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

Question

In an experiment with 15 observations on \(x\), the following results were available: \(\Sigma x^{2}=2830, \Sigma x=170\) One observation that was 20 was found to be wrong and was replaced by the correct value 30. The corrected variance is (a) \(8.33\) (b) \(78.00\) (c) \(188.66\) (d) \(177.33\)

Step-by-Step Solution

Answer

The corrected variance is 78.00.

Step 1: Correct the Sum of Observations

The original sum of observations is given as \( \Sigma x = 170 \). Since the observation 20 is replaced by 30, we need to add the difference to the sum. Thus, the corrected sum is \( 170 - 20 + 30 = 180 \).

Step 2: Correct the Sum of Squared Observations

The original sum of squared observations is given as \( \Sigma x^2 = 2830 \). We remove the square of the incorrect observation and add the square of the correct observation: \( 2830 - 20^2 + 30^2 = 2830 - 400 + 900 = 3330 \).

Step 3: Calculate the Mean

The mean of the observations is the total sum of the observations divided by the number of observations. This is \( \bar{x} = \frac{180}{15} = 12 \).

Step 4: Calculate the Variance

Variance is calculated using the formula \( s^2 = \frac{\Sigma x^2}{n} - \bar{x}^2 \). Substitute the known values: \( s^2 = \frac{3330}{15} - 12^2 = 222 - 144 = 78 \).

Step 5: Identify the Correct Answer

Based on our calculation, the corrected variance is \( 78.00 \), which corresponds to option (b).

Key Concepts

Mean

The concept of the "Mean," often referred to as the average, is a foundational element in statistics. It helps us understand the central tendency of a data set. Calculating the mean involves three straightforward steps:

• Add up all the observations: In the provided exercise, the observations initially summed to 170. However, one value was corrected, which altered the sum to 180.
• Count the number of observations: In the exercise, there were 15 observations.
• Divide the total sum by the number of observations: By dividing 180 by 15, we find the mean is 12.

The mean offers a quick snapshot of the data's norm. It's essential for calculating variance, as it serves as a baseline for measuring how data points deviate from this central value.

Sum of Squares

The "Sum of Squares" is vital in statistics because it measures variability within a data set. It is a key component in determining the variance. To adjust for any updated observations, follow these steps:

• Remove the square of the incorrect observation: For this problem, the square of 20 was originally included, contributing 400 to the sum of squares.
• Add the square of the correct observation: Replace the squared value of the incorrect data point with the correct one, which is 30 in this instance, contributing 900 to the sum of squares.

This gives us an updated total sum of squares of 3330.
Using this corrected sum ensures more accurate calculations of overall variability through the variance formula.

Statistical Observations

"Statistical Observations" represent the data points gathered in an experiment or survey. In our exercise, there are 15 such observations. This number is central to computing statistics like the mean and variance.

• Identify the initial observation value: Originally, all values were tallied to provide sums and squared sums, which were incorrect due to an error.
• Recognize errors and correct them: Adjustments to individual observations, as seen in replacing 20 with 30, lead to updated statistics.

These observations are the foundational elements on which statistical analysis is built. As you compute statistics, always ensure your observations are accurate to maintain the integrity of your conclusions.