Overview
This lecture covers how to organize and summarize collected data using frequency distributions, including definitions, step-by-step construction, and how to represent these distributions in graphs.
Frequency Distributions
- A frequency distribution organizes data into groups (classes) and counts (frequencies).
- Classes are groups of data values; frequencies are the counts for each class.
- Example: Hair color grouped as blonde, brown, black with counts for each.
- Frequency distribution shows trends and patterns not easily seen in raw data.
Constructing a Frequency Distribution
- Determine the number of classes (groups) appropriate for the data set.
- Calculate class width: (maximum value - minimum value) ÷ number of classes, then round up to the nearest whole number.
- Choose a starting point (usually the minimum data value) for the first lower class limit.
- Create classes by adding the class width to each lower class limit.
- Upper class limit is one less than the next lower class limit.
- Always round class width up to ensure all data is covered.
Key Components of Frequency Distributions
- Lower class limit: Smallest value in a class.
- Upper class limit: Largest value in a class.
- Class midpoint: (Lower class limit + Upper class limit) ÷ 2.
- Class boundary: Value between an upper class limit and the next lower class limit (used for histograms).
Relative and Cumulative Frequency Distributions
- Relative frequency = class frequency ÷ total number of data items (often shown as a percentage).
- Cumulative frequency adds up frequencies as you move through the classes, showing the running total.
Graphical Representations
- Histograms are bar charts with touching bars, displaying frequencies or relative frequencies for each class.
- Use class midpoints or boundaries for the horizontal axis, frequencies for the vertical.
- Relative frequency histograms swap counts for percentages.
- Cumulative frequency histograms plot cumulative totals.
- Normal distribution has a central peak with data rising and then falling symmetrically; not all data sets are normal.
Key Terms & Definitions
- Frequency Distribution — Table showing classes of data and corresponding counts.
- Class — Group or range within a data set.
- Class Width — Difference between two lower class limits.
- Lower Class Limit — Smallest value in a class.
- Upper Class Limit — Largest value in a class.
- Class Midpoint — Average of a class’s lower and upper limits.
- Class Boundary — Value between upper limit of one class and lower limit of the next, used to prevent gaps in histograms.
- Relative Frequency — Proportion of data within a class (as a percentage).
- Cumulative Frequency — Running total of frequencies up to a given class.
- Histogram — Bar graph with adjoining bars representing frequency distributions.
- Normal Distribution — Data distribution with a central peak and symmetrical rise and fall.
Action Items / Next Steps
- Complete Section 1.5, page 34–35, problems 1, 2, and 5–25 by Monday.
- Review how to construct and interpret frequency, relative frequency, and cumulative frequency distributions.
- Practice making histograms from sample data using both class midpoints and boundaries.