Problem 45 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

5 students of a class have an average height \(150 \mathrm{~cm}\) and variance \(18 \mathrm{~cm}^{2}\). A new student, whose height is \(156 \mathrm{~cm}\), joined them. The variance (in \(\mathrm{cm}^{2}\) ) of the height of these six students is: [Jan. 9, 2019 (I)] (a) 16 (b) 22 (c) 20 (d) 18

Step-by-Step Solution

Answer

The variance of the heights for the six students is 20 cm² (option c).

Step 1: Calculate the Total Sum of Heights

To find the total sum of the heights for the original five students, we need to use the average. The average height is 150 cm, so the total sum of the heights is calculated as: \[ 5 \times 150 = 750 \text{ cm} \]

Step 2: Compute the Initial Variance Formula

We know the variance for the original five students is 18 \( \text{cm}^2 \). We use the formula for variance: \[ \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2 \] Thus, for the initial five students, \[ 18 = \frac{1}{5} \sum_{i=1}^{5} (x_i - 150)^2 \] \[ \sum_{i=1}^{5} (x_i - 150)^2 = 90 \]

Step 3: Add New Student and Recalculate the Sum of Heights

Now, a new student with a height of 156 cm joins. Update the total sum of the heights for all six students: \[ 750 + 156 = 906 \text{ cm} \]

Step 4: Calculate the New Average Height

With the new total, calculate the new average height for all six students: \[ \bar{x}_{new} = \frac{906}{6} = 151 \text{ cm} \]

Step 5: Compute the New Variance

To find the new variance, we calculate: \[ \frac{1}{6} \left( 90 + (156 - 151)^2 + 5 \times (151 - 150)^2 \right) \] Breaking steps:- The deviation of the new student's height is \((156 - 151)^2 = 25\).- Each of the original five student's height deviation from the new mean is \((150 - 151)^2 = 1\).Thus, \[ \frac{1}{6} (90 + 25 + 5 \times 1) = \frac{1}{6} \times 120 = 20 \text{ cm}^2 \]

Step 6: Conclusion

The calculated variance for the heights of the six students is 20 cm². Refer to the available choices, and (c) 20 matches our calculation.

Key Concepts

Average Height

In any group, the average height provides a central value representing the typical height of individuals within that group. It's calculated by summing up all individual heights and dividing by the number of individuals.
For instance, in the case of the five students with an average height of 150 cm, we calculate this by:

• Total sum of their heights: 5 students multiplied by their average of 150 cm, leading to 750 cm.
• This average tells us each student is, on average, 150 cm tall.

Calculating average height helps to understand general patterns and compare new data. When a new student joins with a height of 156 cm, the average adjusts to accommodate this additional information. This change results in a new average of 151 cm, representing a slight increase in the typical height due to the taller new student.

Standard Deviation

Standard deviation is a key concept in statistics that measures the amount of variation or dispersion in a set of values. In simpler terms, it tells us how spread out individual heights are compared to the average.
It is closely related to variance, the square of standard deviation, which we calculated as 18 cm² for the original group of students.

• For the original five students, their standard deviation, the square root of the variance, would be approximately 4.24 cm.
• With the inclusion of the new student, the variance changes to 20 cm², leading to a new standard deviation of about 4.47 cm.

Such calculations show us that while the students are fairly consistent in height, there is still some variability.
This tool is crucial for understanding how a new entry, like the student with 156 cm height, affects the spread of data.

Mathematics Problem Solving

Problem-solving in mathematics involves applying logical thought processes and methodical approaches to reach a solution. This exercise showcases these steps beautifully:

• Understanding what is given: Start by absorbing details like the average height and variance of initial data.
• Utilizing formulas: The variance formula \[ \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2 \]helps in organizing thoughts and calculations.
• Step-by-step processing: Break problems into smaller, manageable parts, like recalculating the sum with a new student and finding the new average.
• Final calculations: Accurately compute results, such as variance, to align with the available solution options.

Each step of this problem harnesses core mathematical skills to ensure accurate and efficient problem-solving.
This approach is not just applicable to height averages but can be extended to various mathematical challenges, underscoring the importance of clear, sequential thinking.