Measures of Dispersion Class 11 NCERT: Complete Chapter 6 Guide

In CBSE Class 11 Statistics Chapter 6, Measures of Dispersion, you learn how spread out or scattered the observations of a data set are. An average such as mean or median gives one central value, but it does not tell whether the values are close together or widely different. For example, two classes may have the same average marks, yet one class may be much more consistent than the other. This chapter builds the tools to measure such variation using range, quartile deviation, mean deviation, variance, standard deviation and coefficient of variation. You will also learn when to use absolute measures and when to use relative measures for comparison. On YoLearn AI Tutor, with voice explanations and sketchpad-style working, this page helps you master formulas, interpretation, solved examples, common CBSE mistakes and practice problems aligned with measures of dispersion class 11 ncert.

Why Dispersion Is Needed Along With Averages

A measure of central tendency summarizes data by one representative value, but it can hide important differences. Suppose Series A is 48, 49, 50, 51, 52 and Series B is 10, 30, 50, 70, 90. Both have mean 50, but Series A is tightly clustered while Series B is highly scattered. Dispersion measures this scatter. In examinations, always connect the formula to its meaning: a smaller dispersion means greater uniformity, consistency or reliability. A larger dispersion means greater variability. Class 11 Statistics mainly uses population-style formulas for the given data: divide by N, not by N - 1, unless the question specifically says sample variance.

Key Definitions and Formulas

Dispersion
The extent to which observations vary from a central value or from one another.
Range
The difference between the largest and smallest observation: Range = L - S. It is quick but uses only two values.
Quartile Deviation
Half of the interquartile range: QD = (Q3 - Q1)/2. It measures spread of the middle 50% of the data.
Mean Deviation
The arithmetic mean of absolute deviations from a chosen average: MD = Σ|x - A|/N or Σf|x - A|/N.
Variance
The mean of squared deviations from mean: σ² = Σ(x - x̄)²/N or Σf(x - x̄)²/N.
Standard Deviation
The positive square root of variance: σ = √σ². It is in the same unit as the original data.
Coefficient of Variation
A relative measure of dispersion: CV = (Standard Deviation / Mean) × 100. Lower CV means greater consistency.

How to Choose the Correct Measure in a Question

  1. Check what the question asks to compare — If the question asks which series is more consistent, use coefficient of variation, not just standard deviation. CV compares dispersion relative to the mean.
  2. Identify the data type — For individual data, work directly with observations. For discrete frequency data, multiply each calculation by frequency f. For grouped continuous data, use class marks as x values.
  3. Choose the centre correctly — Mean deviation may be calculated about mean, median or mode as asked. If no centre is specified, read the question carefully; NCERT problems often state the base explicitly.
  4. Use absolute or squared deviations properly — Mean deviation uses absolute deviations, so negative signs are ignored. Variance and standard deviation use squared deviations, so never cancel positive and negative deviations.

Worked Examples with Steps

  • Example 1: Find range and coefficient of range for 12, 15, 18, 21, 24.
    Step 1: Largest value L = 24 and smallest value S = 12.
    Step 2: Range = L - S = 24 - 12 = 12.
    Step 3: Coefficient of range = (L - S)/(L + S) = 12/(24 + 12) = 12/36 = 1/3.
    Final answer: Range = 12 and coefficient of range = 0.333.

  • Example 2: Find mean deviation about median for x: 10, 20, 30, 40 with frequencies 3, 5, 4, 2.
    Step 1: Total frequency N = 3 + 5 + 4 + 2 = 14. Cumulative frequencies are 3, 8, 12, 14.
    Step 2: Median is the average position around N/2. The 7th and 8th observations both lie at x = 20, so median = 20.
    Step 3: Compute f|x - M|: 3|10 - 20| = 30, 5|20 - 20| = 0, 4|30 - 20| = 40, 2|40 - 20| = 40. Total = 110.
    Step 4: Mean deviation about median = 110/14 = 7.86.
    Final answer: Mean deviation about median = 7.86.

  • Example 3: Find variance and standard deviation for grouped data with classes 0-10, 10-20, 20-30, 30-40 and frequencies 2, 3, 4, 1.
    Step 1: Class marks x are 5, 15, 25, 35 and N = 10.
    Step 2: Σfx = 2(5) + 3(15) + 4(25) + 1(35) = 190, so mean = 190/10 = 19.
    Step 3: Σfx² = 2(25) + 3(225) + 4(625) + 1(1225) = 4450.
    Step 4: Variance = Σfx²/N - x̄² = 4450/10 - 19² = 445 - 361 = 84.
    Step 5: Standard deviation = √84 = 9.17.
    Final answer: Variance = 84 and standard deviation = 9.17.

