Question 2 of 21intermediate🔧 ApplyNumerical3 marks

Consider the following distribution of daily wages of 50 workers of a factory.

Daily wages (in ₹)500 - 520520-540540 - 560560 - 580580 - 600
Number of workers12148610

Find the mean daily wages of the workers of the factory by using an appropriate method.

Correct Answer

To find the mean daily wages, we will use the Direct Method. This method is appropriate here as the class sizes are uniform and the values of the class marks (xix_i) are manageable integers.

First, we calculate the class mark (xix_i) for each class interval using the formula xi=Upper limit+Lower limit2x_i = \frac{\text{Upper limit} + \text{Lower limit}}{2}. Then, we compute the product fixif_i x_i for each class.

Table Calculation:

Daily wages (in ₹)Number of workers (fif_i)Class mark (xix_i)fixif_i x_i
500 - 520125106120
520 - 540145307420
540 - 56085504400
560 - 58065703420
580 - 600105905900
TotalΣfi=50\Sigma f_i = 50Σfixi=27260\Sigma f_i x_i = 27260

Mean Calculation: Using the formula from the Direct Method:

xˉ=ΣfixiΣfi\bar{x} = \frac{\Sigma f_i x_i}{\Sigma f_i}

Substituting the values:

xˉ=2726050=545.2\bar{x} = \frac{27260}{50} = 545.2

Thus, the mean daily wages of the workers is ₹ 545.20.

Exercise: EXERCISE 13.1 | Q: 2 | (Page 11)
For More Understanding

Explanation

The solution follows the Direct Method for finding the mean of grouped data, as illustrated in Table 13.3 of the provided context. This method is chosen because the class marks (xix_i) are small integers, making the calculation of fixif_i x_i straightforward without needing the step-deviation method. The class mark is calculated as the average of the upper and lower limits for each class interval. The sum of the frequencies (Σfi\Sigma f_i) is 50, and the sum of the products of frequency and class mark (Σfixi\Sigma f_i x_i) is 27260. Dividing the total wages by the total number of workers yields the mean.

Solution Steps

  1. Step 1: Identify the method. Since the class marks are small integers, the Direct Method is appropriate.

  2. Step 2: Calculate class marks (xix_i) for each wage interval (e.g., for 500-520, xi=(500+520)/2=510x_i = (500+520)/2 = 510).

  3. Step 3: Multiply the number of workers (fif_i) by the class mark (xix_i) for each row to get fixif_i x_i.

  4. Step 4: Sum the fixif_i x_i values to get Σfixi=27260\Sigma f_i x_i = 27260 and sum the frequencies to get Σfi=50\Sigma f_i = 50.

  5. Step 5: Apply the mean formula xˉ=ΣfixiΣfi=2726050=545.2\bar{x} = \frac{\Sigma f_i x_i}{\Sigma f_i} = \frac{27260}{50} = 545.2.

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