Exploring Data with Tables and Graphs: Frequency Distributions

Overview of Data Exploration and Frequency Distributions

• Core Topics in Data Exploration:

  • 2-1 Frequency Distributions for Organizing and Summarizing Data: Methods to partition, organize, and summarize raw datasets into structured frequency tables.

  • 2-2 Histograms: Visual representations of frequency distributions that simplify data analysis.

  • 2-3 Graphs that Enlighten and Graphs that Deceive: Identifying effective graphical presentations versus misleading visual statistics.

• Purpose of Frequency Distributions:

  • When dealing with large datasets, a frequency distribution (or frequency table) serves as an essential tool to organize and summarize raw data.

  • Frequency distributions help reveal the fundamental nature and shape of a dataset's distribution.

  • Partitioning Data: A frequency distribution shows how data values are partitioned among several distinct categories (or classes) by listing each category alongside its corresponding frequency (the total count of data values belonging to that category).

Core Terminology of Frequency Distributions

• Lower Class Limits:

  • The smallest numbers that can physically belong to each of the distinct classes.

• Upper Class Limits:

  • The largest numbers that can physically belong to each of the distinct classes.

• Class Boundaries:

  • The precise numbers used to separate adjacent classes without leaving gaps created by the class limits.

• Class Midpoints:

  • The specific values located at the exact middle of each class.

  • Calculation Formula:     $\text{Class Midpoint} = \frac{\text{Lower Class Limit} + \text{Upper Class Limit}}{2}$

• Class Width:

  • The numerical difference between two consecutive lower class limits (or two consecutive lower class boundaries) in a frequency distribution.

Procedure for Constructing a Frequency Distribution

• Step 1: Select the Number of Classes:

  • Determine the desired number of classes for the table, typically between $5$ and $20$.

• Step 2: Calculate the Class Width:

  • Use the class width formula and round up to find a convenient, workable increment:     $\text{Class width} \approx \frac{\text{maximum data value} - \text{minimum data value}}{\text{number of classes}}$

  • It is standard practice to round up to ensure all data points fit within the defined boundaries.

• Step 3: Determine the First Lower Class Limit:

  • Choose a starting value for the first lower class limit by taking either the exact minimum data value or a convenient value lower than the minimum.

• Step 4: Generate All Lower Class Limits:

  • Add the calculated class width to the first lower class limit to produce the second lower class limit. Repeat this addition process until all lower class limits are established.

• Step 5: Set Upper Class Limits:

  • List the generated lower class limits vertically in a column, then identify and enter the non-overlapping upper class limits for each class.

• Step 6: Tally Data and Compute Frequencies:

  • Place each raw data value into its corresponding class row using tally marks, then sum the tally marks to determine the total frequency for each class.

Worked Example: Los Angeles Daily Commute Times

• Problem Objective:

  • Construct a frequency distribution containing $7$ classes using a sample dataset of $50$ daily commute times (in minutes) for commuters in Los Angeles.

• Raw Dataset ($50$ Values in Minutes):

  • $18$, $25$, $45$, $75$, $60$, $40$, $25$, $8$, $50$, $10$, $10$, $30$, $15$, $25$, $50$, $20$, $30$

  • $20$, $45$, $30$, $60$, $30$, $20$, $15$, $30$, $60$, $30$, $15$, $35$, $40$, $5$, $30$, $40$, $20$

  • $10$, $45$, $30$, $15$, $25$, $25$, $5$, $90$, $30$, $15$, $60$, $20$, $60$, $30$, $25$, $25$

• Step-by-Step Construction:

  • Target Classes: Select $7$ as the total number of desired classes.

  • Class Width Calculation:

  • Minimum value = $5$

  • Maximum value = $90$     $\text{Class width} \approx \frac{90 - 5}{7} = \frac{85}{7} = 12.14 \approx 15$

  • Rounding $12.14$ up to $15$ provides a convenient and practical interval for commute time measurements.

  • First Lower Class Limit Selection: Select $0$ as the first lower class limit because $0$ is less than the minimum value $5$ and forms an intuitive baseline for commute duration.

  • Lower Class Limits Listing: Add the class width of $15$ sequentially to generate seven lower class limits:

  • $0$, $15$, $30$, $45$, $60$, $75$, $90$

  • Upper Class Limits Identification:

  • Upper limits are set to avoid overlap: $14$, $29$, $44$, $59$, $74$, $89$, $104$

• Tallying Frequencies:

  • $0-14$ minutes: Frequency = $6$

  • $15-29$ minutes: Frequency = $18$

  • $30-44$ minutes: Frequency = $14$

  • $45-59$ minutes: Frequency = $5$

  • $60-74$ minutes: Frequency = $5$

  • $75-89$ minutes: Frequency = $1$

  • $90-104$ minutes: Frequency = $1$

  • Total Sample Size ($N$) = $6 + 18 + 14 + 5 + 5 + 1 + 1 = 50$

Relative Frequency Distributions

• Definition:

  • A Relative Frequency Distribution (or Percentage Frequency Distribution) replaces raw frequencies with proportions or percentages.

• Formulas:   $\text{Relative Frequency for a Class} \approx \frac{\text{Frequency for a Class}}{\text{Sum of All Frequencies}}$   $\text{Percentage for a Class} \approx \frac{\text{Frequency for a Class}}{\text{Sum of All Frequencies}} \times 100\%$

• Summation Rule:

  • The sum of all relative frequencies must equal approximately $1$ (or $100\%$ for percentages), allowing for minor variations due to rounding errors.

