Chapter 2: Organizing, Displaying, and Describing Data

Objectives of Data Organization, Display, and Description

• Fundamental Statistical Understanding:

  • Discuss the nature, purpose, and types of statistics available for analyzing data.

  • Recognize and define mathematical symbols commonly used throughout statistical analysis.

  • Discuss the fundamental principles of data handling.

• Measurement Scales and Statistical Alignment:

  • Correctly identify the measurement scale of any given variable.

  • Explain the direct relationship between the measurement scale of a variable and the correct statistical procedure to utilize.

• Data Visualization and Summarization:

  • Construct and interpret various data displays:

    • Frequency distribution tables

    • Bar charts

    • Histograms

    • Stem-and-leaf plots

    • Frequency polygons

    • Cumulative frequency polygons

  • Describe variables utilizing appropriate measures across four primary dimensions:

    • Measures of central tendency

    • Measures of dispersion

    • Distribution shape

    • Skewness

  • Explain the practical use and interpretation of percentiles.

Variables and Measurement Scales

• Definition of a Variable:

  • A variable is defined as any characteristic that can and does assume different values for different people, objects, or events being studied.

• The Four Measurement Scales:

  • Nominal Scale:

    • Numbers are used simply as arbitrary codes or labels to represent distinct characteristics.

    • There is no inherent order, rank, or hierarchy among the categories.

    • The numerical assignments to categories are completely arbitrary.

    • Examples:

      • Gender: coded as $1$ for Male, $2$ for Female.

      • Ethnicity: coded as $1$ for African American, $2$ for Caucasian, $3$ for Hispanic.

  • Ordinal Scale:

    • Numbers represent categories that can be placed in a meaningful numerical order (such as from lowest to highest or least to most).

    • There is no information provided regarding the precise size of the interval between the different values.

    • The actual size of the interval may vary significantly between adjacent categories.

    • There is no "true" zero point.

    • Almost all subjective measurement scales—including satisfaction, pain, and depression scales—are categorized as ordinal scales.

    • Examples:

      • Pain Scale: rated as $1$ for No pain, $2$ for A little pain, $3$ for Some pain, $4$ for A lot of pain.

      • Military Rank: ranked sequentially as private, corporal, sergeant, and master sergeant.

  • Interval Scale:

    • Numbers can be placed in a meaningful order.

    • The distances (intervals) between adjacent numbers are strictly equal.

    • Mathematical operations of addition and subtraction are valid across an interval scale.

    • There is no true zero point (zero does not indicate the total absence of the phenomenon being measured); consequently, ratios cannot be calculated.

    • Examples:

      • Fahrenheit temperature

      • SAT scores

      • GRE scores

      • LSAT scores

      • MCAT scores

  • Ratio Scale:

    • Numbers can be placed in a meaningful order.

    • The intervals between adjacent numerical values are equal.

    • There is an absolute, "true" zero determined by nature, representing the complete absence of the phenomenon.

    • Ratios can be meaningfully computed and interpreted.

    • Almost all biomedical measures belong to the ratio scale.

    • Examples:

      • Body weight

      • Age

      • Number of minutes spent exercising

      • Cholesterol level

      • Number of weeks pregnant

Principles and Visual Displays of Data

• Core Goals of Data Display:

  • The primary objective is to obtain an intuitive and comprehensive understanding of the distribution of the data.

  • Key distributional characteristics evaluated include:

    • Central tendency: Identification of the most frequently occurring values.

    • Dispersion: Determination of how the values are spread out.

    • Shape and skewness: Assessment of the symmetry or asymmetry inherent in the distribution of values.

    • Outliers: Detection of unusual values that deviate from the general pattern of the data.

• Frequency Distribution Tables:

  • Structured tabular summaries that organize data systematically.

  • Standard components displayed in a frequency table include:

    • Possible values or grouped intervals of the variable.

    • Raw frequency ($\text{No.}$): The actual number of cases/observations within that value or interval.

    • Relative frequency ($\%\text{ of Total Sample}$): The percentage of cases falling within that value or interval.

    • Cumulative frequency ($\text{Cumulative }\%$): The running total percentage of cases having up to and including a given value or interval.

  • Table 2-3: Frequency Distribution of Body Weight (Class Intervals of $20\,\text{lb}$, Total $N = 39$):

    • $100-119\,\text{lb}$: Raw Frequency = $2$, Relative Frequency = $5.1\%$, Cumulative Frequency = $5.1\%$

    • $120-139\,\text{lb}$: Raw Frequency = $4$, Relative Frequency = $10.3\%$, Cumulative Frequency = $15.4\%$

    • $140-159\,\text{lb}$: Raw Frequency = $5$, Relative Frequency = $12.8\%$, Cumulative Frequency = $28.2\%$

    • $160-179\,\text{lb}$: Raw Frequency = $6$, Relative Frequency = $15.4\%$, Cumulative Frequency = $43.6\%$

    • $180-199\,\text{lb}$: Raw Frequency = $11$, Relative Frequency = $28.2\%$, Cumulative Frequency = $71.9\%$

    • $200-219\,\text{lb}$: Raw Frequency = $4$, Relative Frequency = $10.3\%$, Cumulative Frequency = $82.1\%$

    • $220-239\,\text{lb}$: Raw Frequency = $3$, Relative Frequency = $7.7\%$, Cumulative Frequency = $89.7\%$

    • $240-259\,\text{lb}$: Raw Frequency = $2$, Relative Frequency = $5.1\%$, Cumulative Frequency = $94.9\%$

    • $260-279\,\text{lb}$: Raw Frequency = $1$, Relative Frequency = $2.6\%$, Cumulative Frequency = $97.4\%$

    • $280-299\,\text{lb}$: Raw Frequency = $0$, Relative Frequency = $0.0\%$, Cumulative Frequency = $97.4\%$

    • $300-319\,\text{lb}$: Raw Frequency = $1$, Relative Frequency = $2.6\%$, Cumulative Frequency = $100\%$

    • Total: Raw Frequency = $39$, Relative Frequency = $100.0\%$

• Histograms:

  • Graphical visual displays representing continuous or ordered categorical data.

