Overview
This lecture covers how to visually display quantitative (numerical) variables using frequency distributions, relative frequency distributions, and histograms, and introduces common shapes of distributions.
Quantitative Variables and Their Displays
- Quantitative variables are numerical, e.g., height, age, weight, number of rooms.
- Bar charts and pie charts display categorical variables, not quantitative ones.
- Frequency distribution shows how many times each value of a variable occurs.
- Relative frequency distribution shows the proportion of each value (frequency divided by total).
- The sum of all relative frequencies should equal 1, but may differ slightly due to rounding.
Histograms: Construction and Purpose
- Histograms visually display distributions of quantitative variables.
- Histograms use "classes" or "bins," which are ranges of values; each bar shows the number in each class.
- Histograms are similar to bar charts, but with no gaps between bars and data must have a natural order.
- The y-axis may show frequencies (counts) or relative frequencies (proportions).
- Classes (or bins) should typically be the same width for clarity (e.g., 10-unit intervals).
- When constructing classes, start below the lowest data value and ensure all intervals cover the full data range.
Examples of Creating Histograms
- Example: Number of available cars in 50 households creates classes for each car count.
- Example: Eruption times of Old Faithful geyser use 10-second intervals as class widths.
- Tally marks are a helpful way to count observations in each class when constructing frequency tables.
Distribution Shapes
- Uniform distribution: All classes have about equal frequencies (e.g., rolling a fair die).
- Bell-shaped (normal) distribution: Most observations are in the center, with fewer at extremes; symmetric.
- Skewed-right distribution: Tail extends to the right; many values are low, few are high.
- Skewed-left distribution: Tail extends to the left; many values are high, few are low.
- Symmetric distributions (uniform, bell-shaped) have even balance; skewed distributions do not.
Key Terms & Definitions
- Quantitative Variable — a variable measured numerically.
- Frequency Distribution — a table showing the count of observations for each value or class.
- Relative Frequency Distribution — a table showing proportions for each value or class.
- Histogram — a graph for quantitative data using adjacent bars to represent class frequencies.
- Class/Bin — a range of values grouped together in a histogram.
- Uniform Distribution — frequencies are evenly spread across classes.
- Bell-shaped Distribution — symmetric, mound-shaped distribution with most values near the center.
- Skewed Distribution — asymmetric distribution with a longer tail on one side.
Action Items / Next Steps
- Practice creating frequency and relative frequency distributions from sample quantitative data.
- Construct histograms using data sets by creating appropriate classes and tallying frequencies.
- Review textbook examples of different distribution shapes.