Prove the following data calculate standard deviation and its coefficient – Age (less than) 10 20 30 40 50 60 70 80 No. of person 15 30 53 75 100 110 115 12?

Question

Answer

Calculating Standard Deviation and Coefficient of Age Data

Step 1: Calculate Mean

First, we need to calculate the mean of the data. We can do this by adding up all the ages and dividing by the total number of people:

Mean = (10 x 15) + (20 x 30) + (30 x 53) + (40 x 75) + (50 x 100) + (60 x 110) + (70 x 115) + (80 x 120) / 618 = 50.22

Step 2: Calculate Deviation

Next, we need to calculate the deviation of each age from the mean:

Deviation = Age - Mean

For example, the deviation for age 10 is:

Deviation = 10 - 50.22 = -40.22

Step 3: Calculate Squared Deviation

Now we need to calculate the squared deviation for each age:

Squared Deviation = Deviation^2

For example, the squared deviation for age 10 is:

Squared Deviation = (-40.22)^2 = 1618.48

Step 4: Calculate Variance

Next, we need to calculate the variance of the data. We can do this by adding up all the squared deviations and dividing by the total number of people:

Variance = (1618.48 + 249.48 + 6.28 + 439.68 + 250 + 179.6 + 93.1 + 1148.89) / 618 = 421.22

Step 5: Calculate Standard Deviation

Finally, we can calculate the standard deviation by taking the square root of the variance:

Standard Deviation = sqrt(421.22) = 20.52

Step 6: Calculate Coefficient of Variation

The coefficient of variation is a measure of relative variability, calculated as the standard deviation divided by the mean and expressed as a percentage:

Coefficient of Variation = (20.52 / 50.22) x 100% = 40.9%

Explanation

The above data represents the age distribution of a group of people. We first calculated the mean of the data, which is the average age of the group. Next, we calculated the deviation of each age from the mean, which gives us an idea of how spread out the data is. We then squared the deviations to get rid of negative signs and to give more weight to larger deviations. The variance is the average of the squared deviations, which gives us a measure of the overall spread of the data. Finally, we took the square root of the variance to get the standard deviation, which is a commonly used measure of variability. The coefficient of variation is another measure of variability that takes into account the mean of the data. It tells us how much variability there is relative to the mean, expressed as

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