When dealing with a large amount of data, organizing it into a grouped frequency distribution makes it easier to analyze. This lesson focuses on how to calculate the mean (the average) from such a table, which gives us a single, representative value for the entire data set.

1. Core Concepts & Vocabulary

• Grouped Frequency Distribution: A table that organizes data into groups called class intervals.

• Class Interval: The range for each group (e.g., 10 - 20).

• Frequency (f): The number of data values in each class interval.

• Mid-value (x): The middle value of a class interval, used to represent all data in that group. Mid-value = (Lower Value + Upper Value) / 2.

• Modal Class: The class interval with the highest frequency.

• Mean: The average of the data. For grouped data, it's an estimate because we use mid-values.

2. Calculating the Mean: Two Methods

Method 1: The Direct Method (Best for smaller numbers)

Use this method when the mid-values are easy to work with.

Formula: Mean = Σfx / Σf

• Σf is the total number of data points (sum of frequencies).

• Σfx is the sum of the (mid-value × frequency) products.

Step-by-Step:

1. Create a 4-column table: Class Interval, Mid-value (x), Frequency (f), fx.

2. Calculate the mid-value (x) for each class.

3. Calculate fx for each row by multiplying f × x.

4. Find the totals: Add up the f column to get Σf. Add up the fx column to get Σfx.

5. Divide: Mean = Σfx / Σf.

Method 2: The Assumed Mean Method (Best for large numbers)

This method simplifies calculations by letting you work with smaller numbers.

Formula: Mean = A + (Σfd / Σf)

• A is the Assumed Mean (a mid-value you choose).

• d is the Deviation (d = x - A).

• Σfd is the sum of the (frequency × deviation) products.

Step-by-Step:

1. Create a 5-column table: Class Interval, Mid-value (x), Frequency (f), Deviation (d), fd.

2. Choose an Assumed Mean (A): Pick a mid-value from near the middle of the table (often from the modal class). This makes the deviation numbers smaller.

3. Calculate the deviation (d) for each row: d = x - A. (Note: d will be 0 for the row you chose A from, and negative for rows above it).

4. Calculate fd for each row by multiplying f × d.

5. Find the totals: Add up the f column to get Σf. Add up the fd column (paying attention to negative signs) to get Σfd.

6. Apply the formula: Mean = A + (Σfd / Σf).

3. Exam Tips & Tricks

• When to Use Which Method?

  • If the mid-values are simple (e.g., 5, 15, 25), use the Direct Method.

  • If the mid-values are large and complex (e.g., 550, 650, 750), use the Assumed Mean Method to avoid difficult multiplication and potential errors.

• The Σf Check: The first thing you should do is add up the f column. This Σf value must match the total number of items mentioned in the question (e.g., "data from 50 farmers"). If it doesn't, you've misread the table.

• The Deviation Pattern Check: In the Assumed Mean method, if all your class intervals have the same width, the d column will have a constant pattern (e.g., -20, -10, 0, 10, 20). This is a great way to check if your mid-values are correct.

• Keep Tables Neat: A well-organized table is crucial. Use a ruler to draw your columns and write numbers clearly to prevent calculation mistakes.

4. Important Points to Remember

• The Mean is an Estimate: Because we use mid-values, the calculated mean is a very good estimate, not an exact value.