Descriptive Statistics: Frequency Distributions and Their Graphs

Chapter 2: Descriptive Statistics

Section 2.1: Frequency Distributions and Their Graphs

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

Frequency Distribution

• Definition: A frequency distribution is 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.

• 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: $\text{Class Width} = \frac{\text{Range}}{\text{Number of Classes}}$ Round up to the next convenient number.

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

Constructing a Frequency Distribution

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

2. Find the class width using the formula above.

3. Determine class limits:

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

   • Add the class width to find subsequent lower limits.

   • Upper limit of a class is one less than the lower limit of the next class.

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

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

Example: Constructing a Frequency Distribution

Given cell phone screen times (in minutes) for 30 adults, construct a frequency distribution with 7 classes. Minimum value: 155, Maximum value: 405.

• Range: $405 - 155 = 250$

• Class width: $\frac{250}{7} \approx 35.7$ (rounded up to 36)

• First lower limit: 155; subsequent lower limits found by adding 36.

• First upper limit: 190; subsequent upper limits found by adding 36.

Midpoints, Relative Frequency, and Cumulative Frequency

• Midpoint: The average of the lower and upper class limits. $\text{Midpoint} = \frac{\text{Lower Class Limit} + \text{Upper Class Limit}}{2}$

• Relative Frequency: The proportion or percentage of 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: Calculating Midpoints, Relative and Cumulative Frequencies

• Midpoints are calculated for each class using the formula above.

• Relative frequencies are calculated as the frequency divided by the total number of data entries.

• Cumulative frequencies are calculated by summing frequencies up to each class.

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

Graphs of Frequency Distributions

• Frequency Histogram: A bar graph representing the frequency distribution. The horizontal scale is quantitative, and consecutive bars must touch.

• Class Boundaries: Numbers that separate classes without forming gaps. Used so histogram bars touch.

• Frequency Polygon: A line graph emphasizing continuous change in frequencies. Begins and ends on the horizontal axis, extended one class width beyond the first and last midpoints.

• Relative Frequency Histogram: Similar to a frequency histogram, but the vertical scale measures relative frequencies.

• Cumulative Frequency Graph (Ogive): A line graph displaying cumulative frequency at each upper class boundary. 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).

Constructing Graphs

1. For histograms, mark the horizontal scale at class boundaries or midpoints.

2. For frequency polygons, use class midpoints and frequencies, extending the graph as described.

3. For ogives, plot points at upper class boundaries and corresponding cumulative frequencies, connecting them left to right.

Technology for Constructing Histograms

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

• Enter midpoints and frequencies, select histogram plot, and use features to trace and analyze bars.

Additional info: The included image is the cover of the textbook "Elementary Statistics" by Ron Larson, which is directly relevant as it visually identifies the source and context of the study notes.