2. Descriptive Statistics (1)

Data Organization via Tables and Frequency Distributions

• Data organization and summarization systematically begin with frequency distributions.

• Ordered Array:

  • Definition: A simple arrangement of individual observations in order of magnitude (ascending or descending).
  • Limitation: Becomes very difficult and unwieldy to construct or read with large sample sizes.
  • Example dataset arranged as an ordered array ($n = 24$):
  • 12, 15, 17, 18, 19, 22, 23, 26, 27, 31, 31, 34, 36, 39, 41, 41, 42, 43, 44, 54, 59, 61, 65, 67.

• Frequency Distribution (Tables):

  • Definition: A tabular arrangement listing each possible value (or category/interval) that data can assume, alongside the exact number of times (frequency) each value occurs.
  • Application for Nominal and Ordinal Data:
  • Frequency distributions serve as a primary summary tool.
  • Frequently includes the relative frequency (percentage or proportion of total observations for each category).
  • Tables facilitate clear visualization of overall data distribution.
  • Example (Gender Distribution, $n = 46$):
    • Male: Frequency = 30, Relative Frequency = $65.2\%$
    • Female: Frequency = 16, Relative Frequency = $34.8\%$
    • Total: Frequency = 46, Relative Frequency = $100.0\%$

• Grouping Discrete and Continuous Data:

  • Data values are organized into non-overlapping class intervals, usually of equal width.
  • Individual frequency distributions of continuous variables (such as exact age) yield too many unique single-occurrence values and lack summarizeable insight.
  • Grouping values into structured intervals (e.g., 10-year age groups) reveals hidden pattern distributions.
  • Example (Age Group Distribution, $n = 45$):
  • Age Group $< 20$: Frequency = 10, Relative Frequency = $21.7\%$, Cumulative Relative Frequency = $21.7\%$
  • Age Group 20–55: Frequency = 25, Relative Frequency = $54.4\%$, Cumulative Relative Frequency = $76.1\%$
  • Age Group $> 55$: Frequency = 11, Relative Frequency = $23.9\%$, Cumulative Relative Frequency = $100.0\%$
  • Total: Frequency = 45, Relative Frequency = $100.0\%$

• Qualitative Variable Frequency Summaries:

  • Count the absolute number of cases falling into each discrete categorical class.
  • Example 1 (ICU Admission Type for $n = 25$ Patients):
  • Medical: Frequency = 12, Relative Frequency = 0.48
  • Surgical: Frequency = 6, Relative Frequency = 0.24
  • Cardiac: Frequency = 5, Relative Frequency = 0.20
  • Other: Frequency = 2, Relative Frequency = 0.08
  • Total: Frequency = 25, Relative Frequency = 1.00
  • Example 2 (Smoker Study by Gender, $n = 234$):
  • Male (coded as 1): Frequency = 110, Relative Frequency = $47.0\%$
  • Female (coded as 2): Frequency = 124, Relative Frequency = $53.0\%$
  • Total: Frequency = 234, Relative Frequency = $100.0\%$

• Quantitative Variables and Interval Construction:

  • Select a set of continuous, non-overlapping intervals such that every single observation falls into exactly one interval.
  • Sturge's Rule:
  • A statistical formula used to determine the ideal number of class intervals ($K$) and class interval width ($W$) based on total observations ($n$):       $K = 1 + 3.322(\log_{10}(n))$$W = \frac{L - S}{K}$
  • Variable Definitions:
    • $K$ = Number of class intervals
    • $W$ = Width of class interval
    • $n$ = Total number of observations
    • $L$ = Largest value in the dataset
    • $S$ = Smallest value in the dataset
  • Worked Example using Sturge's Rule:
  • Dataset (Weekly leisure time in hours for $n = 40$ college students):       23, 24, 18, 14, 20, 36, 24, 26, 23, 21, 16, 15, 19, 20, 22, 14, 13, 10, 19, 27, 29, 22, 38, 28, 34, 32, 23, 19, 21, 31, 16, 28, 19, 18, 12, 27, 15, 21, 25, 16.
  • Calculation of Intervals ($K$):       $K = 1 + 3.322(\log_{10}(40)) = 1 + 3.322(1.602) = 6.32 \approx 6$
  • Identification of Extremes:       $L = 38$$S = 10$
  • Calculation of Width ($W$):       $W = \frac{38 - 10}{6} = \frac{28}{6} = 4.66 \approx 5$
  • Resulting Frequency Table:
    • Interval 10–14: Frequency = 5, Relative Frequency = 0.125, Cumulative Relative Frequency = 0.125
    • Interval 15–19: Frequency = 11, Relative Frequency = 0.275, Cumulative Relative Frequency = 0.400
    • Interval 20–24: Frequency = 12, Relative Frequency = 0.300, Cumulative Relative Frequency = 0.700
    • Interval 25–29: Frequency = 7, Relative Frequency = 0.175, Cumulative Relative Frequency = 0.875
    • Interval 30–34: Frequency = 3, Relative Frequency = 0.075, Cumulative Relative Frequency = 0.950
    • Interval 35–39: Frequency = 2, Relative Frequency = 0.050, Cumulative Relative Frequency = 1.000
    • Total: Frequency = 40, Relative Frequency = 1.000

