Descriptive Statistics and Graphical Summaries

Fundamentals of Descriptive Statistics and Data Visualization

  • Descriptive Statistics:
    • Defined as graphical and numerical summaries of data.
    • Serves as a fundamental tool used to explore the main features and underlying patterns within data.
    • Used to paint a picture of how the values of a variable are distributed among the observational units.
  • Distribution of a Variable:
    • Describes the specific values that a variable takes on.
    • Details how often (the frequency with which) the variable takes those specific values.
  • Graphs and Plots:
    • Considered essential in data analysis; data should ALWAYS be plotted.
    • Used to display the distribution of a variable in an easily readable and interpretable manner.

Types of Graphical Representations

  • Bar Charts:
    • Variable Type: Used for categorical variables.
    • Axes Construction:
      • The x-axis depicts the possible values of the variable.
      • The y-axis depicts their corresponding relative frequencies.
  • Dotplots:
    • Variable Type: Used for quantitative variables with a small number of discrete values.
    • Construction Procedure:
      1. Draw a number line along the x-axis.
      2. Vertically stack dots, placing one dot per observational unit corresponding to the observed value of the variable.
    • Data Preparation: Sometimes values are rounded before plotting.
  • Histograms:
    • Variable Type: Used for quantitative variables when there are many observational units and a wide range of values.
    • Construction Procedure:
      1. Divide the range of observed values of the variable into intervals (or bins) of equal width.
        • Bin Width Consideration: Too few bins will over-summarize the data; too many bins will obscure the overall pattern.
      2. Draw a number line along the x-axis.
      3. Plot vertical bars corresponding to the number of units within each bin.
  • Boxplots:
    • Variable Type: Used for all types of quantitative variables.
    • Construction Procedure:
      1. Compute the five key values of the data: minimum, maximum, median, lower quartile, and upper quartile.
        • Lower Quartile: The value where 25%25\% of the data lies below and 75%75\% lies above.
        • Upper Quartile: The value where 75%75\% of the data lies below and 25%25\% lies above.
      2. Calculate the Inter-Quartile Range (IQRIQR):             IQR=upper quartile−lower quartileIQR = \text{upper quartile} - \text{lower quartile}
      3. Identify outliers as points located more than 1.5×IQR1.5 \times IQR above the upper quartile or below the lower quartile.
      4. Draw the boxplot according to standard structural boxplot diagrams.

Describing Distributions

  • Center (Central Tendency):
    • Addresses the question: What value of the variable is most central or "typical"?
    • Sample Mean:
      • Represents the arithmetic average of the observed values.
      • Usually denoted by xˉ\bar{x} (pronounced "x-bar").
      • For nn observed values (x1,x2,…,xn)(x_1, x_2, \dots, x_n), calculated using the formula:             xˉ=x1+x2+⋯+xnn\bar{x} = \frac{x_1 + x_2 + \dots + x_n}{n}
      • Serves as the physical balance point (center of gravity) of the distribution.
    • Sample Median:
      • Represents the middle value when observed values are arranged in numerical order.
      • Divided equally such that 50%50\% of observed values are less than the median and 50%50\% are greater than the median.
      • Corresponds to the "equal areas" point of the distribution.
  • Shape:
    • Symmetric: A distribution is symmetric if the shape of values below the center is a rough mirror image of the shape of values above the center.
    • Skewed to the Right: A distribution is skewed to the right if values above the center are distributed over a larger range than values below the center (characterized by a long right tail).
    • Skewed to the Left: A distribution is skewed to the left if values below the center are distributed over a larger range than values above the center (characterized by a long left tail).
    • Mode:
      • Refers to a "peak" of the distribution, corresponding to values that occur with highest frequency.
      • A graph may have one mode (one peak) or multiple modes (more than one peak).
    • Choosing Center Measures:
      • When a distribution is skewed or contains extreme values, the median might be a more appropriate measure of center than the mean.
      • The mean is sensitive to extreme values (both extremely large and extremely small); the median is not sensitive to extreme values.
  • Unusual Observations:
    • Involves identifying any notable departures from the overall pattern.
    • Values that do not fit the pattern might indicate outliers.
    • Outlier Protocol: Always investigate outliers, identify their source, and consider their impact on the study results. Do not just eliminate them!
  • Variability (Spread):
    • Range:
      • Serves as a rough measure of variability.
      • Calculated as:             range=maximum observed value−minimum observed value\text{range} = \text{maximum observed value} - \text{minimum observed value}
    • Sample Standard Deviation (SDSD):
      • A commonly used measure of variability, usually denoted by ss.
      • Measures how far observed values are away from the mean, on average.
      • For variables on the same scale, a larger SDSD indicates a greater amount of variability.
      • For many variables, most observed values will lie within one or two standard deviations of the mean.
      • The units of the SDSD are identical to the measurement units of the variable.