Problem 8 - A Complete Resource Book in Mathematics for JEE Main

Question

The sum of squares of deviations for 10 observations taken from mean 50 is \(250 .\) The coefficient of variation is (A) \(50 \%\) (B) \(10 \%\) (C) \(40 \%\) (D) None of these

Step-by-Step Solution

Answer

The coefficient of variation is (B) 10%.

Step 1: Understand the Given Information

We are given that the sum of squares of deviations from the mean is \(250\) for 10 observations and the mean is \(50\). We need to find the coefficient of variation.

Step 2: Calculate Variance

The variance for the observations can be calculated using the formula: \[ \text{Variance} = \frac{\text{Sum of squares of deviations}}{\text{Number of observations}} \] Substituting the given values, we have \( \text{Variance} = \frac{250}{10} = 25. \)

Step 3: Find the Standard Deviation

The standard deviation is the square root of the variance. So, we calculate: \[ \text{Standard Deviation} = \sqrt{25} = 5. \]

Step 4: Calculate the Coefficient of Variation

The coefficient of variation (CV) is calculated using the formula: \[ CV = \left( \frac{\text{Standard Deviation}}{\text{Mean}} \right) \times 100\% \] Substituting the calculated values, we have \( CV = \left( \frac{5}{50} \right) \times 100\% = 10\%. \)

Key Concepts

Understanding Variance

Variance is a measure that describes how much the observations in a data set differ from the mean. It highlights the spread of data points. The variance helps in understanding the level of consistency across a set of numbers. For example:

• A higher variance indicates that the data points are more spread out from the mean.
• A lower variance suggests that the data points are close to the mean, indicating more consistency.

The formula to calculate variance is given by: \[ \text{Variance} = \frac{\text{Sum of squares of deviations}}{\text{Number of observations}} \] In the original exercise, the sum of squared deviations from the mean is 250 for 10 observations, leading to a variance of 25. This calculation provides insight into the spread of the data points around the mean value of 50.

Decoding Standard Deviation

Standard deviation is a pivotal statistical measure that quantifies the amount of variation or dispersion in a set of values. A higher standard deviation indicates that the values are more dispersed from the mean, whereas a lower standard deviation signifies that values are closer to the mean. Here's why it matters:

• Standard deviation is particularly useful when comparing the spread between different data sets.
• It's more interpretable than variance because it is expressed in the same unit as the data.

To calculate the standard deviation, we take the square root of the variance: \[ \text{Standard Deviation} = \sqrt{\text{Variance}} \] In the provided example, the variance is 25, so the standard deviation is \( \sqrt{25} = 5 \). This tells us that on average, the observations deviate from the mean by 5 units.

Navigating the Mean

The mean, often referred to as the average, represents a central value for a set of numbers. It is calculated by summing all the values and dividing by the number of values. Knowing the mean is crucial because:

• It provides a baseline comparison for understanding variance and standard deviation.
• The mean serves as a reference point that other measurements are compared against.

For the given data set, the mean is stated to be 50. This central point is vital for calculating deviations, which in turn are used for computing both variance and standard deviation. Understanding the mean helps in comprehending how much individual data points differ from this central value, indicating their spread or dispersion within the data set.