• Key Frequency Concepts:

  • Cumulative Frequency: The running total sum of frequencies across successive class intervals.
  • Cumulative Relative Frequency: The percentage or proportion of total observations with values equal to or less than the upper boundary of a specified interval.
  • Mid-point: The central value lying midway between lower and upper class limits:     $\text{Mid-point} = \frac{\text{Lower Limit} + \text{Upper Limit}}{2}$
  • True Limits: Boundaries that make continuous variable class intervals seamless across adjacent classes (eliminating discrete gaps between intervals).
  • Method: Subtract 0.5 from the lower limit and add 0.5 to the upper limit of each class interval.
  • Table of True Limits and Mid-points (College Student Leisure Time Data):
    • Interval 10–14: True Limit = 9.5–14.5, Mid-point = 12, Frequency = 5
    • Interval 15–19: True Limit = 14.5–19.5, Mid-point = 17, Frequency = 11
    • Interval 20–24: True Limit = 19.5–24.5, Mid-point = 22, Frequency = 12
    • Interval 25–29: True Limit = 24.5–29.5, Mid-point = 27, Frequency = 7
    • Interval 30–34: True Limit = 29.5–34.5, Mid-point = 32, Frequency = 3
    • Interval 35–39: True Limit = 34.5–39.5, Mid-point = 37, Frequency = 2

Tabular Presentation Types and Best Practices

• Simple Frequency Distribution Table:

  • Displays frequencies for a single categorical or grouped continuous variable.
  • Example (Primary and Secondary Syphilis Morbidity Cases by Age, 1989):
  • 0–14 years: 230 cases ($0.5\%$
  • 15–19 years: 4,378 cases ($10.0\%$
  • 20–24 years: 10,405 cases ($23.6\%$
  • 25–29 years: 9,610 cases ($21.8\%$
  • 30–34 years: 8,648 cases ($19.6\%$
  • 35–44 years: 6,901 cases ($15.7\%$
  • 45–54 years: 2,631 cases ($6.0\%$
  • $> 44$ years (older sub-group): 1,278 cases ($2.9\%$
  • Total: 44,081 cases ($100\%$

• Two-Variable Table (Crosstabulation):

  • Displays frequencies cross-classified across two distinct variables simultaneously.
  • Example (Syphilis Morbidity Cases by Age and Sex, 1989):
  • 0–14 years: Male = 40, Female = 190, Total = 230
  • 15–19 years: Male = 1,710, Female = 2,668, Total = 4,378
  • 20–24 years: Male = 5,120, Female = 5,285, Total = 10,405
  • 25–29 years: Male = 5,301, Female = 4,306, Total = 9,610
  • 30–34 years: Male = 5,537, Female = 3,111, Total = 8,648
  • 35–44 years: Male = 5,004, Female = 1,897, Total = 6,901
  • 45–54 years: Male = 2,144, Female = 487, Total = 2,631
  • $> 44$ years: Male = 1,147, Female = 131, Total = 1,278
  • Total: Male = 26,006, Female = 18,075, Overall Total = 44,081

• Multi-Variable Tables:

  • Capable of summarizing three or more variables in structured row/column blocks (e.g., Sex, Age, Religion, Occupation).

