Histograms, Distribution Shapes, Skewness, and Normal Quantile Plots

Key Concept

  • A histogram is often easier to interpret than a frequency table for understanding data distribution.

Histogram basics

  • Histogram: bars of equal width drawn adjacent (unless gaps in the data).
  • Horizontal axis: classes of quantitative data values.
  • Vertical axis: frequencies.
  • Heights of bars correspond to frequencies.

Relative Frequency Histogram

  • Same shape and horizontal scale as a histogram.
  • Vertical axis shows relative frequencies.
  • Relative frequency for a class i: rf<em>i=f</em>iNrf<em>i = \frac{f</em>i}{N} where fif_i is the class frequency and NN is the total number of items.

Uses of a Histogram

  • Visually displays the shape of the distribution.
  • Shows the center (location) of the data.
  • Shows the spread (variability).
  • Identifies outliers.

Distribution Shapes

  • Bell-shaped (Normal) distribution.
  • Uniform distribution.
  • Skewed to the right (positive skew).
  • Skewed to the left (negative skew).

Normal Distribution

  • Data that form a roughly bell-shaped histogram are said to have a normal distribution.

Skewness (1 of 3)

  • Skewness: a distribution that is not symmetric and extends more to one side than the other.
  • Right-skewed (positive): longer right tail; left side is shorter.
  • Left-skewed (negative): longer left tail; right side is shorter.
  • Mnemonic: Skewed left resembles toes on the left foot; skewed right resembles toes on the right foot.

Detecting Skewness by Mean vs. Median

  • Right-skew (positive): \text{Median} < \text{Mean}
  • Symmetric: Mean=Median\text{Mean} = \text{Median}
  • Left-skew (negative): \text{Mean} < \text{Median}

Assessing Normality with Normal Quantile Plots (1 of 5)

  • Criteria for a normal distribution: the pattern of points is reasonably close to a straight line and shows no non-straight systematic pattern.

Assessing Normality with Normal Quantile Plots (2 of 5)

  • Not Normal if the points do not lie near a straight-line pattern or show a systematic non-straight pattern.

Assessing Normality with Normal Quantile Plots (3 of 5)

  • Normal distribution criterion: points are reasonably close to a straight line and no other systematic pattern.

Assessing Normality with Normal Quantile Plots (4 of 5)

  • Not Normal if the points do not lie close to a straight line.

Assessing Normality with Normal Quantile Plots (5 of 5)

  • Not Normal if the points show a systematic pattern that is not a straight line.

Quick Reference: Visual and Interpretive Cues

  • Use histograms to gauge shape, center, spread, and outliers at a glance.
  • Use relative frequency histograms to compare distributions with different totals.
  • Check skewness to decide whether mean or median is a better measure of center.
  • Use normal quantile plots to assess normality before applying methods that assume normality.