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अध्याय 13: सांख्यिकी (Statistics)

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CBSE • कक्षा XI • Mathematics • अध्याय 13

अनुमानित समय: 45 Mins

प्रगति: अध्ययनरत

सांख्यिकी (Statistics)

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

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प्रस्तावना अध्याय अवधारणाएँ अध्याय सारांश स्व-मूल्यांकन अभ्यास टेस्ट दें

📊 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.

यह अध्याय क्यों महत्वपूर्ण है

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

अध्ययन से पूर्व (आवश्यक ज्ञान)

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

इस अध्याय के लक्ष्य

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 ($CV = \frac{\sigma}{\bar{x}} \times 100$).

अध्याय रूपरेखा एवं प्रगति

1 1. Mean Deviation

2 2. Variance & Standard Deviation ($...

3 3. Coefficient of Variation (CV)

सम्पूर्ण सैद्धांतिक एवं वैचारिक अध्ययन

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.

चित्रात्मक व्याख्या एवं मॉडल

Statistics Master Matrix

Conceptual framework, core mechanisms, and analytical relationships

Academic Architecture

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

अध्याय का सार संक्षेप एवं 10 मुख्य निष्कर्ष

मुख्य बिंदु 1

Dispersion: Measure of scatter or spread of observations around central tendency.

मुख्य बिंदु 2

Mean Deviation: Average distance of observations from mean or median.

मुख्य बिंदु 3

Standard Deviation: Universal root-mean-square dispersion metric.

मुख्य बिंदु 4

Variance: Average of squared deviations from the arithmetic mean.

मुख्य बिंदु 5

Coefficient of Variation (CV): Dimensionless percentage index comparing consistency across data sets.

स्व-मूल्यांकन अभ्यास (Check Your Understanding)

मूल वैचारिक स्पष्टता की जांच के लिए नैदानिक प्रश्न। पहले स्वयं हल करें, फिर उत्तर देखें।

Find the mean deviation about the mean for the data: $4, 7, 8, 9, 10, 12, 13, 17$.

उत्तर एवं व्याख्या देखें

उत्तर: 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.

Find the variance and standard deviation for the numbers: $6, 7, 10, 12, 13, 4, 8, 12$.

उत्तर एवं व्याख्या देखें

उत्तर: 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.

What is the Coefficient of Variation? Why is it useful?

उत्तर एवं व्याख्या देखें

उत्तर: 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.

Between two cricket players A and B with batting averages 40 and 40, and standard deviations 5 and 8, who is more consistent?

उत्तर एवं व्याख्या देखें

उत्तर: $CV_A = \frac{5}{40} \times 100 = 12.5\%$. $CV_B = \frac{8}{40} \times 100 = 20\%$. Player A has a lower CV and is significantly more consistent.
Player A is more consistent (lower CV).

What is the effect on the standard deviation if each observation in a data set is multiplied by a positive constant $k$?

उत्तर एवं व्याख्या देखें

उत्तर: 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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