Comprehensive guide and practice questions for CBSE Class 11 Applied Mathematics curriculum (2026-27).
Comprehensive Concepts & Principles: Standard Deviation & Variance
Introduced by Karl Pearson in 1893, the Standard Deviation ($sigma$) is the premier, mathematically rigorous measure of statistical dispersion. Defined as the positive square root of the arithmetic mean of squared deviations from the arithmetic mean, standard deviation overcomes the algebraic limitations of mean deviation by squaring deviations $(x_i - \bar{X})^2$ rather than discarding signs with absolute values.
Coefficient of Variation ($CV$): The universal relative measure of variability:
$$CV = \frac{\sigma}{\bar{X}} \times 100\%$$
A series with a LOWER $CV$ is more consistent, more homogeneous, and more stable.
Invariance Properties: Origin vs. Scale
Standard deviation exhibits distinct mathematical responses to linear transformations $y_i = a + b x_i$:
• Independent of Change of Origin: Adding or subtracting a constant $a$ from every observation leaves standard deviation completely unchanged: $\sigma_{x \pm a} = \sigma_x$.
• Dependent on Change of Scale: Multiplying every observation by a constant $b$ scales standard deviation by $|b|$: $\sigma_{bx} = |b| \cdot \sigma_x$, and scales variance by $b^2$: $\sigma_{bx}^2 = b^2 \cdot \sigma_x^2$.
Worked Benchmark Examples
Example 1 (Variance & SD of Individual Series):
Find the Variance and Standard Deviation of the first 5 positive multiples of 3:
$$3, \quad 6, \quad 9, \quad 12, \quad 15$$
The batting records of two cricketers A and B across a season are:
• Batsman A: Mean $= 50$ runs, $\sigma = 12$ runs
• Batsman B: Mean $= 40$ runs, $\sigma = 8$ runs
(i) Who is the higher run-scorer on average?
(ii) Who is the more consistent batsman?
Show Step-by-Step Solution
• (i) Since $\bar{X}_A = 50 > \bar{X}_B = 40$, Batsman A is the higher run-scorer on average.
• (ii) To evaluate consistency, compute the Coefficient of Variation ($CV$):
$$CV_A = \frac{\sigma_A}{\bar{X}_A} \times 100 = \frac{12}{50} \times 100 = \mathbf{24\%}$$
$$CV_B = \frac{\sigma_B}{\bar{X}_B} \times 100 = \frac{8}{40} \times 100 = \mathbf{20\%}$$
• Conclusion: Batsman B is more consistent because $CV_B (20\%) < CV_A (24\%)$.
CHECK YOUR PROGRESS 10.3
Graded textbook problems on Variance, Standard Deviation, and Coefficient of Variation based on CBSE Class 11 Applied Mathematics.
Problem 1Raw Data SD
Calculate the Variance and Standard Deviation of the numbers:
$$6, \quad 8, \quad 10, \quad 12, \quad 14$$
Prove that the variance of the first $n$ natural numbers $1, 2, 3, \dots, n$ is given by $\sigma^2 = \frac{n^2 - 1}{12}$, and compute the standard deviation for the first 10 natural numbers.
A dataset has a standard deviation of 6.
(a) If 5 is added to every observation, what is the new standard deviation?
(b) If every observation is multiplied by 3, what are the new standard deviation and variance?
Show Step-by-Step Solution
• (a) Adding a constant shifts the origin without affecting spread: New $\sigma = \mathbf{6}$.
• (b) Multiplying by 3 scales the spread by 3:
- New $\sigma = 3 \times 6 = \mathbf{18}$.
- New Variance $\sigma^2 = 18^2 = \mathbf{324}$ (which is $3^2 \times 36$).
Problem 4Step-Deviation Calculation
Using the step-deviation method with $h = 10$ and $A = 35$, the following summary values were obtained for 50 items:
$\sum f_i u_i = 10, \quad \sum f_i u_i^2 = 98$.
Calculate the Mean and Standard Deviation of the distribution.
Two industrial factories A and B report weekly wages:
• Factory A: Mean weekly wage = ₹5,000, $\sigma = ₹400$
• Factory B: Mean weekly wage = ₹6,000, $\sigma = ₹450$
Which factory has greater wage inequality (variability in wages)?
Show Step-by-Step Solution
• $CV_A = \frac{400}{5000} \times 100 = \mathbf{8\%}$.
• $CV_B = \frac{450}{6000} \times 100 = \mathbf{7.5\%}$.
• Conclusion: Factory A has greater wage variability ($8\% > 7.5\%$), meaning wages are relatively more unequally distributed in Factory A.
Comprehensive Practice Worksheet: Standard Deviation & Variance
Graded problems classified into 1-Mark, 2-Mark, and 4-Mark sections based on CBSE Applied Mathematics curriculum standards.
Q4: Why is Standard Deviation considered superior to Mean Deviation in higher mathematical statistics?
Show Answer & Scheme
Explanation [2 Marks]:
(1) Standard deviation squares deviations algebraically instead of artificially discarding signs with absolute value bars, making it continuously differentiable.