CBSE Exam Tips and Common Traps

Do not compare two series only by standard deviation when their means are different; use coefficient of variation. In mean deviation, never add signed deviations because positives and negatives can cancel to zero; use absolute values. In grouped data, always convert class intervals to class marks before applying mean, variance or standard deviation formulas. For quartiles, first arrange individual data in ascending order, then apply the correct positional formula. Also remember that standard deviation is never negative, while variance is expressed in squared units.

Must-Remember Points for Chapter 6

  • Range is simple but unstable because it depends only on the extreme values.
  • Quartile deviation is useful when extreme values are present because it focuses on the middle half of the distribution.
  • Mean deviation is easiest to interpret because it uses actual distances, but absolute signs make algebraic treatment limited.
  • Standard deviation is the most widely used measure because it considers every observation and has strong mathematical properties.
  • Coefficient of variation is unit-free and is used to compare consistency between two or more series.
  • Lower dispersion usually means higher consistency, but always mention the measure used for comparison.

Practice Questions with Solutions

  • Q: Find the standard deviation of 4, 6, 8, 10, 12.
    A: Step 1: Mean = (4 + 6 + 8 + 10 + 12)/5 = 40/5 = 8.
    Step 2: Deviations from mean are -4, -2, 0, 2, 4.
    Step 3: Squared deviations are 16, 4, 0, 4, 16. Their sum is 40.
    Step 4: Variance = 40/5 = 8, so standard deviation = √8 = 2.83.
    Final answer: Standard deviation = 2.83.

  • Q: Two series have means and standard deviations as follows: Series A mean = 50, SD = 5; Series B mean = 80, SD = 12. Which series is more consistent?
    A: Step 1: Use coefficient of variation because means are different.
    Step 2: CV of A = (5/50) × 100 = 10%.
    Step 3: CV of B = (12/80) × 100 = 15%.
    Step 4: Lower CV means greater consistency.
    Final answer: Series A is more consistent.

  • Q: For the data 3, 5, 7, 8, 10, 12, 13, 15, 18, find quartile deviation and its coefficient.
    A: Step 1: Data is already in ascending order and N = 9.
    Step 2: Q1 position = (N + 1)/4 = 10/4 = 2.5th item, so Q1 = (5 + 7)/2 = 6.
    Step 3: Q3 position = 3(N + 1)/4 = 30/4 = 7.5th item, so Q3 = (13 + 15)/2 = 14.
    Step 4: Quartile deviation = (Q3 - Q1)/2 = (14 - 6)/2 = 4.
    Step 5: Coefficient of QD = (Q3 - Q1)/(Q3 + Q1) = 8/20 = 0.4.
    Final answer: Quartile deviation = 4 and coefficient of QD = 0.4.

  • Q: Find mean deviation about mean for x values 10, 20, 30 with frequencies 2, 3, 5.
    A: Step 1: Total frequency N = 2 + 3 + 5 = 10.
    Step 2: Mean = Σfx/N = [2(10) + 3(20) + 5(30)]/10 = 230/10 = 23.
    Step 3: Calculate f|x - mean|: 2|10 - 23| = 26, 3|20 - 23| = 9, 5|30 - 23| = 35.
    Step 4: Total absolute deviation = 26 + 9 + 35 = 70.
    Step 5: Mean deviation about mean = 70/10 = 7.
    Final answer: Mean deviation about mean = 7.

Frequently Asked Questions

What is the main idea of measures of dispersion in Class 11 Statistics?

Measures of dispersion show how far data values are spread around an average or within a distribution. They help judge consistency, reliability and variability, which averages alone cannot reveal.

Which measure of dispersion is most important for CBSE exams?

Standard deviation and coefficient of variation are very important because they are used in many comparison-based questions. However, CBSE can also ask direct problems on range, quartile deviation and mean deviation.

When should I use coefficient of variation?

Use coefficient of variation when comparing the consistency of two or more series, especially when their means are different. The series with lower CV is considered more consistent.

Is variance the same as standard deviation?

No. Variance is the average of squared deviations, while standard deviation is the square root of variance. Standard deviation is easier to interpret because it has the same unit as the original observations.