Statistics Class 11 Math Quick Recall / Short Notes (2026-27)

Statistics Formulas and Measures of Dispersion Concepts

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Mean Deviation (M.D.): $\frac{1}{n} \sum |x_i - \text{mean/median}|$ .

Variance ( $\sigma^2$ ): $\frac{1}{n} \sum (x_i - \bar{x})^2$ .

Standard Deviation ( $\sigma$ ): Positive square root of Variance.

Shortcut for Variance: $\frac{\sum x_i^2}{n} - (\bar{x})^2$ .

Lower M.D./S.D.: Indicates more consistent (less dispersed) data. 📥 Download 1-Page Short Notes PDF (Zero-Friction)

Introduction

Statistics is the science of data analysis, focusing on Measures of Dispersion which describe how information is spread around a central value. Master Mean Deviation, Variance, and Standard Deviation to excel in data science foundations and probability modeling. This Class 11 Math Chapter 15 guide provides all essential formulas for JEE and CBSE success. Statistics is the science of collecting, organizing, and analyzing data to draw meaningful conclusions.

1. Measures of Dispersion

Dispersion refers to the scattering of data around a central value. Two sets of data can have the same mean but look completely different based on how far the values are from the mean.

Key Measures:

Range: The difference between the maximum and minimum values (Max - Min).
Quartile Deviation: (Not in latest NCERT syllabus, but useful for competition).
Mean Deviation: The arithmetic mean of the absolute deviations of the observations from an average.
Standard Deviation: The most stable and widely used measure of dispersion.

2. Mean Deviation (M.D.)

Mean deviation can be calculated about the Mean or the Median.

Calculation Steps:

Find the Mean ( $\bar{x}$ ) or Median ( $M$ ) of the data.
Find the absolute differences $|x_i - \bar{x}|$ or $|x_i - M|$ .
Calculated the average of these absolute differences.

Formula for Ungrouped Data: $M.D. (\bar{x}) = \frac{\sum |x_i - \bar{x}|}{n}$

Formula for Grouped Data: $M.D. (\bar{x}) = \frac{\sum f_i |x_i - \bar{x}|}{N}$ (where $N = \sum f_i$ )

3. Variance and Standard Deviation

While Mean Deviation uses absolute values, Variance uses squares of deviations to avoid negative signs.

Variance ( $\sigma^2$ ):

The average of the squared deviations from the mean.

Formula: $\sigma^2 = \frac{1}{n} \sum (x_i - \bar{x})^2$

Standard Deviation ( $\sigma$ ):

The square root of the variance. It is preferred because it shares the same units as the original data.

Short Method for Discrete Frequency Distribution: $\sigma = \frac{1}{N} \sqrt{N \sum f_i x_i^2 - (\sum f_i x_i)^2}$

4. Analysis of Frequency Distributions

Sometimes we need to compare two different series (like marks of two students in different subjects) to see which is more consistent.

Coefficient of Variation (C.V.):

To compare dispersion between two sets with different means or units, we use C.V.

Formula: $C.V. = \frac{\sigma}{\bar{x}} \times 100$
Consistency Rule: The series with a lower C.V. is said to be more stable or consistent.

Comprehensive Exam Strategy (Q&A)

Q1: Find the mean deviation about the mean for the data: 6, 7, 10, 12, 13, 4, 8, 12. Answer:

Sum = 72, n = 8.
Mean ( $\bar{x}$ ) = 72/8 = 9.
Absolute Deviations: $|6-9|=3, |7-9|=2, |10-9|=1, |12-9|=3, |13-9|=4, |4-9|=5, |8-9|=1, |12-9|=3$ .
Sum of absolute deviations = 3+2+1+3+4+5+1+3 = 22.
M.D.( $\bar{x}$ ) = 22/8 = 2.75.

Q2: If the variance of 10 observations is 16, what will be the new variance if each observation is multiplied by 3? Answer:

Property: If each observation $x_i$ is multiplied by $k$ , the new variance becomes $k^2$ times the original variance.
New Variance = $3^2 \times 16 = 9 \times 16 = \mathbf{144}$ .

Q3: Which measure is better: Mean Deviation or Standard Deviation? Answer: Standard Deviation is generally better for advanced mathematical analysis because it is based on squared values (avoiding the non-algebraic "absolute" signs) and is more sensitive to outliers.

Related Revision Notes

Chapter 16: Probability
Chapter 12: Three Dimensional Geometry
[External Reference: NCERT Class 11 Math Chapter 15 (Authoritative Source)]

Conclusion

Statistics in Class 11 moves beyond just finding averages to understanding the reliability of data. Mastering Mean Deviation and Variance allows you to quantify "risk" and "consistency"—skills used in everything from weather forecasting to the stock market. Keep your calculations precise, and remember: consistency is key (both in data and in your study routine)!