Describing Data with Frequency Distributions and Graphs

Describing Data with Tables and Graphs

Section 2.1: Frequency Distributions and Their Graphs

This section introduces the concept of frequency distributions and their graphical representations, which are foundational tools for organizing and summarizing quantitative data in statistics.

• Frequency Distribution: A table that displays classes or intervals of data along with the count (frequency) of values in each class.

• Purpose: To organize raw data into a more interpretable format, revealing patterns and trends.

Key Terms and Definitions

• Class: A category or interval into which data values are grouped.

• Frequency (f): The number of data values 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 difference between the lower limits (or upper limits) of consecutive classes. $\text{Class Width} = \text{Lower Limit of 2nd Class} - \text{Lower Limit of 1st Class}$

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

Constructing a Frequency Distribution

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

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

3. Find the lower limit of the first class (often the minimum value or a convenient value below it).

4. Determine the remaining lower class limits by adding the class width successively.

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

6. Tally the data into classes and count the frequencies.

Example Table: Frequency Distribution

Class	Frequency
191-226	3
227-262	4
263-298	7
299-334	8
335-370	5
371-406	3

Determining the Midpoint

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

• Example: For the class 190-226, $\text{Midpoint} = \frac{190 + 226}{2} = 208$

Relative Frequency

• The proportion or percentage of data values in each class. $\text{Relative Frequency} = \frac{\text{Class Frequency}}{\text{Sample Size}}$

Cumulative Frequency

• The sum of the frequencies for that class and all previous classes.

• The cumulative frequency of the last class equals the sample size.

Example Table: Frequency, Relative Frequency, and Cumulative Frequency

Class	Frequency	Relative Frequency	Cumulative Frequency
191-226	3	0.10	3
227-262	4	0.13	7
263-298	7	0.23	14
299-334	8	0.27	22
335-370	5	0.17	27
371-406	3	0.10	30

Graphs of Frequency Distributions

• Histogram: A bar graph representing the frequency of classes. The horizontal axis shows class boundaries or midpoints, and the vertical axis shows frequency.

• Frequency Polygon: A line graph connecting the midpoints of each class at their respective frequencies. The line starts and ends at the horizontal axis.

• Relative Frequency Histogram: Similar to a histogram, but the vertical axis represents relative frequencies (percentages or decimals).

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

Example Table: Cumulative Frequency (Ogive)

Upper Class Boundary	Cumulative Frequency
226.5	3
262.5	7
298.5	14
334.5	22
370.5	27
406.5	30

Applications and Examples

• Frequency distributions and their graphs are used to summarize large data sets, making it easier to identify patterns such as skewness, modality, and outliers.

• Example: Analyzing cell phone screen times for a group of adults to determine usage patterns.

Summary of Steps for Constructing Frequency Distributions and Graphs

1. Organize raw data into classes.

2. Calculate class width and limits.

3. Tally data into classes and compute frequencies.

4. Calculate midpoints, relative frequencies, and cumulative frequencies as needed.

5. Construct appropriate graphs (histogram, frequency polygon, relative frequency histogram, ogive).

Additional info: These notes provide foundational skills for later topics such as descriptive statistics, probability distributions, and inferential statistics.