Describing Shapes of Distributions and Measures of Central Tendency

Baseline Visualizations and the Normal Distribution

• Descriptive Shapes Baseline:

  • When describing the shape of an empirical data distribution, it is implicitly or explicitly contrasted with the theoretical normal distribution (commonly called the bell curve).
  • Hallmark of Normality:
    • A large concentration of scores is clustered in the middle of the distribution.
    • The frequency of scores decreases symmetrically and parallelly as values move higher or lower away from the center.

• Structural Sections of a Distribution:

  • Center: The middle region of the distribution containing the highest density of scores.
  • Tails: The extreme upper and lower bounds at the outer ends of the distribution.
  • Shoulder: The transitional region located between the center peak and the outer tails (getting closer to the top of the curve).

• Departures from Normality:

  • If an empirical distribution perfectly matches the bell curve, it is simply described as normal.
  • Describing a distribution in words involves detailing the exact ways in which its shape departs from the default baseline of normality.
  • Shape descriptions evaluate three primary characteristics: Number of Modes, Skewness, and Kurtosis.

Characteristic 1: Number of Modes

• Definition of Mode:

  • The mode is the most frequently occurring value in a dataset.
  • On a frequency polygon, a mode corresponds to a distinct highest point or peak.

• Modal Classification Types:

  • Unimodal:
    • A distribution possessing exactly one prominent peak ($1$ mode).
    • Unimodal is the default expected state for a normal distribution.
  • Bimodal / Approximately Bimodal:
    • A distribution possessing two distinct peaks ($2$ modal values).
    • Qualifier Usage ( ext{Approximately}): If one peak is technically slightly higher than another peak, but both clearly stand out relative to the rest of the distribution, the accurate descriptor is approximately bimodal rather than unimodal.
    • Exclusion of Minor Peaks: A small peak located far away in the tail end of a distribution is not counted as a modal value (e.g., it does not make a distribution trimodal) if its height is negligible compared to the primary peaks.
  • Rectangular / Approximately Rectangular / Multimodal:
    • A frequency polygon whose overall contour roughly forms a rectangle.
    • Occurs when frequencies are essentially uniform across the entire range of possible values of $x$.
    • In a rectangular distribution, virtually every value functions as a mode; thus, it is also referred to as multimodal.

Characteristic 2: Skewness

• Definition of Skewness:

  • Skewness refers specifically to the asymmetry or lack of symmetry in a distribution shape (not to be confused with general statistical bias).
  • Symmetry Test: If a distribution graph were folded directly in half onto itself, a symmetrical distribution would match up identically on both sides.

• Classifications of Skewness:

  • Symmetrical / Approximately Symmetrical:
    • The left and right halves of the distribution are roughly mirror images of each other.
  • Positive Skew (Skewed to the Right):
    • The majority of scores cluster at the lower (left) end of the scale.
    • A long, extended tail points toward the positive (right) end of the $x$-axis.
    • Naming Convention Rule: Distributions are named after the direction in which the long, elongated tail points, not where the majority of scores are clustered.
  • Negative Skew (Skewed to the Left):
    • The majority of scores cluster at the higher (right) end of the scale.
    • A long, extended tail points toward the negative (left) end of the $x$-axis.
    • Visually creates a lopsided curve pointing to the left.

Characteristic 3: Kurtosis

• Definition of Kurtosis:

  • Etymologically derived from the Greek word for "curve" or "arch".
  • Refers to the relative peakedness and tail weight of a distribution curve compared to a standard normal distribution.

• Classifications of Kurtosis:

  • Mesokurtic:
    • The prefix meso- means "middle".
    • Represents a standard normal distribution with a balanced, intermediate peak height and expected tail proportions.
  • Leptokurtic:
    • The prefix lepto- means "slender".
    • Characterized by a tall, narrow/slender central peak.
    • Features heavy tails, meaning there are more scores residing in the extreme tails than found in a normal distribution.
  • Platykurtic:
    • The prefix platy- derives from flat features (e.g., the bill of a platypus).
    • Characterized by a broad, flat curve with a low peak height.
    • Features light tails, containing very few extreme scores at the outer ends.

