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CBSE • Class XI • Mathematics • Ch 13
Estimated Time: 45 Mins
Study Progress: In Progress

Statistics

In Class 11 Mathematics, "Statistics" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

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📊 Have You Ever Wondered?

Two cricket batsmen can both have an identical batting average of 50 runs, yet one scores a steady 50 in every match while the other alternates betwee...

Two cricket batsmen can both have an identical batting average of 50 runs, yet one scores a steady 50 in every match while the other alternates between a century and a duck! Measures of Dispersion measure consistency and volatility.

Why This Chapter Matters

In Class 11 Mathematics, "Statistics" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

Before You Begin (Prerequisites)

  • Mean, Median, and Mode from Class 10.
  • Grouped frequency tables.
  • Absolute values.

What You Will Learn (Core Objectives)

  • Explain the limitation of central tendency and need for Measures of Dispersion.
  • Calculate Mean Deviation about Mean and Median for ungrouped and grouped frequency distributions.
  • Define Variance ($\sigma^2$) and Standard Deviation ($\sigma$) as root mean square deviations.
  • Apply shortcut formula for Variance: $\sigma^2 = \frac{1}{N}\sum f_i x_i^2 - (\bar{x})^2$.
  • Compare variability of distributions using Coefficient of Variation ($CV = \frac{\sigma}{\bar{x}} \times 100$).

Chapter Roadmap & Progression

1 1. Mean Deviation
2 2. Variance & Standard Deviation ($...
3 3. Coefficient of Variation (CV)

Complete Concept Guide (100% Curriculum Coverage)

1. Mean Deviation

Mean Deviation is the arithmetic mean of the numerical values of the deviations of items from central tendency: $$\mathbf{MD(\bar{x}) = \frac{\sum f_i |x_i - \bar{x}|}{N}} \quad \text{and} \quad \mathbf{MD(M) = \frac{\sum f_i |x_i - M|}{N}}$$

2. Variance & Standard Deviation ($\sigma$)

Overcomes absolute values by squaring deviations:
• Variance ($\sigma^2$): $$\mathbf{\sigma^2 = \frac{\sum f_i (x_i - \bar{x})^2}{N} = \frac{\sum f_i x_i^2}{N} - (\bar{x})^2}$$
• Standard Deviation ($\sigma$): $\mathbf{\sigma = \sqrt{\text{Variance}}}$

3. Coefficient of Variation (CV)

To compare consistency between two series with different units or means: $$\mathbf{CV = \frac{\sigma}{\bar{x}} \times 100}$$ A series with smaller CV is more consistent, stable, and uniform; larger CV indicates greater variability.

Visual Learning & Conceptual Map

Statistics Master Matrix

Conceptual framework, core mechanisms, and analytical relationships
Academic Architecture

1. Mean Deviation • 2. Variance & Standard Deviation ($\sigma$)

Chapter Summary & 10 Key Takeaways

Takeaway 1
Dispersion: Measure of scatter or spread of observations around central tendency.
Takeaway 2
Mean Deviation: Average distance of observations from mean or median.
Takeaway 3
Standard Deviation: Universal root-mean-square dispersion metric.
Takeaway 4
Variance: Average of squared deviations from the arithmetic mean.
Takeaway 5
Coefficient of Variation (CV): Dimensionless percentage index comparing consistency across data sets.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

1
Find the mean deviation about the mean for the data: $4, 7, 8, 9, 10, 12, 13, 17$.
Reveal Answer & Explanation
Answer: Mean $\bar{x} = \frac{4+7+8+9+10+12+13+17}{8} = \frac{80}{8} = 10$. Deviations $|x_i - 10|$: $6, 3, 2, 1, 0, 2, 3, 7$. Sum $= 24$. $MD(\bar{x}) = \frac{24}{8} = 3$.
3.
2
Find the variance and standard deviation for the numbers: $6, 7, 10, 12, 13, 4, 8, 12$.
Reveal Answer & Explanation
Answer: Sum $= 72, n = 8 \implies \bar{x} = 9$. Deviations $(x_i - 9)^2$: $9, 4, 1, 9, 16, 25, 1, 9$. Sum $= 74$. Variance $\sigma^2 = \frac{74}{8} = 9.25$. Standard deviation $\sigma = \sqrt{9.25} \approx 3.04$.
Variance = 9.25, SD ≈ 3.04.
3
What is the Coefficient of Variation? Why is it useful?
Reveal Answer & Explanation
Answer: The Coefficient of Variation is $CV = \frac{\sigma}{\bar{x}} \times 100$. It is a normalized dimensionless measure of dispersion used to compare consistency, stability, or risk between two data sets having different means or different units.
CV = (σ/x̄) × 100; compares consistency across datasets.
4
Between two cricket players A and B with batting averages 40 and 40, and standard deviations 5 and 8, who is more consistent?
Reveal Answer & Explanation
Answer: $CV_A = \frac{5}{40} \times 100 = 12.5\%$. $CV_B = \frac{8}{40} \times 100 = 20\%$. Player A has a lower CV and is significantly more consistent.
Player A is more consistent (lower CV).
5
What is the effect on the standard deviation if each observation in a data set is multiplied by a positive constant $k$?
Reveal Answer & Explanation
Answer: The standard deviation is also multiplied by $k$ ($\sigma' = k\sigma$). If a constant is merely added to each value, the standard deviation remains unchanged.
Standard deviation is multiplied by k.
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