How to Find Weighted Mean of Grouped Data?

To find the weighted mean of grouped data, you essentially need to calculate a weighted average where the weights are typically frequencies associated with each group or class. Here's a step-by-step guide:

Steps to Calculate Weighted Mean of Grouped Data:

1. Determine the Midpoint of Each Class (x_i): For each class interval, calculate the midpoint. This is done by averaging the upper and lower limits of the class.

   • Midpoint (x_i) = (Lower Limit + Upper Limit) / 2

2. Identify the Frequency of Each Class (f_i): Note the frequency (f_i), which represents the number of observations within each class interval.

3. *Multiply the Midpoint by the Frequency for Each Class (x_i f_i):** For each class, multiply the midpoint (x_i) by its corresponding frequency (f_i). This gives you a weighted value for each class.

4. *Sum the Weighted Values (∑(x_i f_i)):** Add up all the values calculated in the previous step. This is the sum of the weighted values.

5. Sum the Frequencies (∑f_i): Add up all the frequencies. This gives you the total number of observations in the dataset.

6. Divide the Sum of Weighted Values by the Sum of Frequencies: Divide the result from Step 4 by the result from Step 5. This gives you the weighted mean.

   • Weighted Mean = ∑(x_i * f_i) / ∑f_i

Formula:

The weighted mean (x̄_w) is calculated using the following formula:

x̄_w = ∑(x_i * f_i) / ∑f_i

Where:

• x̄_w = Weighted mean
• x_i = Midpoint of the i-th class
• f_i = Frequency of the i-th class
• ∑ = Summation

Example:

Let's say we have the following grouped data:

Class Interval	Frequency (fi)
10-20	5
20-30	8
30-40	12
40-50	5

1. Midpoints (x_i):

   • (10+20)/2 = 15
   • (20+30)/2 = 25
   • (30+40)/2 = 35
   • (40+50)/2 = 45

2. *Multiply Midpoint by Frequency (x_i f_i):**

   • 15 * 5 = 75
   • 25 * 8 = 200
   • 35 * 12 = 420
   • 45 * 5 = 225

3. *Sum of Weighted Values (∑(x_i f_i)):**

   • 75 + 200 + 420 + 225 = 920

4. Sum of Frequencies (∑f_i):

   • 5 + 8 + 12 + 5 = 30

5. Weighted Mean:

   • 920 / 30 = 30.67

Therefore, the weighted mean for this grouped data is 30.67.

Conclusion:

Calculating the weighted mean of grouped data involves finding the midpoint of each class, weighting it by its frequency, summing these weighted values, and then dividing by the total frequency. This provides a more accurate representation of the average when dealing with grouped data compared to a simple average.