2.3.2. Computing Mode for Grouped Data

The mean is also computed for the grouped data using either direct or indirect method.

When dealing with grouped data, we cannot use individual values directly since they are summarized into classes. Therefore, we represent these values using midpoints of each class range. The mean can be calculated using either a direct method or an indirect method.

Imagine you and your friends are pooling your weekly allowances into different jars based on the amount. Instead of counting each coin, you simply note down the total amount per jar. To find the average allowance, you would use the average amount in each jar (the midpoint) instead of individual coins.

When scores are grouped into a frequency distribution, the individual values lose their identity. These values are represented by the midpoints of the class intervals in which they are located. While computing the mean from grouped data using direct method, the midpoint of each class interval is multiplied with its corresponding frequency (f); all values of fx are added to obtain ∑fx that is finally divided by the number of observations i.e., N. Hence, mean is calculated using the following formula: ∑fx / N.

In the direct method, first, we find the midpoint for each class interval. Then, we multiply this midpoint by the frequency of that class to get a value called fx. Finally, we sum all these fx values together (∑fx) and divide by the total number of observations (N) to calculate the mean.

Think of a classroom where students are grouped by grades. Instead of knowing each student's exact score, you know how many scored in ranges (like 50-60, 60-70, etc.). You calculate the average score by taking the average score for each range (midpoint), multiplying it by how many students are in that range, and then averaging those for a class average.

The following formula can be used for the indirect method for grouped data: ∑fd / N. This formula involves calculating deviations from an assumed mean which simplifies calculations.

In the indirect method, we first assume a mean (A) that is easy for calculations, usually the midpoint of the class where we expect the mean to fall. We then calculate the deviation of each class's midpoint from this assumed mean and multiply these deviations by their respective frequencies (f). All of these products are summed to get ∑fd and divided by N to find the mean.

Imagine you're picking an average score from a range of tests. Instead of averaging directly, you decide to use zero as a reference to see how far each test score is above or below it (deviation). You then tally those differences, and by averaging them, you find out your average performance compared to zero. It’s like taking a step back to see the whole picture, instead of just focusing on the scores.

Example 2.2: Compute the average wage rate of factory workers using data given in Table 2.2: Wage Rate (Rs./day) Number of workers (f) Classes: 50 - 70 (10) 70 - 90 (20) 90 - 110 (25) 110 - 130 (35) 130 - 150 (9)

To find the average wage rate, we first calculate the midpoints of each class interval (like 60 for 50-70). Then, we multiply each class midpoint by its frequency to get fx. For instance, for the 50-70 range with 10 workers, fx = 60 * 10 = 600. We sum all fx values and divide by total number of workers to get the mean wage. The sum of fx will give us the total earnings, and when that total is divided by the number of workers, we obtain the average wage.

Imagine a bakery where each type of pastry represents a wage class. Instead of adding the price of each piece, you categorize them by type and simply note how much each type has in total sales. Then, rather than buying each pastry individually, you calculate the average price by finding deals on total batches instead.

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