  • The underlying data must be at least ordinal in scale (applicable to ordinal, interval, and ratio scales; inappropriate for nominal data).

  • Rules for Histogram Construction:

    • The values or class intervals of the variable are placed on the horizontal x-axis.

    • Class intervals must be mutually exclusive, exhaustive, and constructed with uniform (even) widths.

    • The vertical bars of the histogram must touch, reflecting the continuous nature of the scale.

Measures of Central Tendency

• Purpose:

  • To describe the center of a data set and identify the single value that best represents the most frequently occurring or central value.

• The Mode:

  • Definition: The most frequently recurring value within a dataset.

  • Properties: A distribution can have no mode, a single mode, or multiple modes (e.g., bimodal, multimodal).

  • Applicability: Statistically valid for nominal, ordinal, interval, and ratio level data.

• The Median:

  • Definition: The value that lies precisely in the middle of an ordered distribution; corresponds to the $50\text{th}$ percentile.

  • Properties: Divides the upper half of the data from the lower half; robust to extreme values and outliers.

  • Applicability: Appropriate for ordinal, interval, and ratio level data.

• The Mean ($\bar{x}$):

  • Definition: The arithmetic average of all values in the distribution.

  • Calculation: Computed as the sum of all individual values of the variable divided by the total sample size ($n$):

$\bar{x} = \frac{\sum x_i}{n}$

• Applicability: Most appropriate for interval and ratio level data; occasionally utilized for ordinal data.

Measures of Dispersion

• Purpose:

  • To quantify the degree of spread, variability, or scatter among the data values around the central point.

• Range:

  • Definition: The mathematical difference between the highest value and the lowest value in a data set.

  • Applicability: Appropriate for ordinal, interval, and ratio level data.

• Interquartile Range:

  • Defined contextually as the middle section of the distribution associated with the $50\text{th}$ percentile.

  • Applicability: Appropriate for ordinal, interval, and ratio level data.

• Variance ($S^2$):

  • Definition: The sum of the squared deviations of each score from the sample mean, divided by the sample size minus one ($n - 1$):

$S^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}$

*   Applicability: Designed for interval and ratio data.

• Standard Deviation ($s$):

  • Definition: The average distance of each data point from the mean; mathematically calculated as the positive square root of the variance:

$s = \sqrt{S^2}$

• Applicability: Most appropriate for interval and ratio level data.

Distribution Shape, Symmetry, and Skewness

• Symmetrical Distributions:

  • Values are evenly and equally distributed on both sides of the center.

  • There is an equal amount of data positioned to the left and to the right of the central point.

  • Important Caveat: Not all symmetrical distributions are "normal" distributions (for example, rectangular and U-shaped distributions are symmetrical but non-normal).

• Skewed (Asymmetrical) Distributions:

  • Data values are not evenly distributed around the central tendency.

  • Positive Skew (Right-Skew): The tail of the distribution extends farther to the right toward positive values; majority of scores cluster at the lower end.

  • Negative Skew (Left-Skew): The tail of the distribution extends farther to the left toward lower values; majority of scores cluster at the higher end.

• Recognized Distribution Shapes:

  • Normal Distribution: Bell-shaped, unimodal, symmetrical.

  • Rectangular Distribution: Uniform frequency across all intervals, symmetrical.

  • Bimodal Distribution (b1-model): Displays two distinct peaks or modes.

  • U-shaped Distribution: High frequencies at both extremes with low frequency in the center, symmetrical.

  • Positive (Right-Skew) Distribution: Tail trailing off to the right.

  • Negative (Left-Skew) Distribution: Tail trailing off to the left.

Questions & Discussion

• Question 1: Measurement Scale of Military Rank

  • Question: What measurement level is the variable "military rank"? (Nominal, Ordinal, Interval, or Ratio?)

  • Answer: Ordinal.

  • Rationale: Military rank does not have an absolute true zero or equal intervals between steps, but the categories can be arranged in a clear hierarchical order: private, corporal, sergeant, and master sergeant.

• Question 2: Suitability of Histograms for Categorical Variables

  • Question: True or False: A histogram would be a good way to show the distribution of ethnicity among the participants of a study.

  • Answer: False.

  • Rationale: Ethnicity is a nominal level variable. Histograms require continuous or ordered data and are exclusively used for ordinal, interval, and ratio level variables. A frequency table or bar chart would be a better choice.

• Question 3: Central Tendency for Child Height

  • Question: What would be an appropriate measure of central tendency to use when looking at height of children at age $5$? (Mode, Median, Mean, or All of these?)

  • Answer: All of these.

  • Rationale: Height is a ratio level variable (possessing an absolute, natural true zero). Because ratio data satisfy the criteria for all lower levels of measurement, the mode, median, and mean are all statistically appropriate to use.