Descriptive Statistics: Frequency Distributions and Graphs

Descriptive Statistics

Frequency Distributions and Their Graphs

Descriptive statistics involves methods for organizing and summarizing data. One of the foundational tools is the frequency distribution, which helps visualize how data values are spread across different intervals or classes.

• Frequency Distribution: A table showing classes or intervals of data with a count of the number of entries in each class. The frequency (f) of a class is the number of data entries in that class.

• Class Limits: Each class has a lower class limit (least number that can belong to the class) and an upper class limit (greatest number that can belong to the class).

• Class Width: The distance between lower (or upper) limits of consecutive classes. Calculated as: $\text{Class Width} = \frac{\text{Range}}{\text{Number of Classes}}$

• Range: The difference between the maximum and minimum data entries. $\text{Range} = \text{Maximum Value} - \text{Minimum Value}$

Constructing a Frequency Distribution

To construct a frequency distribution, follow these steps:

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

2. Find the class width by dividing the range by the number of classes and rounding up.

3. Determine the class limits, starting with the minimum data entry as the lower limit of the first class.

4. Find the upper class limits (one less than the lower limit of the next class).

5. Make tally marks for each data entry in the appropriate class and count the total frequency for each class.

Example: Constructing a Frequency Distribution

Given cell phone screen times for 30 U.S. adults, construct a frequency distribution with seven classes. The minimum value is 155, and the class width is rounded up to 36.

• First lower limit: 155

• First upper limit: 190

• Continue adding class width to find remaining limits.

Measures Derived from Frequency Distributions

• Midpoint: The midpoint of a class is calculated as: $\text{Midpoint} = \frac{\text{Lower Class Limit} + \text{Upper Class Limit}}{2}$

• Relative Frequency: The portion or percentage of the data that falls in a particular class: $\text{Relative Frequency} = \frac{\text{Frequency of Class}}{\text{Sample Size}}$

• Cumulative Frequency: The sum of the frequency for that class and all previous classes. The cumulative frequency of the last class equals the sample size.

Example: Finding Midpoints, Relative and Cumulative Frequencies

For each class, calculate the midpoint, relative frequency, and cumulative frequency. Patterns may emerge, such as the most common range or the proportion of data below a certain value.

Frequency Histograms

A frequency histogram is a bar graph representing the frequency distribution. The horizontal scale is quantitative and measures data values, while the vertical scale measures frequencies. Consecutive bars must touch.

• Class Boundaries: Numbers that separate classes without forming gaps between them. Bars begin and end at class boundaries to ensure continuity.

Example: Constructing a Frequency Histogram

Draw a frequency histogram for the cell phone screen times. Mark the horizontal scale at either midpoints or class boundaries.

Frequency Polygon

A frequency polygon is a line graph that emphasizes the continuous change in frequencies. It is constructed using class midpoints and frequencies, and the graph begins and ends on the horizontal axis, extended one class width beyond the first and last midpoints.

Example: Constructing a Frequency Polygon

Plot the points representing midpoints and frequencies, connect them with line segments, and extend the graph as described.

Relative Frequency Histogram

A relative frequency histogram has the same shape and horizontal scale as the frequency histogram, but the vertical scale measures relative frequencies instead of absolute frequencies.

Example: Constructing a Relative Frequency Histogram

For the cell phone screen times, the histogram shows the proportion of adults in each class interval.

Cumulative Frequency Graph (Ogive)

An ogive is a line graph displaying the cumulative frequency of each class at its upper class boundary. The horizontal axis marks upper class boundaries, and the vertical axis marks cumulative frequencies.

• To construct an ogive, plot points for each upper class boundary and its cumulative frequency, connect them, and start at the lower boundary of the first class (cumulative frequency zero).

Example: Constructing an Ogive

The ogive for cell phone screen times shows how many adults had screen times below certain thresholds, with the steepest increase indicating the most common interval.

Technology for Constructing Histograms

Statistical software and calculators (e.g., TI-84 Plus, Minitab, Excel, StatCrunch) can be used to construct histograms and other graphs. Enter midpoints and frequencies, select the appropriate graph type, and interpret the results.

Example: Using Technology

Instructions for TI-84 Plus: Enter midpoints in L1, frequencies in L2, turn on Plot 1, select Histogram, and use Xlist: L1, Freq: L2. Use TRACE to view frequencies for each class.