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Exercise Miscellaneous - Statistics NCERT Solutions | Mathematics (Maths) Class 11 - Commerce PDF Download

Q1: The mean and variance of eight observations are 9 and 9.25, respectively. If six of the observations are 6, 7, 10, 12, 12 and 13, find the remaining two observations.
Ans:
Let the remaining two observations be x and y.
Therefore, the observations are 6, 7, 10, 12, 12, 13, x, y.
Exercise Miscellaneous - Statistics NCERT Solutions | Mathematics (Maths) Class 11 - Commerce
Exercise Miscellaneous - Statistics NCERT Solutions | Mathematics (Maths) Class 11 - Commerce
Exercise Miscellaneous - Statistics NCERT Solutions | Mathematics (Maths) Class 11 - Commerce
From (1), we obtain
x2 + y+ 2xy = 144 …(3)
From (2) and (3), we obtain
2xy = 64 … (4)
Subtracting (4) from (2), we obtain
x2 + y2 – 2xy = 80 – 64 = 16
⇒ x – y = ± 4 …(5)
Therefore, from (1) and (5), we obtain
x = 8 and y = 4, when x – y = 4
x = 4 and y = 8, when x – y = –4
Thus, the remaining observations are 4 and 8.

Q2: The mean and variance of 7 observations are 8 and 16, respectively. If five of the observations are 2, 4, 10, 12 and 14. Find the remaining two observations.
Ans: 
Let the remaining two observation be x and y
The observation are 2 , 4 , 10 , 12 , 14 , x , y
Exercise Miscellaneous - Statistics NCERT Solutions | Mathematics (Maths) Class 11 - Commerce
Exercise Miscellaneous - Statistics NCERT Solutions | Mathematics (Maths) Class 11 - Commerce
Exercise Miscellaneous - Statistics NCERT Solutions | Mathematics (Maths) Class 11 - Commerce
From (1), we obtain
x2 + y2 + 2xy = 196 ....(3)
From (2) and (3), we obtain
2xy = 196 − 100
⇒ 2xy = 96 ...........(4)
subtracting (4) from (2), we obtain
x2 + y2 − 2xy = 100 − 96
⇒ (x − y)2 = 4
⇒ x − y = ± 2 ............(5)
Therefore, from (1) and (5) we obtain
x = 8 and y = 6 when x − y = 2
x = 6 and y = 8 when x − y = − 2
Thus the remaining observation are 6 and 8

Q3: The mean and standard deviation of six observations are 8 and 4, respectively. If each observation is multiplied by 3, find the new mean and new standard deviation of the resulting observations.
Ans:

Exercise Miscellaneous - Statistics NCERT Solutions | Mathematics (Maths) Class 11 - Commerce
Exercise Miscellaneous - Statistics NCERT Solutions | Mathematics (Maths) Class 11 - Commerce
We know that,
Exercise Miscellaneous - Statistics NCERT Solutions | Mathematics (Maths) Class 11 - Commerce
Exercise Miscellaneous - Statistics NCERT Solutions | Mathematics (Maths) Class 11 - Commerce

Q4: Given that Exercise Miscellaneous - Statistics NCERT Solutions | Mathematics (Maths) Class 11 - Commerce is the mean and σ2 is the variance of n observations x1, x2 … xn. Prove that the mean and variance of the observations ax1, ax2, ax3 …axn are ax and a2 σ2, respectively (a ≠ 0).
Ans: 
The given n observations are x1, x2 … xn.
Mean =Exercise Miscellaneous - Statistics NCERT Solutions | Mathematics (Maths) Class 11 - Commerce
Variance = σ2
Exercise Miscellaneous - Statistics NCERT Solutions | Mathematics (Maths) Class 11 - Commerce
If each observation is multiplied by a and the new observations are yi, then

Exercise Miscellaneous - Statistics NCERT Solutions | Mathematics (Maths) Class 11 - Commerce
Therefore, mean of the observations, ax1, ax2 … axn, isExercise Miscellaneous - Statistics NCERT Solutions | Mathematics (Maths) Class 11 - Commerce
Substituting the values of xiand Exercise Miscellaneous - Statistics NCERT Solutions | Mathematics (Maths) Class 11 - Commercein (1), we obtain

Exercise Miscellaneous - Statistics NCERT Solutions | Mathematics (Maths) Class 11 - Commerce
Thus, the variance of the observations, ax1, ax… axn, is a2 σ2.

