The marks obtained by 40 students are...

MathematicsVariance and Standard DeviationJEE Advanced 1982Moderate

The marks obtained by 40 students are grouped in a frequency table in class intervals of 10 marks each. The mean and the variance obtained from this distribution are found to be 40 and 49 respectively. It was later discovered that two observations belonging to the class interval (21-30) were included in the class interval (31-40) by mistake. Find the mean and the variance after correcting the error.

Answer:new mean = 39.5, new variance = 49.25

Visualized Solution (Hindi)

Initial Data and Error Identification

Total students $N = 40$
Given Mean $\overset{x}{ˉ} = 40$
Given Variance $σ^{2} = 49$
Error: 2 observations moved from $(21 - 30)$ to $(31 - 40)$

Defining Midpoints of Intervals

Midpoint of $(21 - 30)$ is $x_{1} = 25$
Midpoint of $(31 - 40)$ is $x_{2} = 35$
Correction: Replace two $35$ s with two $25$ s

Formula for Mean .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } \overset{x}{ˉ}

Mean formula: $\overset{x}{ˉ} = \frac{\sum x _{i}}{N}$
Rearranging: $\sum x_{i} = N \times \overset{x}{ˉ}$

Calculating Old Sum of Observations

Old $\sum x_{i} = 40 \times 40 = 1600$

Correcting the Sum .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } \sum x_{i}

Corrected $\sum x_{i} = 1600 - 2 (35) + 2 (25)$
Corrected $\sum x_{i} = 1600 - 70 + 50 = 1580$

Calculating New Mean .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } \overset{x}{ˉ}_{n e w}

New Mean $\overset{x}{ˉ}_{n e w} = \frac{1580}{40}$
$\overset{x}{ˉ}_{n e w} = 39.5$

Formula for Variance .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } σ^{2}

Variance formula: $σ^{2} = \frac{\sum x _{i}^{2}}{N} - (\overset{x}{ˉ})^{2}$
Rearranging: $\sum x_{i}^{2} = N (σ^{2} + \overset{x}{ˉ}^{2})$

Calculating Old Sum of Squares .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } \sum x_{i}^{2}

Old $\sum x_{i}^{2} = 40 (49 + 4 0^{2})$
Old $\sum x_{i}^{2} = 40 (49 + 1600) = 40 (1649) = 65960$

Correcting Sum of Squares .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } \sum x_{i}^{2}

Corrected $\sum x_{i}^{2} = 65960 - 2 (3 5^{2}) + 2 (2 5^{2})$
Corrected $\sum x_{i}^{2} = 65960 - 2450 + 1250 = 64760$

Calculating New Variance .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } σ_{n e w}^{2}

New Variance $σ_{n e w}^{2} = \frac{64760}{40} - (39.5)^{2}$
$σ_{n e w}^{2} = 1619 - 1560.25 = 58.75$

Final Results Summary

New Mean: $39.5$
New Variance: $58.75$
Key Takeaway: Always update the sum and sum of squares separately before recalculating statistics.

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Visualized Solution (Hindi)

Initial Data and Error Identification

Defining Midpoints of Intervals

Formula for Mean $\overset{x}{ˉ}$

Calculating Old Sum of Observations

Correcting the Sum $\sum x_{i}$

Calculating New Mean $\overset{x}{ˉ}_{n e w}$

Formula for Variance $σ^{2}$

Calculating Old Sum of Squares $\sum x_{i}^{2}$

Correcting Sum of Squares $\sum x_{i}^{2}$

Calculating New Variance $σ_{n e w}^{2}$

Final Results Summary

Conceptually Similar Problems

In calculating the mean and variance of 10 readings, a student wrongly used the figure 52 for the correct figure of 25. He obtained the mean and variance as 45.0 and 16.0 respectively. Determine the correct mean and variance.

The mean of the data set comprising of 16 observations is 16. If one of the observation valued 16 is deleted and three new observations valued 3, 4 and 5 are added to the data, then the mean of the resultant data, is:

In an experiment with 15 observations on x, the following results were available: $\sum x^{2} = 2830, \sum x = 170$ . One observation that was 20 was found to be wrong and was replaced by the correct value 30. The corrected variance is

All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even after the grace marks were given ?

The average marks of boys in class is 52 and that of girls is 42. The average marks of boys and girls combined is 50. The percentage of boys in the class is

The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of the set is increased by 2, then the median of the new set

In a class of 100 students there are 70 boys whose average marks in a subject are 75. If the average marks of the complete class is 72, then what is the average marks of the girls?

