Problem 21 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The mean and variance of 7 observations are 8 and 16 , respectively. If five observations are \(2,4,10,12,14\), then the absolute difference of the remaining two observations is: (a) 1 (b) 4 (c) 2 (d) 3

Step-by-Step Solution

Answer

The absolute difference is 2.

Step 1: Understanding Mean and Total Sum

The mean of the observations is given as 8, and there are 7 observations in total. Therefore, the total sum of all observations is calculated as follows:\[ \text{Total Sum} = \text{Mean} \times \text{Number of observations} = 8 \times 7 = 56. \]

Step 2: Calculate Sum of Given Observations

We have 5 observations: 2, 4, 10, 12, and 14. We first find their total sum:\[ 2 + 4 + 10 + 12 + 14 = 42. \]

Step 3: Determine Sum of Remaining Observations

We need to find the sum of the last two observations. Since the total sum of all observations is 56, we find the sum of the last two observations by subtracting the sum of the given observations from the total sum:\[ \text{Sum of two observations} = 56 - 42 = 14. \]

Step 4: Set Equations for Two Observations

Let the two unknown observations be \(x\) and \(y\). Based on Step 3, we have:\[ x + y = 14. \] Given the variance of the observations is 16, the formula for variance is:\[ \text{Variance} = \frac{\sum{x_i^2}}{n} - \left(\text{Mean}\right)^2. \] From this, the second equation derived from the variance is\[ \frac{(2^2 + 4^2 + 10^2 + 12^2 + 14^2 + x^2 + y^2)}{7} - 8^2 = 16. \]

Step 5: Simplify and Solve for Difference

Calculate the square of the given observations:\[ 2^2 + 4^2 + 10^2 + 12^2 + 14^2 = 4 + 16 + 100 + 144 + 196 = 460. \]Substitute this into the variance equation:\[ \frac{460 + x^2 + y^2}{7} - 64 = 16. \]Simplifying, we get:\[ \frac{460 + x^2 + y^2}{7} = 80 \Rightarrow 460 + x^2 + y^2 = 560 \Rightarrow x^2 + y^2 = 100. \]We have two equations: \(x + y = 14\) and \(x^2 + y^2 = 100\). Using the identity:\[ (x+y)^2 = x^2 + y^2 + 2xy \Rightarrow 14^2 = 100 + 2xy \Rightarrow 196 = 100 + 2xy \Rightarrow 2xy = 96 \Rightarrow xy = 48. \] The difference \(|x-y|\) can be calculated using:\[ |x - y| = \sqrt{(x+y)^2 - 4xy} = \sqrt{14^2 - 4 \times 48} = \sqrt{196 - 192} = \sqrt{4} = 2. \]

Step 6: Conclusion

The absolute difference between the two remaining observations is 2.

Key Concepts

Observations

In statistics, 'observations' refer to individual pieces of data that have been collected through surveys, experiments, or other research methods. These data points are used to perform further analysis or draw conclusions. Each observation contributes to the overall understanding of the dataset.

For the exercise, the observations are measurements or data points that we're analyzing to calculate both mean and variance. In this example, we know that there are 7 observations in total with a given mean of 8 and variance of 16. Five of these observations are explicitly provided: 2, 4, 10, 12, and 14. Our task is to determine the two remaining observations, knowing their properties contribute to the mean and variance calculations as a whole.

• These seven observations allow us to build equations based on the known mean and variance.
• Each observation influences the overall statistical properties of the data.

Variance Calculation

Variance is a measure of how spread out the observations in a dataset are, around the mean.
It's calculated using the formula:
\[ \text{Variance} = \frac{\sum{(x_i - \overline{x})^2}}{n} \]where \(x_i\) represents each observation, \(\overline{x}\) is the mean, and \(n\) is the total number of observations.

In our exercise, the variance is given as 16, and the mean is 8. To align the set with this variance, each observation's square (\(x_i^2\)) needs to be considered.

• The variance equation observes how much each observation varies from the mean, squared.
• To solve for the variance accurately, sum of squared observations is divided by the total number of observations.
• This relation is crucial for finding the unknown values, because their squared values also need to satisfy the total variance.

Absolute Difference

The absolute difference refers to the non-negative difference between two values.
It's often calculated as \(|x - y|\).

In this exercise, we have identified the remaining two observations by setting up equations with their sum and variance criteria.
The equations were \(x + y = 14\) and \(x^2 + y^2 = 100\), obtained from variance calculations.

The absolute difference then comes into play when solving for \(|x - y|\), helping us confirm the accurate pair of observations. \[ |x - y| = \sqrt{(x+y)^2 - 4xy} \] Using this formula, computation shows that the absolute difference is 2.

• The absolute difference gives insight into how varied the two unknown values are.
• Unlike standard differences, absolute difference is always positive, making it suitable for determining distance between values.
• For our two observations, their calculations ensure cohesion with the original dataset's statistical measures.