Q5: The mean and standard deviation of 20 observations are found to be 10 and 2, respectively. On rechecking, it was found that an observation 8 was incorrect. Calculate the correct mean and standard deviation in each of the following cases:
(i) If wrong item is omitted.
(ii) If it is replaced by 12.
Ans:
(i) Number of observations (n) = 20
Incorrect mean = 10
Incorrect standard deviation = 2

Exercise Miscellaneous - Statistics NCERT Solutions | Mathematics (Maths) Class 11 - Commerce
That is, incorrect sum of observations = 200
Correct sum of observations  = 200 - 8 = 192
Correct mean = (Correct sum )/19 = 192/19 = 10.1
Exercise Miscellaneous - Statistics NCERT Solutions | Mathematics (Maths) Class 11 - Commerce
Exercise Miscellaneous - Statistics NCERT Solutions | Mathematics (Maths) Class 11 - Commerce
= √4.09
= 2.02
(ii) When 8 is replaced by 12,
Incorrect sum of observations = 200
Correct sum of observations = 200 - 8 + 12 = 204
∴ Correct mean = (Correct sum)/20 = 204/20 = 10.2
Exercise Miscellaneous - Statistics NCERT Solutions | Mathematics (Maths) Class 11 - Commerce
Exercise Miscellaneous - Statistics NCERT Solutions | Mathematics (Maths) Class 11 - Commerce

Q6: The mean and standard deviation of a group of 100 observations were found to be 20 and 3, respectively. Later on it was found that three observations were incorrect, which were recorded as 21, 21 and 18. Find the mean and standard deviation if the incorrect observations are omitted.
Ans: 
Number of observations (n) = 100
Incorrect mean Exercise Miscellaneous - Statistics NCERT Solutions | Mathematics (Maths) Class 11 - Commerce = 20Incorrect standard deviation (σ) = 3
Exercise Miscellaneous - Statistics NCERT Solutions | Mathematics (Maths) Class 11 - Commerce
Incorrect sum of observations  = 2000
Correct sum of observations  = 2000 - 21 - 21 - 18 = 2000 - 60 = 1940
∴ Correct mean = (correct sum)/(100 - 3) = 1940/97 = 20
Exercise Miscellaneous - Statistics NCERT Solutions | Mathematics (Maths) Class 11 - Commerce
Exercise Miscellaneous - Statistics NCERT Solutions | Mathematics (Maths) Class 11 - Commerce

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FAQs on Exercise Miscellaneous - Statistics NCERT Solutions - Mathematics (Maths) Class 11 - Commerce

1. What is the importance of statistics in our daily lives?
Ans. Statistics play a crucial role in various aspects of our daily lives, including analyzing data, making informed decisions, predicting trends, conducting research, and understanding patterns and relationships in different fields.
2. How can statistics help in business decision-making?
Ans. Statistics can assist businesses in making informed decisions by analyzing market trends, customer preferences, sales data, financial performance, and other crucial factors that impact the success and growth of the organization.
3. What are the different types of statistical data?
Ans. The different types of statistical data include nominal data (categories without any specific order), ordinal data (categories with a specific order), interval data (numeric data with equal intervals), and ratio data (numeric data with a true zero point).
4. How is statistical analysis used in research studies?
Ans. Statistical analysis is essential in research studies to analyze data, test hypotheses, identify patterns, draw conclusions, and make predictions based on the findings. It helps researchers make sense of complex data and draw reliable conclusions.
5. How can statistics be used to interpret survey results?
Ans. Statistics can be used to interpret survey results by calculating frequencies, percentages, averages, correlations, and other statistical measures to analyze the responses, identify trends, draw comparisons, and draw meaningful insights from the survey data.
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