Central Tendency: The Mean

• Purpose of Central Tendency:

  • Central tendency summarizes an entire dataset by identifying a single numerical value representing the middle or center of the distribution.
  • Serves as a fundamental descriptive metric and integrates directly into inferential hypothesis testing.

• Definition and Formula of the Mean:

  • The mean is the arithmetic average of all scores in a distribution.
  • Formula:         $\bar{x} = \frac{\sum x}{N}$
    • $x$ represents individual raw scores.
    • $\sum$ (Capital Greek letter Sigma) denotes the summation operator (instruction to sum all values).
    • $N$ represents the total number of scores in the sample.

• Notation Standards:

  • $\bar{x}$ (X-bar) is standard notation in general mathematics and introductory statistics.
  • Capital $M$ (often underlined as $\underline{M}$) is the official standard format required by APA Style (American Psychological Association) for reporting sample averages.

• Calculation Walkthrough:

  • Dataset of stress scores ranging from $0$ to $10$ ($N = 16$):
    • Sum of scores: $\sum x = 109$
    • Sample size: $N = 16$
    • Mean calculation:             $M = \frac{109}{16} = 6.8$
  • An average of $6.8$ provides an accurate center estimate for scores clustered around $7$.

• Limitations and Outlier Sensitivity:

  • The mean is the default measure of central tendency.
  • Major Drawback: The mean is highly sensitive to outliers (extreme values far removed from the rest of the data).
  • Real-World Example (Household Income):
    • Income data is bounded at the lower end by $0$ dollars (cannot earn negative income).
    • Income is completely unbounded at the upper end, allowing extreme high earners to pull the curve far to the right (positive skew).
    • Including extreme high incomes artificially inflates the mean, making average household income look substantially higher than what a typical family earns.

Central Tendency: The Median

• Definition and Core Advantage:

  • The median is the exact positional middle score in an ordered distribution, dividing the upper $50\%$ of scores from the lower $50\%$
  • Primary Virtue: The median is robust and uninfluenced by extreme outliers or heavy skewness.

• Step-by-Step Calculation Procedure:

  1. Arrange all raw scores in ascending order from lowest to highest.
  2. Iteratively cross off/eliminate paired scores starting simultaneously from the highest and lowest ends.
  3. Odd Number of Scores: The single score remaining in the exact center is the median.
  4. Even Number of Scores: Two central scores will remain. Compute the arithmetic average of these two middle values:        $\text{Median} = \frac{x_1 + x_2}{2}$

• Dataset Execution Example:

  • Arranged data ($N = 16$): $3, 4, 4, 5, 6, 7, 7, 7, 7, 7, 8, 8, 9, 9, 10$
  • Sequential outer elimination yields two center scores: $7$ and $7$.
  • Median calculation:         $\text{Median} = \frac{7 + 7}{2} = 7$

• Comparison: Mean vs. Median:

  • Why the Mean is Preferred over the Median as Default:
    1. Algebraic Integration: Calculating a mean relies on simple arithmetic operations ($\sum x / N$), which integrate seamlessly into complex mathematical formulas. Automating positional sorting for medians within algebraic formulas is far more difficult.
    2. Information Retention: The mean incorporates the exact quantitative value and distance of every single score in the dataset. The median ignores magnitude differences above and below the center point, reflecting only relative position.

Central Tendency: The Mode

• Definition and Identification:

  • The mode is the score or category value corresponding to the maximum absolute frequency in a distribution.
  • Identified visually as the peak on a polygon or by finding the highest count in a frequency table.

• Primary Advantage:

  • The mode is the only measure of central tendency that can be calculated for non-numeric, nominal, or categorical data.
  • Example: Eye color responses cannot be arithmetically averaged (e.g., one cannot calculate an average eye color value), but one can identify the modal eye color (the single most common eye color category present in the sample).