• Essential Guidelines for Constructing Tables:

  • Keep tables simple; limit variables per table to three or fewer.
  • Ensure every table is completely self-explanatory.
  • Provide a descriptive title detailing what, when, and where the data was collected.
  • Label all rows and columns clearly.
  • State explicit measurement units.
  • Define all codes, symbols, and abbreviations in footnotes.
  • Always report grand totals.
  • Include data source attributions in footnotes if data is secondary or adapted.

Diagrammatic Presentation and Categorical Data Graphs

• Importance of Diagrammatic Representation:

  • Diagrams command greater visual attention than numeric tables.
  • Provide an immediate summary of overall distribution trends.
  • Possess high memory retention value.
  • Facilitate comparisons across datasets.
  • Aid in recognizing patterns and trajectory trends.
  • Poorly designed graphs fail to communicate and actively mislead viewers.

• Overview of Specific Graph Types by Data Class:

  • Nominal & Ordinal Data: Bar graph, Pie chart.
  • Quantitative Data: Histogram, Stem-and-leaf plot, Box plot, Scatter plot, Line graph, Ogive curve.

• Bar Charts (Bar Graphs):

  • Structure: Categories placed on horizontal axis (X-axis); frequencies or relative frequencies on vertical axis (Y-axis).
  • Bar Height: Directly proportional to frequency or relative frequency.
  • Construction Rules:
  • All bars must feature equal width.
  • Bars must be separated by uniform gaps (bars must not touch).
  • Bars must rest on a common baseline.
  • Axes must be explicitly labeled.

• Example Data (Distribution of Hospital Referral Sources, $n = 1906$):
  • General Practitioner (GP): 769 patients ($40.3\%$
  • Out-Patient Department (OPD): 623 patients ($32.7\%$
  • Casualty: 256 patients ($13.4\%$
  • Other: 161 patients ($8.5\%$
  • Other Hospital: 97 patients ($5.1\%$

• Sub-divided (Component) Bar Charts:
  • Sub-divides single bars into sub-components based on their proportional ratio to the whole.
  • Component ordering must remain identical across all bars in the chart.
  • Variants: Stacked component bar charts and $100\%$ component bar charts.

• Multiple Bar Graphs:
  • Displays relationships across multiple variables simultaneously.
  • Can be drawn vertically or horizontally.
  • Horizontal bar charts are effective for display when category titles are lengthy (e.g., FGM info sources).

• Pie Charts:
  • Represents relative frequencies of a single categorical variable by dividing a circle into proportional sectors.
  • Construction Steps:
  1. Build frequency table.
  2. Calculate percentage ($P$) for each category.
  3. Convert percentages into degrees ($D$):        $\text{Degree} = \text{Percentage} \times 3.6^\circ = \frac{\text{Frequency}}{\text{Total}} \times 360^\circ$
  4. Plot circle sectors using calculated angles.
  • Example Data (Female Causes of Death in England and Wales, 1989, $n = 236,000$):
  • Circulatory system: 100,000 deaths ($42\%$
  • Neoplasms: 70,000 deaths ($30\%$
  • Respiratory system: 30,000 deaths ($13\%$
  • Others: 20,000 deaths ($8\%$
  • Digestive system: 10,000 deaths ($4\%$
  • Injury and Poisoning: 6,000 deaths ($3\%$

Graphs for Quantitative Data and Exploratory Data Analysis

• Histograms:
  • Graphical display of continuous frequency distributions.
  • X-axis represents class interval boundaries; Y-axis represents frequencies or relative frequencies.
  • Adjacent bars touch without gaps because class intervals represent continuous numeric ranges.
  • Area of each bar is directly proportional to observation frequency in that interval.

• Limitations of Histograms:

  • Difficult to construct manually.

  • Raw individual data values inside intervals are completely lost and cannot be reconstructed.

  • Stem-and-Leaf Plot:

• An exploratory data analysis method that creates a visual frequency distribution (similar to a histogram) while preserving full raw data values.

• Best suited for small to moderate data sizes; not used for formal publication reporting.

• Data Point Structure:

  • Stem: Consists of initial leading digit(s).
  • Leaf: Consists of the final rightmost single digit.
  • Example: For number 483, stem = 48, leaf = 3.

• Steps to Construct:

  1. Identify smallest and largest stems.
  2. List stems vertically in ascending order.     3.