2.1.2 Frequency Tables

Frequency tables are a fundamental tool in statistics for organising and summarising data. They show how often each value or range of values (class intervals) appears in a dataset. Frequency tables are particularly useful in both descriptive statistics and for laying the groundwork for further statistical analysis.

Basic Structure of a Frequency Table

A frequency table typically includes the following columns:

1. Data Values or Class Intervals:

• For Discrete Data: This column lists each distinct value in the dataset.
• For Continuous Data: This column shows ranges (class intervals) that group the data values.

2. Frequency:

• The number of times each value or interval occurs in the dataset.

3. Cumulative Frequency (optional):

• This column shows the running total of frequencies up to each value or interval, helping to identify medians and quartiles.

4. Relative Frequency (optional):

• The proportion of the total frequency that each value or interval represents. It is often expressed as a percentage.

Constructing a Frequency Table

Step 1: Collect Data

• Gather the raw data that needs to be summarised. Step 2: Determine the Range

• Find the minimum and maximum values in your data. Step 3: Choose Class Intervals (for Continuous Data)

• For continuous data, divide the range into equal intervals. The number of intervals (often called classes) depends on the dataset size and distribution. Step 4: Tally the Frequencies

• Count how many data points fall into each value or class interval and record these counts in the frequency column. Step 5: Calculate Cumulative and Relative Frequencies (if needed)

• Cumulative Frequency: Add the frequency of each interval to the sum of the previous frequencies.

• Relative Frequency: Divide each frequency by the total number of data points and multiply by $100$ to get a percentage.

Example of a Frequency Table

Interpreting Frequency Tables

• Identifying the Mode: The mode is the value or interval with the highest frequency.
• Calculating the Median: For discrete data, locate the median by finding the middle value(s) in the cumulative frequency column. For continuous data, use the cumulative frequency to identify the class interval containing the median and then interpolate within that interval if necessary.
• Understanding Data Distribution: The shape of the frequency distribution (e.g., symmetric, skewed) can be inferred from the table, especially when visualised as a histogram or frequency polygon.

Advantages of Frequency Tables

• Data Organisation: They provide a clear summary of large datasets, making it easier to see patterns.
• Facilitates Further Analysis: Frequency tables are often the first step in data analysis, leading to the calculation of measures of central tendency (mean, median, mode), dispersion (range, variance, standard deviation), and graphical representations (histograms, cumulative frequency curves).

Summary

Frequency tables are essential for organising and summarising data. They serve as a foundation for more complex analyses, clearly showing how data points are distributed. Whether dealing with discrete or continuous data, constructing and interpreting frequency tables is a key skill in statistical analysis.