Descriptive Statistics: Frequency Distribution Tables and Histograms

Descriptive Statistics: Frequency Distribution Tables and Histograms

Introduction

Descriptive statistics provide methods to summarize and present raw data in a meaningful way. Two main functions are to organize data and to display it for easier interpretation. This section focuses on frequency distribution tables and histograms, which are foundational tools in statistics for visualizing and summarizing data sets.

Frequency Distribution Tables & Histograms

Frequency Distribution Table

A frequency distribution table summarizes data by showing how many times each value or range of values (class) occurs in a data set. This helps in understanding the distribution and patterns within the data.

• Raw data: Data in its original, unorganized form.

• Grouped frequency distribution: Data is organized into classes or intervals, and the frequency of data points in each class is recorded.

• Ungrouped frequency distribution: Each unique value is listed with its frequency.

Histogram

A histogram is a graphical representation of a frequency distribution. It uses adjacent bars to show the frequency of data within each class interval. The height of each bar corresponds to the frequency of the class it represents.

• Histograms are used for quantitative data.

• They help visualize the shape, spread, and central tendency of the data.

Mechanics for Making a Grouped Frequency Table

To construct a grouped frequency table, follow these steps:

1. Decide on the class limits and class width.

2. Count the number of observations in each class limit (this is the frequency).

3. Determine the number of classes (should be between 5 and 20).

4. Calculate the class width using the formula:

$\text{class width} = \frac{\text{maximum number} - \text{minimum value}}{\text{number of classes}}$

5. Round your class width to the nearest integer.

6. Choose the minimum data value or a convenient value below it as the first lower class limit.

7. Using the first lower class limit and class width, proceed to list the other lower class limits.

8. List the upper class limits in vertical column and proceed to enter the upper class limits to the right of each lower class limit.

9. Using each individual data value, put a tally mark in the appropriate class as a preliminary count.

10. Add all the tally marks to get the frequency.

Variations of Frequency Distribution Table

• Relative Frequency: The proportion of data values in each class relative to the total number of data values.

• Cumulative Frequency: The sum of frequencies for all classes up to and including the current class.

Other Types of Graphs That Enlighten

Besides histograms, several other graphical methods are used to display data distributions and relationships:

Graph Type	Description
Stem-and-leaf	Used to quickly get a sense of the distribution of data; retains original data values.
Time-Series graph	Shows how a variable changes over time; useful for trends and forecasting.
Dotplot	Similar to a histogram but uses dots instead of bars to record occurrences.
Pareto chart	Orders bars so that they decrease in height; helps highlight the most frequent categories.
Pie chart	Shows the whole into sectors proportional to the respective frequencies.
Scatterplot	Graph that relates two variables to each other, such as height and weight.

Example: Constructing a Grouped Frequency Table

• Suppose you have the following data set: 2, 5, 7, 8, 10, 12, 13, 15, 18, 20.

• Step 1: Decide on the number of classes (e.g., 4).

• Step 2: Calculate class width: $\frac{20 - 2}{4} = 4.5$ (round to 5).

• Step 3: Classes: 2-6, 7-11, 12-16, 17-21.

• Step 4: Count frequencies for each class.

Additional info:

• Relative frequency is calculated as $\frac{\text{frequency of class}}{\text{total number of data points}}$.

• Cumulative frequency is useful for determining medians and percentiles.