Question

For 100 observations \(\sum(x i-30)=0\) and \(\sum(x i-30)^{2}=10000\) then C.V. (coefficient of variance) is \(\%\) (a) 10 (b) 100 (c) \(33.33\) (d) 30

Short answer

The coefficient of variation (C.V.) is \(33.33\%\).

Step-by-step solution

Find the mean (µ)

Since we know that \(\sum(x_i - 30) = 0\), we can write this as \(\sum x_i - \sum 30 = 0\) Now, since there are 100 observations, \(\sum 30 = 30\times 100\), which means \(\sum x_i - 3000 = 0\) So, the sum of all x_i is \(\sum x_i = 3000\) Now, we can find the mean as follows: \(\mu = \frac{\sum x_i}{n}\) Where n is the number of observations (100 in this case). \(\mu = \frac{3000}{100} = 30\).

Find the variance (σ^2)

We have given that \(\sum(x_i - 30)^2 = 10000\). Now, let's find the variance, σ^2. Variance, σ^2 is defined as: \(\sigma^2 = \frac{\sum(x_i - \mu)^2}{n}\) We know that: - n = 100 (number of observations) - µ = 30 (mean of the dataset) Substituting the values: \(\sigma^2 = \frac{10000}{100} = 100\)

Find the standard deviation (σ)

To find the standard deviation (σ), we need to find the square root of the variance (σ^2). \(\sigma = \sqrt{100} = 10\)

Compute the Coefficient of Variation (C.V.)

Now that we have both the mean (µ) and standard deviation (σ), we can calculate the coefficient of variation (C.V.) as follows: C.V. = \(\frac{\sigma}{\mu} \times 100\%\) C.V. = \(\frac{10}{30} \times 100\%\) C.V. = \(33.33 \%\) So, the correct answer is (c) \(33.33\%\).

Key concepts

Understanding Mean

The mean, often known as the average, is a primary measure of central tendency in statistics. It provides the central value of a dataset, giving us a sense of the typical number in a group of numbers. For any set of numbers, the mean is calculated by adding up all the numbers, which we denote as \( \sum x_i \), and then dividing this sum by the number of observations, \( n \).

Here's a simple breakdown of the steps involved in calculating the mean:

• Add up all values in the dataset.
• Divide the total by the number of data points.

In the example given, there are 100 observations, making it easy to compute the mean as:\[\mu = \frac{3000}{100} = 30\] The mean plays a crucial role because it acts as a reference point in further statistical calculations like variance and standard deviation.

Exploring Variance

Variance is a measure of how far each number in a dataset is from the mean, which reflects the degree of spread among the numbers. It provides insight into the dispersion of the dataset, helping us understand how much the numbers vary from one another.

To calculate the variance \( \sigma^2 \), follow these steps:

• Subtract the mean from each data point, \( (x_i - \mu) \).
• Square each of these differences to eliminate negative values and accentuate larger deviations.
• Sum up all these squared differences.
• Finally, divide the total by the number of data points (n) to find the average of these squares.

In our exercise, the variance is derived as:\[\sigma^2 = \frac{10000}{100} = 100\]Variance gives a squared measure of spread, making it less intuitive than standard deviation, because of the squared unit.

Demystifying Standard Deviation

Standard deviation, represented as \( \sigma \), simplifies the interpretation of variance by bringing the numbers back to their original units. As the square root of variance, standard deviation helps us better understand the natural spread of numbers relative to the mean.

Key steps to find standard deviation:

• Calculate the variance as discussed.
• Take the square root of the result obtained in variance to get the standard deviation.

In the exercise, we find:\[\sigma = \sqrt{100} = 10\]Standard deviation is extremely useful, as it provides a scale for comparing relative variability between different datasets and is instrumental in computing the coefficient of variation.

JEE Maths and Coefficient of Variation

Coefficient of Variation (C.V.) is a normalized measure of the distribution of data points in a dataset relative to the mean. It's crucial in fields like finance and research where comparisons between datasets with different scales are necessary. In JEE Maths, students frequently encounter C.V. as it helps in evaluating the relative risk or variability of datasets.

To compute the coefficient of variation, use the formula:

• Divide the standard deviation by the mean.
• Multiply by 100 to express it as a percentage.

From the problem assessed:\[C.V. = \frac{10}{30} \times 100\% = 33.33\%\]A lower C.V. indicates less variability relative to the mean, which can be essential when making critical decisions or comparing datasets. In JEE Maths, understanding these concepts forms the basis for tackling data interpretation problems efficiently.