The table shows the distribution of masks obtained by students in an examination. Marks 50 - 54 55 - 59 60 - 64 65 - 69 70 - 74 75 - 79 80 - 84 85 - 89 Freq...

Answer Details

To calculate the mean and standard deviation of the distribution, we need to use the formulae:

Mean = Assumed Mean + (Σfd)/n

Standard Deviation = √(Σf(x-ḿ)²)/n

Where:

• Σfd represents the sum of the products of the frequency (f) and the deviation from the assumed mean (d = x - ḿ)
• n represents the total frequency, which is the sum of all frequencies (Σf)

Using the table and the given assumed mean of 67, we can calculate the values needed for the mean and standard deviation as follows:

a) Mean:

We need to find Σfd and n to plug into the formula. To find Σfd, we need to calculate the deviation from the assumed mean for each class interval and multiply it by the frequency of that interval. For example, for the first interval (50-54), the midpoint is 52, so the deviation from the assumed mean of 67 is -15. We multiply this by the frequency of 5 to get -75. We repeat this process for all intervals and sum up the products to get Σfd = -204. The total frequency n is the sum of all frequencies, which is 5 + 15 + 20 + 28 + 12 + 9 + 7 + 4 = 100. Now we can plug these values into the formula to get:

Mean = 67 + (-204)/100 = 65.8 (correct to one decimal place)

Therefore, the mean of the distribution is 65.8.

b) Standard deviation:

We need to find Σf(x-ḿ)² and n to plug into the formula. To find Σf(x-ḿ)², we need to calculate the squared deviation from the assumed mean for each class interval, multiply it by the frequency of that interval, and sum up the products. For example, for the first interval (50-54), the midpoint is 52, so the deviation from the assumed mean of 67 is -15. We square this to get 225, then multiply by the frequency of 5 to get 1125. We repeat this process for all intervals and sum up the products to get Σf(x-ḿ)² = 8444. The total frequency n is the same as before, 100. Now we can plug these values into the formula to get:

Standard deviation = √(Σf(x-ḿ)²)/n = √(8444/100) = 9.19 (correct to one decimal place)

Therefore, the standard deviation of the distribution is 9.19.

In summary, the mean of the distribution is 65.8 and the standard deviation is 9.19.