Scale 1: Frequency = $2$; Relative Frequency = $2/20 = 0.10$; Cumulative Frequency = $2$

  • Taste Scale 2: Frequency = $3$; Relative Frequency = $3/20 = 0.15$; Cumulative Frequency = $3 + 2 = 5$

  • Taste Scale 3: Frequency = $9$; Relative Frequency = $9/20 = 0.45$; Cumulative Frequency = $9 + 3 + 2 = 14$

  • Taste Scale 4: Frequency = $4$; Relative Frequency = $4/20 = 0.20$; Cumulative Frequency = $4 + 9 + 3 + 2 = 18$

  • Taste Scale 5: Frequency = $2$; Relative Frequency = $2/20 = 0.10$; Cumulative Frequency = $2 + 4 + 9 + 3 + 2 = 18$ (Note: Transcript displays this value as $18$)

  • Totals: Frequency = $20$; Relative Frequency = $1$, or $100\%$; Cumulative Frequency = $20$

Example 3: Binned Exam Scores

• When data sets contain many unique values (like numerical exam scores), data is often grouped into "bins."

• Dataset (20 Raw Scores): $76$, $80$, $78$, $76$, $94$, $75$, $98$, $77$, $84$, $88$, $81$, $72$, $91$, $72$, $74$, $86$, $79$, $88$, $72$, $75$

• Bin Selection Strategy:

  • The scores range from a minimum of $72$ to a maximum of $98$.

  • Bin width was chosen as $5$ points.

  • Bins are defined to avoid overlap and ensure consistent width (e.g., $95$ to $99$, $90$ to $94$, etc.).

• Interpretation of Cumulative Frequency in Binned Data: In this specific case, cumulative frequency is interpreted as the total number of scores in or above that specific bin.

• Frequency Table for Binned Exam Scores (Table 3.5):

  • 95 to 99: Frequency = $1$; Relative Frequency = $1/20 = 0.05$; Cumulative Frequency = $1$

  • 90 to 94: Frequency = $2$; Relative Frequency = $2/20 = 0.10$; Cumulative Frequency = $2 + 1 = 3$

  • 85 to 89: Frequency = $3$; Relative Frequency = $3/20 = 0.15$; Cumulative Frequency = $3 + 2 + 1 = 6$

  • 80 to 84: Frequency = $3$; Relative Frequency = $3/20 = 0.15$; Cumulative Frequency = $3 + 3 + 2 + 1 = 9$

  • 75 to 79: Frequency = $7$; Relative Frequency = $7/20 = 0.35$; Cumulative Frequency = $7 + 3 + 3 + 2 + 1 = 16$

  • 70 to 74: Frequency = $4$; Relative Frequency = $4/20 = 0.20$; Cumulative Frequency = $4 + 7 + 3 + 3 + 2 + 1 = 20$

  • Totals: Frequency = $20$; Relative Frequency = $1$; Cumulative Frequency = $20$