Problem 58 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Mean of 5 observations is 7 . If four of these observations are \(6,7,8,10\) and one is missing then the variance of all the five observations is : [Online April 22, 2013] (a) 4 (b) 6 (c) 8 (d) 2

Step-by-Step Solution

Answer

The variance of all five observations is 4.

Step 1: Understanding Mean

The mean of 5 observations is given as 7. This means when we sum all observations and divide by 5, we get 7. Let's represent the missing observation as \( x \). This gives us the equation: \[ \frac{6 + 7 + 8 + 10 + x}{5} = 7 \]

Step 2: Solve for x

First, solve for the total sum of all observations: \( 6 + 7 + 8 + 10 + x \). Simplify: \( 31 + x \). Set up the equation: \[ \frac{31 + x}{5} = 7 \] Multiply both sides by 5: \[ 31 + x = 35 \] Subtract 31 from both sides to find \( x \): \[ x = 4 \]

Step 3: Find the Variance

First, calculate the mean again using all observations: \( (6, 7, 8, 10, 4) \). The mean is 7. Now find the variance. Calculate each squared deviation: \((6-7)^2, (7-7)^2, (8-7)^2, (10-7)^2, (4-7)^2\) which are \(1, 0, 1, 9, 9\). The total sum of squared deviations is \(20\). Divide by the number of observations (5) to find the variance: \[ \text{Variance} = \frac{1 + 0 + 1 + 9 + 9}{5} = 4 \]

Step 4: Verify the Options

The computed variance is 4. Comparing to the options given, (a) 4 is correct.

Key Concepts

Understanding Mean Calculation

Mean calculation is a key concept in data analysis, helping us summarize a set of observations into a single value. In this exercise, you are given the mean of five observations as 7.
When we say the mean is 7, it means if we add all the observations and divide the sum by the number of observations, i.e., 5 in this case, the result is 7.
To find out more, let's assign one missing observation an unknown value, represented by the variable \( x \). We'll set up an equation:

• Known observations are 6, 7, 8, and 10
• Total observations: 5 (including the unknown one, \( x \))

The equation becomes:\[ \frac{6 + 7 + 8 + 10 + x}{5} = 7 \]This step gives us a complete understanding of how an average helps in balancing the given numerical data and is crucial in many statistical analyses.

Resolving the Missing Value Problem

The missing value problem is often encountered when some data points are unavailable. Solving it is crucial for accurate analysis.
In this case, to find the missing observation \( x \), we start with the equation derived from the mean calculation:\[ \frac{31 + x}{5} = 7 \]Here's how we solve it step-by-step:

• Multiply both sides by 5 to eliminate the fraction:

\[ 31 + x = 35 \]

• Subtract 31 from both sides to isolate \( x \):

\[ x = 4 \]Thus, the missing observation is 4. This technique of deducing unknown values using known information is fundamental in data processing and analysis.

Calculating Squared Deviations

Squared deviations help us understand the spread of data points around the mean, which is essential when computing variance.
In this exercise, you need to calculate each observation's squared deviation from the mean of 7.

• List the observations: 6, 7, 8, 10, and 4.
• Calculate deviations: Find the difference between each observation and the mean.

For example:

• \((6-7)^2 = 1\)
• \((7-7)^2 = 0\)
• \((8-7)^2 = 1\)
• \((10-7)^2 = 9\)
• \((4-7)^2 = 9\)

The squared deviations are 1, 0, 1, 9, and 9. The sum of these squared deviations is 20.
Finally, divide the sum by the number of observations to get the variance:\[ \frac{20}{5} = 4 \]Calculating squared deviations is crucial in statistical variance as it shows the dispersion of data around the mean.