Lesson 2.1 Frequency Table, Tally, Midpoint, Relative Frequency and Cumulative Frequency

Introduction

This tutorial provides a comprehensive guide on creating and understanding frequency tables, tallies, midpoints, relative frequencies, and cumulative frequencies. These concepts are fundamental in statistics for organizing and interpreting data effectively.

Step 1: Understanding Frequency Tables

A frequency table is a way to organize data into categories, showing how often each category occurs.

• Identify your data set: Gather the raw data you want to analyze.
• List categories: Determine the categories that the data can be grouped into.
• Count frequencies: For each category, count how many times it occurs in the data set.

Example

If your data set includes the ages of students: 21, 22, 21, 23, 24, 22, 21, create a frequency table like this:

| Age | Frequency | |-----|-----------| | 21 | 3 | | 22 | 2 | | 23 | 1 | | 24 | 1 |

Step 2: Using Tally Marks

Tally marks are a quick way to record frequencies visually.

• Create tally marks: For each count, draw a vertical line. For every fifth count, cross the first four lines with a diagonal line.

Example

For the age data above, tally marks would look like this:

| Age | Tally | |-----|-------------| | 21 | |||/ | | 22 | ||/ | | 23 | | | | 24 | | |

Step 3: Calculating Midpoints

Midpoints help summarize data in a frequency table by finding the average of the upper and lower boundaries of each class interval.

• Determine class intervals: If your data is grouped (e.g., ages 20-25).

• Calculate midpoints: For each interval, use the formula:

  Midpoint = (Lower boundary + Upper boundary) / 2

Example

For the age group 20-25:

• Midpoint = (20 + 25) / 2 = 22.5

Step 4: Finding Relative Frequency

Relative frequency expresses the frequency of a category as a fraction or percentage of the total.

• Calculate total frequency: Add all the frequencies together.
• Compute relative frequency: For each category, divide its frequency by the total frequency.

Example

Using the previous frequency table:

• Total frequency = 8 (3 + 2 + 1 + 1)
• Relative frequency for age 21 = 3/8 = 0.375 or 37.5%

Step 5: Calculating Cumulative Frequency

Cumulative frequency is the running total of frequencies.

• Start with the first frequency: The cumulative frequency for the first category is the same as its frequency.
• Add subsequent frequencies: Continue adding each category's frequency to the total of the previous categories.

Example

Continuing with the age data:

| Age | Frequency | Cumulative Frequency | |-----|-----------|----------------------| | 21 | 3 | 3 | | 22 | 2 | 5 | | 23 | 1 | 6 | | 24 | 1 | 7 |

Conclusion

In this tutorial, you learned how to create and interpret frequency tables, use tally marks, find midpoints, calculate relative frequencies, and determine cumulative frequencies. These skills are essential for analyzing data effectively.

Next steps include practicing with different data sets to strengthen your understanding and applying these concepts to real-world data analysis tasks.