(2) It possesses well-defined algebraic properties that allow the calculation of combined standard deviation of multiple subgroups, which is impossible with mean deviation.
SECTION C: 4-Mark Comprehensive Numerical Drill
Q5: An agricultural researcher tests the crop yield (in quintals per hectare) from two organic fertilizers Alpha and Beta across 5 experimental plots:
• Fertilizer Alpha: $18, \quad 20, \quad 22, \quad 24, \quad 26$
• Fertilizer Beta: $14, \quad 17, \quad 23, \quad 25, \quad 31$
(i) Find the Mean and Standard Deviation of both fertilizers.
(ii) Compute the Coefficient of Variation for both and determine which fertilizer provides more consistent yields.
If $y = a + bx$:
• $\bar{Y} = a + b\bar{X}$ (Mean responds to BOTH origin and scale)
• $\sigma_y = |b| \cdot \sigma_x$ (Standard deviation responds ONLY to scale $|b|$)
• $\sigma_y^2 = b^2 \cdot \sigma_x^2$ (Variance responds ONLY to squared scale $b^2$)
10-Question Diagnostic Board Mock Examination: Standard Deviation & Variance
Standard timed test: 4 Objective MCQs / Assertion-Reason with instant automatic grading, plus 6 multi-mark subjective problems with CBSE marking rubrics.
PART 1: Objective Section (4 Marks)
Q1: If every value in a dataset is increased by 10, the variance of the new dataset will:
Solution: (C) Adding a constant changes only the origin, leaving variance and standard deviation completely unchanged.
Q2: If the standard deviation of $x$ is 4, then the standard deviation of $y = 3 - 2x$ is:
Solution: (B) $\sigma_y = |-2| \cdot \sigma_x = 2 \times 4 = 8$. Standard deviation is always non-negative.
Q3: A factory produces items with Mean $= 200$ and $CV = 5\%$. The standard deviation of the items is:
Q4 (Assertion-Reason): Assertion (A): When comparing two distributions with different units of measurement, the distribution with the smaller Coefficient of Variation is considered more consistent. Reason (R): The Coefficient of Variation is a dimensionless percentage that normalizes standard deviation by its arithmetic mean.
Solution: (A) Both Assertion and Reason are true and Reason provides the precise normalization logic of $CV$.
PART 2: Subjective Section (Q5 - Q10)
Q52 Marks
Find the variance of the numbers $2, 4, 6, 8, 10$.
Show Step-by-Step Solution & Marking Scheme
• Notice these are $2 \times (1, 2, 3, 4, 5)$.
• Variance of first 5 natural numbers: $\frac{5^2 - 1}{12} = \frac{24}{12} = 2$ [1 Mark].
• By scale property: $\sigma^2(2x) = 2^2 \times \sigma^2(x) = 4 \times 2 = \mathbf{8}$ [1 Mark].
Q62 Marks
The mean of 5 observations is 4.4 and their variance is 8.24. If three of the observations are 1, 2, and 6, find the remaining two observations.
Prove that for any dataset, the Standard Deviation is always greater than or equal to the Mean Deviation about the Mean ($\sigma \ge MD(\bar{X})$).
Show Step-by-Step Solution & Marking Scheme
• Let deviations be $z_i = |x_i - \bar{X}|$.
• The mean of $z_i$ is $\bar{z} = \frac{\sum |x_i - \bar{X}|}{N} = MD(\bar{X})$ [1 Mark].
• The mean of $z_i^2$ is $\frac{\sum |x_i - \bar{X}|^2}{N} = \frac{\sum (x_i - \bar{X})^2}{N} = \sigma^2$ [1 Mark].
• Since the variance of any variable is always non-negative:
$$\text{Var}(z) = \frac{\sum z_i^2}{N} - (\bar{z})^2 \ge 0 \implies \sigma^2 - [MD(\bar{X})]^2 \ge 0$$
$$\sigma^2 \ge [MD(\bar{X})]^2 \implies \mathbf{\sigma \ge MD(\bar{X})} \quad [1 \text{ Mark}]$$
• Equality holds if and only if all deviations are identical in absolute magnitude.
Q83 Marks
For a group of 50 items, $\sum x = 500$ and $\sum x^2 = 6000$. It was subsequently detected that two values originally recorded as 8 and 12 were actually misread, and their correct values are 10 and 10. Compute the corrected Mean and corrected Standard Deviation.
A portfolio manager analyzes two mutual fund assets over a 5-year period. Fund Blue delivered an average annual return of 15% with a standard deviation of 3%. Fund Gold delivered an average annual return of 25% with a standard deviation of 6%.
Q9 (2 Marks): Calculate the Coefficient of Variation for both Fund Blue and Fund Gold. Q10 (2 Marks): Which mutual fund provides higher return per unit of volatility risk, and which fund should a risk-averse conservative investor prefer?
Solution to Q10 [2 Marks]:
• Fund Blue has a lower Coefficient of Variation ($20\% < 24\%$), meaning it has lower relative volatility per unit of return.
• Recommendation: A risk-averse conservative investor should select Fund Blue because its returns are significantly more stable and consistent around the mean.