Chapter 13 of the NCERT Class 11 Mathematics syllabus deals with Statistics a branch of mathematics that focuses on the collecting, analyzing, interpreting and presenting data. In the miscellaneous exercise of this chapter students are introduced to a variety of the problems designed to test their understanding of the key statistical concepts such as the measures of central tendency, dispersion and graphical representation of data.
Statistics
The Statistics is a mathematical discipline that involves the analysis and interpretation of the numerical data. It provides methods to organize and summarize information allowing the meaningful conclusions to be drawn from the datasets. Key statistical measures include the mean, median, mode, variance and standard deviation which help describe the central tendency and variability of the data.
Question 1. 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.
Solution:
Given,
We are provided with six of the observations 6, 7, 10, 12, 12 and 13.
Let us assume the missing observations to be a and b.
Now, Mean
\overline {x} = \frac{\sum{x}}{n} = 99 = (6 + 7 + 10 + 12 + 12 + 13 + a + b)/8
But,
\overline{x} Solving for a + b, we get,
a + b = 12
Also,
Variance
\sigma^2 = \frac{1}{n}\sum{x}^2 - (\overline{x})^2 Equating
\sigma^2 = 26\space and \space \overline{x}=9, n = 8 We have,
9.25 = 1/8(62 + 72 + 102 + 122 + 122 + 132 + a2 + b2) - 92
=> 9.25 + 81 = 1/8(36 + 49 + 100 + 144 + 144 + 169 + a2 + b2)
=> 90.25 * 8 = 642 + a2 + b2
=> a2 + b2 = 80
We have, b = 12 - a
On substituting the value,
a2 + (12 - a)2= 80
=> 2a2 - 24a + 64 = 0
On dividing by 2, we get
a2 - 12a + 32 = 0
Therefore, a = 4, 8
Now, for a = 4, b = 8
And, for a = 8, b = 4
Question 2. The mean and variance of 7 observations are 8 and 16, respectively. If five of the observations are 2, 4, 10, 12, 14. Find the remaining two observations.
Solution:
Given,
We are provided with five of the observations 2, 4, 10, 12, 14.
Let us assume the missing observations to be a and b.
Now, Mean
\overline {x} = \frac{\sum{x_i}}{n} But,
\overline{x} = 8 8 = (2 + 4 + 10 + 12 + 14 + a + b)/(7)
Solving for a + b, we get,
a + b = 14
Also,
Variance =
\sigma^2 = \frac{1}{n}\sum{x}^2 - (\overline{x})^2 Equating
\sigma^2 = 16\space and \space \overline{x}=8, n =7 We have,
16 = 1/7(22 + 42 + 102 + 122 + 142 + a2 + b2) - 64
=> 16 + 64 = 1/7(4 + 16 + 100 + 144 + 196 + a2 + b2)
=> 560 = 460 + a2 + b2
=> a2+b2 = 100
We have, b = 14 - a
On substituting the value, we get
a2 + (14 - a)2= 100
=> 2a2 - 28a + 96 = 0
On dividing by 2, we get
a2 - 14a + 48 = 0
=> (a - 8)(a - 6) = 0
Therefore, a = 6, 8
Now, for a = 6, b = 8
And, for a = 8, b = 6
Question 3. 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.
Solution:
Mean of six observations = 8
Standard deviation of six observations = 4
Let the six observations be x1, x2, x3, x4, x5, x6
Therefore,
Mean of observations,
\overline{x} = (x1 + x2 + x3 + x4 + x5 + x6)/6 = 8If each observation is multiplied by 3 and the resultant observations are yi then,
yi = 3xi
xi = (1/3)yi, where i = 1....6
So, new mean
\bar{y} = (y1 + y2 + y3 + y4 + y5 + y6)/6= 3(x1 + x2 + x3 + x4 + x5 + x6)/6
= 3 * 8
= 24
Standard Deviation
(\sigma) = \sqrt{\frac{1}{n}\sum_{i=1}^{n}X_1 - \frac{1}{n^2}(\sum_{i=1}^{n}X)^2 } \\ (4)^2 = \frac{1}{6}\sum_{i=1}^{6}(x_i-\overline{x})^2 \\ \sum_{i=1}^{6}(x_i-\overline{x})^2 = 96
\overline{y} = 3\overline{x} \\ \overline{x} = \frac{1}{3}\overline {y} Substituting values, we get,
\sum_{i=1}^{6}(y_i - \overline{y})^2 = 864 Therefore,
Variance of new observation = 1/6 x 864
= 144
Standard Deviation = √144
= 12
Question 4. Given that x̅ 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 ax̅ and a2σ2, respectively, (a ≠ 0).
