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 use the Direct Method. First, we calculate the class mark (xix_i) for each class interval by finding the midpoint (average of upper and lower limits).

The calculation table is as follows:

  • For 500-520: xi=510x_i = 510, fixi=12×510=6120f_i x_i = 12 \times 510 = 6120
  • For 520-540: xi=530x_i = 530, fixi=14×530=7420f_i x_i = 14 \times 530 = 7420
  • For 540-560: xi=550x_i = 550, fixi=8×550=4400f_i x_i = 8 \times 550 = 4400
  • For 560-580: xi=570x_i = 570, fixi=6×570=3420f_i x_i = 6 \times 570 = 3420
  • For 580-600: xi=590x_i = 590, fixi=10×590=5900f_i x_i = 10 \times 590 = 5900

The total number of workers is Σfi=50\Sigma f_i = 50 and the sum of the product of values is Σfixi=27260\Sigma f_i x_i = 27260.

Using the formula: Mean (xˉ\bar{x}) = ΣfixiΣfi=2726050=545.2\frac{\Sigma f_i x_i}{\Sigma f_i} = \frac{27260}{50} = 545.2

Therefore, the mean daily wage of the workers is ₹ 545.20.

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

Explanation

The solution follows the Direct Method as illustrated in Table 13.3 of the provided context. This method is appropriate here because the class size is uniform and the numerical values are manageable. We calculate the class mark (xix_i) for each interval, multiply it by the frequency (fif_i), sum these products, and divide by the total frequency to find the mean.

Solution Steps

  1. Step 1: Calculate the class mark (xix_i) for each wage interval (e.g., for 500-520, xi=(500+520)/2=510x_i = (500+520)/2 = 510).

  2. Step 2: Compute the product fixif_i x_i for each class and find the sum Σfixi=6120+7420+4400+3420+5900=27260\Sigma f_i x_i = 6120 + 7420 + 4400 + 3420 + 5900 = 27260.

  3. Step 3: Verify the total frequency Σfi=12+14+8+6+10=50\Sigma f_i = 12 + 14 + 8 + 6 + 10 = 50.

  4. Step 4: 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.