Descriptive Statistics: Frequency Distributions and Graphs

Chapter 2: Descriptive Statistics

2.1 Frequency Distributions and Their Graphs

Descriptive statistics involve organizing, summarizing, and displaying data. One of the foundational tools is the frequency distribution, which helps reveal patterns in data sets by grouping values into classes and displaying their frequencies.

Frequency Distribution

• Frequency Distribution: A table that shows classes or intervals of data with a count of the number of entries in each class.

• Frequency (f): The number of data entries in a class.

• Lower Class Limit: The smallest value that can belong to a class.

• Upper Class Limit: The largest value that can belong to a class.

• Class Width: The distance between lower (or upper) limits of consecutive classes.

• Range: The difference between the maximum and minimum data entries.

Constructing a Frequency Distribution

1. Decide on the number of classes (usually 5–20).

2. Find the class width:

   • Determine the range of the data.

   • Divide the range by the number of classes and round up to the next convenient number.

3. Find the class limits:

   • Use the minimum data entry as the lower limit of the first class.

   • Add the class width to get the lower limits of subsequent classes.

   • Find the upper limit of the first class (one less than the lower limit of the next class), and repeat for the rest.

4. Make tally marks for each data entry in the appropriate class.

5. Count the tally marks to find the total frequency for each class.

Example: Frequency Distribution Construction

• Given a data set of cell phone screen times for 30 adults, construct a frequency distribution with 7 classes.

• Calculate class width, class limits, and tally frequencies as described above.

Midpoints, Relative Frequency, and Cumulative Frequency

• Midpoint: The value halfway between the lower and upper class limits.

  • Formula: $\text{Midpoint} = \frac{\text{Lower Class Limit} + \text{Upper Class Limit}}{2}$

• Relative Frequency: The proportion or percentage of the data that falls in a particular class.

  • Formula: $\text{Relative Frequency} = \frac{\text{Class Frequency}}{\text{Sample Size}}$

• Cumulative Frequency: The sum of the frequency for that class and all previous classes.

Example: Calculating Midpoints, Relative and Cumulative Frequencies

• For each class, calculate the midpoint, relative frequency, and cumulative frequency.

• Patterns can be identified, such as the most common range or the proportion of data below a certain value.

Graphs of Frequency Distributions

Visual representations of frequency distributions help in understanding the data's shape and patterns.

Frequency Histogram

• A bar graph representing the frequency distribution.

• The horizontal axis is quantitative (data values), and the vertical axis is frequency.

• Bars must touch, indicating continuous data.

Class Boundaries

• Numbers that separate classes without forming gaps between them.

• Used to ensure bars in a histogram touch.

Frequency Polygon

• A line graph that emphasizes the continuous change in frequencies.

• Uses class midpoints on the horizontal axis and frequencies on the vertical axis.

• Begins and ends on the horizontal axis, one class width before the first and after the last midpoint.

Relative Frequency Histogram

• Similar to a frequency histogram, but the vertical axis shows relative frequencies instead of absolute frequencies.

Cumulative Frequency Graph (Ogive)

• A line graph displaying the cumulative frequency of each class at its upper class boundary.

• The horizontal axis marks upper class boundaries; the vertical axis marks cumulative frequencies.

• The graph starts at the lower boundary of the first class (cumulative frequency zero) and ends at the upper boundary of the last class (cumulative frequency equals sample size).

Example: Constructing Graphs

• Construct histograms, frequency polygons, relative frequency histograms, and ogives using the frequency distribution data.

• Interpret patterns, such as the most common ranges or cumulative proportions.

Using Technology

• Graphing calculators and statistical software (e.g., TI-84 Plus, Minitab, Excel, StatCrunch) can be used to construct frequency histograms and other graphs.

• Typically, enter midpoints and frequencies, select the appropriate graph type, and interpret the output.

Summary Table: Frequency Distribution Components

Component	Definition	Formula/Notes
Class Limits	Smallest and largest values in a class	Lower and upper class limits
Class Width	Distance between consecutive class limits	$\text{Class Width} = \frac{\text{Range}}{\text{Number of Classes}}$ (rounded up)
Midpoint	Center value of a class	$\text{Midpoint} = \frac{\text{Lower Limit} + \text{Upper Limit}}{2}$
Relative Frequency	Proportion of data in a class	$\text{Relative Frequency} = \frac{f}{n}$
Cumulative Frequency	Sum of frequencies up to a class	Running total of frequencies

Additional info: This summary covers the foundational concepts of frequency distributions and their graphical representations, as outlined in Section 2.1 of an introductory statistics textbook. These tools are essential for organizing and interpreting raw data in statistics.