Solution:
Let us assume the observations to be x1, ...xn
Mean of n observations =
\overline{x} Variance of n observations =
\sigma^2 We know,
Variance(\sigma^2) = \frac{1}{n}\sum_{i=1}^{n}y_1(x_i - \overline{x})^2 y
y_i = ax_i \\ x_i = \frac{1}{a}y_i \\ \overline{y} = \frac{1}{n}\sum_{i=1}^{n}y_i \\ \overline{y} = \frac{1}{n}\sum_{i=1}^{n}ax_i \\ \overline{y} = a\overline{x} Now,
Mean of the observations, ax1, ax2, .....axn =
\overline{x} By substituting values, we get,
\sigma^2 = \frac{1}{n}\sum_{i=1}^{n}(\frac{1}{a}y_1 - \frac{1}{a}\overline{y})^2 \\ a^2\sigma^2 = \frac{1}{n}\sum_{i=1}^{n}(y_1-\overline{y})^2 Variance =
a^2\sigma^2
Question 5. 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
Solution:
(i) On omission of wrong item
n = 20
Incorrect mean = 20
Incorrect SD = 2
\overline{X} = \frac{1}{n}\sum_{i=1}^{20}X_1 \\ 10 = \frac{1}{20}\sum_{i=1}^{20}X_1 \\ \sum_{i=1}^{20}X_1 = 200 Now,
Incorrect sum of observations = 200
Correct sum of observations = 200 - 8 = 192
Therefore,
Correct mean = Correct sum/19
= 192/19
= 10.1
Standard Deviation
(\sigma) = \sqrt{\frac{1}{n}\sum_{i=1}^{n}X_1 - \frac{1}{n^2}(\sum_{i=1}^{n}X)^2 } \\ 2 = \sqrt{\frac{1}{n}\sum_{i=1}^{n}X_1^2 - (\overline{X})^2} 4 = 1/20 Incorrect
\sum_{i=1}^{n}X_1^2 - 100Incorrect
\sum_{i=1}^{n}X_1^2 = 2080Therefore,
Correct
\sum_{i=1}^{n}X_1^2 = Incorrect\sum_{i=1}^{n}X_1^2 - (8)2= 2080 - 64
= 2016
Calculating correct standard deviation,
Correct SD =
\sqrt{\frac{Correct \sum X_1^2}{n}-(Correct Mean)^2} \\ = \sqrt{\frac{2016}{19}-(10.1)^2} = √(1061.1 - 102.1)
= 2.02
(ii) If it is replaced by 12,
Incorrect sum of observations, n = 200
Correct sum of observations n = 200 - 8 + 12
n = 204
Correct Mean = Correct Sum / 20
= 204/20
= 10.2
Standard Deviation
(\sigma) = \sqrt{\frac{1}{n}\sum_{i=1}^{n}X_1 - \frac{1}{n^2}(\sum_{i=1}^{n}X)^2 } \\ 2 = \sqrt{\frac{1}{n}\sum_{i=1}^{n}X_1^2 - (\overline{X})^2} 4 = 1/20 Incorrect
\sum_{i=1}^{n}X_1^2 - 100Incorrect
\sum_{i=1}^{n}X_1^2 = 2080Therefore, Correct
\sum_{i=1}^{n}X_1^2 = Incorrect\sum_{i=1}^{n}X_1^2 - (8)2 + (12)2= 2080 - 64 + 144
= 2160
Calculating correct standard deviation,
Correct SD = \sqrt{\frac{Correct \sum X_1^2}{n}-(Correct Mean)^2} \\ = \sqrt{\frac{2160}{20}-(10.2)^2} = √(108 - 104.04)
= 1.98
Question 6. 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.
Solution:
Given:
n = 100
Incorrect mean, (x̅) = 20
Incorrect standard deviation (σ) = 3
Therefore,
20 = \frac{1}{100}\sum_{i=1}^{100} x_i On solving, we get
\sum_{i=1}^{100}X_1 = 20 * 100 \\ \sum_{i=1}^{100}X_1 = 2000 Incorrect sum of observations = 2000
Now, Correct sum of observations = 2000 - 21- 21 - 18
= 1940
Correct mean = Correct Sum / 97
= 1940/97
= 20
Also,
Standard Deviation
(\sigma) = \sqrt{\frac{1}{n}\sum_{i=1}^{n}X_1 - \frac{1}{n^2}(\sum_{i=1}^{n}X)^2 } \\ 3 = \sqrt{\frac{1}{n}\sum_{i=1}^{n}X_1^2 - (\overline{X})^2} \\ 3 = \sqrt{\frac{1}{100}* Incorrect \sum X_1^2-(20)^2} Incorrect
\sum X_1^2 = 100(9 + 400)Incorrect
\sum X_1^2 = 40900Correct
\sum_{i=1}^{n}X_1^2 = Incorrect\sum_{i=1}^{n}X_1^2 - (21)2 - (21)2 - (18)2= 40900 - 441 - 441 - 324
= 40900 - 1206
= 39694
Therefore,
Correct S.D =
\sqrt{\frac{Correct\sum X_1^2}{n}-(Correct mean)^2} \\ = \sqrt{\frac{39694}{97}-(20)^2} \\ = √(409.216 - 400)
= 3.036
Conclusion
The miscellaneous exercise in Chapter 13 of NCERT Class 11 focuses on the consolidating the understanding of key statistical measures and concepts. By solving a variety of problems students will gain the necessary skills to the analyze data and interpret results effectively. Mastering these statistical techniques is crucial for the tackling more advanced topics in the mathematics and real-world applications.