Let $x_{1}, x_{2}, \dots, x_{n}$ be n observations, and let $\overset{x}{ˉ}$ be their arithmetic mean and $σ^{2}$ be the variance. Statement-1 : Variance of $2 x_{1}, 2 x_{2}, \dots, 2 x_{n}$ is $4 σ^{2}$ Statement-2 : Arithmetic mean $2 x_{1}, 2 x_{2}, \dots, 2 x_{n}$ is $4 \overset{x}{ˉ}$ .

The mean of the numbers a, b, 8, 5, 10 is 6 and the variance is 6.80. Then which one of the following gives possible values of a and b?

The variance of first 50 even natural numbers is

In calculating the mean and variance of 10 readings, a student wrongly used the figure 52 for the correct figure of 25. He obtained the mean and variance as 45.0 and 16.0 respectively. Determine the correct mean and variance.

The mean of the data set comprising of 16 observations is 16. If one of the observation valued 16 is deleted and three new observations valued 3, 4 and 5 are added to the data, then the mean of the resultant data, is:

In an experiment with 15 observations on x, the following results were available: $\sum x^{2} = 2830, \sum x = 170$ . One observation that was 20 was found to be wrong and was replaced by the correct value 30. The corrected variance is

All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even after the grace marks were given ?

The average marks of boys in class is 52 and that of girls is 42. The average marks of boys and girls combined is 50. The percentage of boys in the class is

The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of the set is increased by 2, then the median of the new set

In a class of 100 students there are 70 boys whose average marks in a subject are 75. If the average marks of the complete class is 72, then what is the average marks of the girls?

Let $x_{1}, x_{2}, \dots, x_{n}$ be n observations, and let $\overset{x}{ˉ}$ be their arithmetic mean and $σ^{2}$ be the variance. Statement-1 : Variance of $2 x_{1}, 2 x_{2}, \dots, 2 x_{n}$ is $4 σ^{2}$ Statement-2 : Arithmetic mean $2 x_{1}, 2 x_{2}, \dots, 2 x_{n}$ is $4 \overset{x}{ˉ}$ .

The mean of the numbers a, b, 8, 5, 10 is 6 and the variance is 6.80. Then which one of the following gives possible values of a and b?

The variance of first 50 even natural numbers is

Visualized Solution (Hindi)

Initial Data and Error Identification

Defining Midpoints of Intervals

Formula for Mean .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } xˉ

Calculating Old Sum of Observations

Correcting the Sum .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∑xi​

Calculating New Mean .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } xˉnew​

Formula for Variance .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } σ2

Calculating Old Sum of Squares .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∑xi2​

Correcting Sum of Squares .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∑xi2​

Calculating New Variance .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } σnew2​

Final Results Summary

Conceptually Similar Problems

In calculating the mean and variance of 10 readings, a student wrongly used the figure 52 for the correct figure of 25. He obtained the mean and variance as 45.0 and 16.0 respectively. Determine the correct mean and variance.

The mean of the data set comprising of 16 observations is 16. If one of the observation valued 16 is deleted and three new observations valued 3, 4 and 5 are added to the data, then the mean of the resultant data, is:

All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even after the grace marks were given ?

The average marks of boys in class is 52 and that of girls is 42. The average marks of boys and girls combined is 50. The percentage of boys in the class is

The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of the set is increased by 2, then the median of the new set

In a class of 100 students there are 70 boys whose average marks in a subject are 75. If the average marks of the complete class is 72, then what is the average marks of the girls?

The mean of the numbers a, b, 8, 5, 10 is 6 and the variance is 6.80. Then which one of the following gives possible values of a and b?

The variance of first 50 even natural numbers is

In calculating the mean and variance of 10 readings, a student wrongly used the figure 52 for the correct figure of 25. He obtained the mean and variance as 45.0 and 16.0 respectively. Determine the correct mean and variance.

The mean of the data set comprising of 16 observations is 16. If one of the observation valued 16 is deleted and three new observations valued 3, 4 and 5 are added to the data, then the mean of the resultant data, is:

All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even after the grace marks were given ?

The average marks of boys in class is 52 and that of girls is 42. The average marks of boys and girls combined is 50. The percentage of boys in the class is

The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of the set is increased by 2, then the median of the new set

In a class of 100 students there are 70 boys whose average marks in a subject are 75. If the average marks of the complete class is 72, then what is the average marks of the girls?

The mean of the numbers a, b, 8, 5, 10 is 6 and the variance is 6.80. Then which one of the following gives possible values of a and b?

The variance of first 50 even natural numbers is

Formula for Mean $\overset{x}{ˉ}$

Correcting the Sum $\sum x_{i}$

Calculating New Mean $\overset{x}{ˉ}_{n e w}$

Formula for Variance $σ^{2}$

Calculating Old Sum of Squares $\sum x_{i}^{2}$

Correcting Sum of Squares $\sum x_{i}^{2}$

Calculating New Variance $σ_{n e w}^{2}$