Problem 51 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

For two data sets, each with size 5 , the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4 , respectively. The variance of the combined data set is (A) \(\frac{11}{2}\) (B) 6 (C) \(\frac{13}{2}\) (D) \(\frac{5}{2}\)

Step-by-Step Solution

Answer

The variance of the combined data set is \( \frac{11}{2} \).

Step 1: Understand the Given Information

We have two data sets. Set 1 has variance \( \sigma_1^2 = 4 \) and mean \( \mu_1 = 2 \), with 5 data points. Set 2 has variance \( \sigma_2^2 = 5 \) and mean \( \mu_2 = 4 \), also with 5 data points.

Step 2: Recall the Formula for Combined Variance

The formula to find the variance of a combined data set with independent samples and equal weights is:\[\sigma^2_{combined} = \frac{n_1\sigma_1^2 + n_2\sigma_2^2 + n_1(\mu_1-\mu_{combined})^2 + n_2(\mu_2-\mu_{combined})^2}{n_1+n_2}\]where \( n_1 \) and \( n_2 \) are the sample sizes, \( \mu_{combined} \) is the mean of the combined data set.

Step 3: Calculate the Combined Mean \( \mu_{combined} \)

The combined mean is calculated by:\[\mu_{combined} = \frac{n_1\mu_1 + n_2\mu_2}{n_1+n_2} \]Substitute the values: \( n_1 = 5 \), \( \mu_1 = 2 \), \( n_2 = 5 \), and \( \mu_2 = 4 \):\[\mu_{combined} = \frac{5 \times 2 + 5 \times 4}{5 + 5} = \frac{10 + 20}{10} = \frac{30}{10} = 3\]Thus, \( \mu_{combined} = 3 \).

Step 4: Substitute into the Combined Variance Formula

Substitute the known values into the combined variance formula:\[\sigma^2_{combined} = \frac{5 \times 4 + 5 \times 5 + 5(2-3)^2 + 5(4-3)^2}{10}\]This simplifies to:\[\sigma^2_{combined} = \frac{20 + 25 + 5(1) + 5(1)}{10} = \frac{20 + 25 + 5 + 5}{10}\]

Step 5: Final Calculation

Calculate the simplified equation:\[\sigma^2_{combined} = \frac{55}{10} = 5.5\]This matches option \( \frac{11}{2} \).

Key Concepts

Data Set Variance

Variance is a crucial statistical measure that tells us about the spread of data points in a data set. It essentially measures how far each number in the set is from the mean (average) and thus from every other number in the set.
For a better grasp, consider a simple data set: {1, 2, 3}. To get the variance, you first compute the mean (in this case, 2), then find the difference of each data point from the mean. Square these differences, sum them up, and divide by the number of data points to get the variance.

• Small variance indicates that the data points are close to the mean.
• Larger variance signals that the data points are spread out over a larger range of values.

Remember, in statistical analysis, understanding variance helps you better understand data variability, which is integral for further statistical applications.

Mean of Combined Data Set

When dealing with multiple data sets, finding the mean of the combined data can provide a single measure of center for all the data points involved.
For our combined data set, we're essentially looking at the combined influence of all data points from both sets. The mean of the combined data set is calculated using the formula \( \mu_{combined} = \frac{n_1\mu_1 + n_2\mu_2}{n_1+n_2} \).
Here's why this formula works:

• The term \( n_1\mu_1 \) represents the total sum of all observations in the first data set.
• Likewise, \( n_2\mu_2 \) represents the total sum of the second data set.
• Adding these gives the overall sum of all observations, and dividing by the total number of observations (_1 + n_2) offers the mean for the combined data set.

This approach makes sure every data point in both sets contributes proportionally to the combined mean.

Formula for Combined Variance

The combined variance of two or more datasets gives us a comprehensive measure of how much variation exists when considering both sets collectively.
The formula for the combined variance, \( \sigma^2_{combined} \) is: \[ \sigma^2_{combined} = \frac{n_1\sigma_1^2 + n_2\sigma_2^2 + n_1(\mu_1-\mu_{combined})^2 + n_2(\mu_2-\mu_{combined})^2}{n_1+n_2} \]

• The terms \( n_1\sigma_1^2 \) and \( n_2\sigma_2^2 \) are contributions of variances from each data set.
• The parts \( n_1(\mu_1-\mu_{combined})^2 \) and \( n_2(\mu_2-\mu_{combined})^2 \) adjust for the differences in means between the individual data sets and the combined mean. This helps account for the spread relative to the new average.

This formula is particularly useful when the sample sizes or variances of the data sets are not equal. By applying this, one can understand not only individual and collective data variation but also how the interaction between datasets affects overall variance.