• Relative Frequency Breakdown for LA Commute Times ($N = 50$):

  • $0-14$ minutes: $12\%$ (calculated as $6 \div 50$)

  • $15-29$ minutes: $36\%$ (calculated as $18 \div 50$)

  • $30-44$ minutes: $28\%$ (calculated as $14 \div 50$)

  • $45-59$ minutes: $10\%$ (calculated as $5 \div 50$)

  • $60-74$ minutes: $10\%$ (calculated as $5 \div 50$)

  • $75-89$ minutes: $2\%$ (calculated as $1 \div 50$)

  • $90-104$ minutes: $2\%$ (calculated as $1 \div 50$)

Comparative Data Analysis: New York vs. Boise Commute Times

• Utility of Relative Frequency Distributions:

  • Combining two or more relative frequency distributions into a single comparative table allows direct visual and analytical comparisons between datasets of vastly different sample sizes.

• Comparison Table (Sample Size $N = 1000$ for NY, NY and $N = 1000$ for Boise, ID):

  • $0-14$ minutes: NY = $8.6\%$, Boise = $30.3\%$

  • $15-29$ minutes: NY = $20.3\%$, Boise = $45.5\%$

  • $30-44$ minutes: NY = $24.8\%$, Boise = $17.0\%$

  • $45-59$ minutes: NY = $17.2\%$, Boise = $3.5\%$

  • $60-74$ minutes: NY = $18.5\%$, Boise = $2.2\%$

  • $75-89$ minutes: NY = $3.3\%$, Boise = $0.3\%$

  • $90-104$ minutes: NY = $4.7\%$, Boise = $0.3\%$

  • $105-119$ minutes: NY = $0.1\%$, Boise = $0.0\%$

  • $120-134$ minutes: NY = $0.0\%$, Boise = $0.0\%$

  • $135-149$ minutes: NY = $2.5\%$, Boise = $0.9\%$

• Comparative Insights:

  • Commute times in Boise, ID are significantly lower overall than commute times in New York, NY.

  • In Boise, $75.8\%$ of all commutes fall within the lowest two time brackets ($0-29$ minutes), whereas only $28.9\%$ of New York commutes fall within those same intervals ($8.6\% + 20.3\% = 28.9\%$

  • This statistical disparity aligns with expectations based on the relative physical geographic size, population density, and infrastructure differences between the two cities.

Cumulative Frequency Distributions

• Definition:

  • A Cumulative Frequency Distribution tabulates the accumulated total frequencies for each class combined with all preceding classes.

  • Class descriptions are adjusted to express upper limit thresholds (e.g., "Less than [upper boundary]").

• Cumulative Frequency Table for LA Commute Times:

  • Less than $15$ minutes: $6$

  • Less than $30$ minutes: $24$ (calculated as $6 + 18$)

  • Less than $45$ minutes: $38$ (calculated as $24 + 14$)

  • Less than $60$ minutes: $43$ (calculated as $38 + 5$)

  • Less than $75$ minutes: $48$ (calculated as $43 + 5$)

  • Less than $90$ minutes: $49$ (calculated as $48 + 1$)

  • Less than $105$ minutes: $50$ (calculated as $49 + 1$)

Analyzing Distributions: Normal Distributions and Gaps

• Identifying Normal Distributions:

  • Frequency tables are frequently analyzed to determine whether a dataset follows a normal distribution.

  • Criteria for a Normal Distribution:

  1. Frequency Trend: Class frequencies start low, increase gradually to one or two maximum peak frequencies, and then decrease back to low frequencies.

  2. Symmetry: The distribution exhibits approximate symmetry. Frequencies listed prior to the maximum frequency roughly mirror those following the maximum.

  • Example Frequency Table of a Normal Distribution:

  • $0-14$ minutes: Frequency = $2$ (Starts low)

  • $15-29$ minutes: Frequency = $4$

  • $30-44$ minutes: Frequency = $10$

  • $45-59$ minutes: Frequency = $18$ (Reaches peak maximum)

  • $60-74$ minutes: Frequency = $10$

  • $75-89$ minutes: Frequency = $4$

  • $90-104$ minutes: Frequency = $2$ (Decreases back to low)

• Understanding Gaps in Data:

  • The presence of prominent gaps (classes with frequencies of zero bounded by non-zero classes) indicates that data may be sampled from two or more separate populations.

  • Important Caveat: While gaps signal multiple underlying populations, the inverse statement is not always true; combined data from distinct populations does not guarantee a gap will appear.

  • Case Study: Penny Weight Distribution:

  • $2.40-2.49\,\text{g}$: Frequency = $18$

  • $2.50-2.59\,\text{g}$: Frequency = $19$

  • $2.60-2.69\,\text{g}$: Frequency = $0$

  • $2.70-2.79\,\text{g}$: Frequency = $0$

  • $2.80-2.89\,\text{g}$: Frequency = $0$

  • $2.90-2.99\,\text{g}$: Frequency = $2$

  • $3.00-3.09\,\text{g}$: Frequency = $25$

  • $3.10-3.19\,\text{g}$: Frequency = $8$

  • Interpretation: A clear frequency gap exists between $2.60\,\text{g}$ and $2.89\,\text{g}$. This gap reflects two distinct coin compositions manufactured across different time periods:

    • Pennies minted prior to $1983$ are composed of $95\%$ copper and $5\%$ zinc (resulting in heavier weights).

    • Pennies minted after $1983$ are composed of $2.5\%$ copper and $97.5\%$ zinc (resulting in lighter weights).

• Transition to Histograms:

  • Although frequency distribution tables effectively group and quantify numerical data, a histogram provides a graphical view that is far easier to analyze and interpret than raw tables of figures.