Class 11 Statistics Notes Chapter 6 (Measures of dispersion) – Statistics For Economics Book

Detailed Notes with MCQs of a very important chapter for your exams: Chapter 6 - Measures of Dispersion from your Statistics for Economics book.

We've already learned about measures of central tendency (like mean, median, mode), which give us a single value representing the center of a dataset. However, averages alone can be misleading. Two datasets might have the same average, but the spread or scatter of their individual values could be vastly different. This 'spread' or 'variability' is what we call Dispersion.

Measuring dispersion helps us understand how much the data points deviate from the central value, giving us a more complete picture of the dataset.

Why Study Dispersion? (Purpose/Significance)

1. Determine Reliability of an Average: A low dispersion indicates high uniformity and suggests the average is quite representative of the data. High dispersion means the average is less reliable.
2. Compare Variability: Measures of dispersion allow us to compare the spread of two or more different series. For example, comparing the consistency of two batsmen based on their scores, or two products based on their quality characteristics.
3. Basis for Statistical Quality Control: In industry, controlling the variability of a product's quality is crucial. Dispersion measures help set tolerance limits.
4. Basis for Further Statistical Analysis: Measures like Standard Deviation are fundamental for more advanced statistical techniques like correlation, regression, hypothesis testing, etc.
5. Understanding Inequality: Measures like the Lorenz Curve help visualize and quantify economic inequalities (e.g., income or wealth distribution).

Properties of a Good Measure of Dispersion:

• Easy to understand and calculate.
• Rigidly defined.
• Based on all observations.
• Not unduly affected by extreme values.
• Capable of further algebraic treatment.

Types of Measures of Dispersion

There are two main types:

A) Absolute Measures: These express dispersion in the same units as the original data (e.g., rupees, kilograms, marks). They are useful for describing the variability within a single dataset but not for comparing datasets with different units or significantly different averages.

B) Relative Measures: These are expressed as ratios or percentages (unit-free numbers). They are used for comparing the variability of two or more datasets, even if they have different units or different average values. Generally, a relative measure is obtained by dividing the absolute measure by an appropriate average.

1. Absolute Measures of Dispersion

(i) Range (R)

• Definition: The difference between the largest (L) and the smallest (S) observation in the data.
• Formula: R = L - S
• Merits:
  • Simplest to understand and calculate.
• Demerits:
  • Based only on two extreme values; ignores all other data points.
  • Highly affected by fluctuations in sampling (extreme values can change drastically).
  • Cannot be calculated for open-ended distributions (where the lowest or highest class limit is not defined).

(ii) Quartile Deviation (Q.D.) or Semi-Interquartile Range

• Concept: It measures the average range of the middle 50% of the data, thus overcoming the limitation of Range being affected by extreme values. It's based on the upper quartile (Q3) and the lower quartile (Q1).
• Interquartile Range (IQR): The difference between the third quartile (Q3) and the first quartile (Q1). IQR = Q3 - Q1. This represents the range containing the central 50% of observations.
• Quartile Deviation (Q.D.): Half of the Interquartile Range.
• Formula: Q.D. = (Q3 - Q1) / 2
  • Recall Calculation of Q1 and Q3:
    • Individual/Discrete Series: Arrange data. Q1 = Size of (N+1)/4 th item; Q3 = Size of 3(N+1)/4 th item.
    • Continuous Series: Q1 = Size of N/4 th item; Q3 = Size of 3N/4 th item. Then use the formula: Q = l + [(iN/4 - cf) / f] * h (where i=1 for Q1, i=3 for Q3; l=lower limit, cf=cumulative frequency of preceding class, f=frequency of quartile class, h=class interval).
• Merits:
  • Better than Range as it considers the middle 50% of data.
  • Not affected by extreme values.
  • Can be calculated for open-ended distributions.
• Demerits:
  • Ignores 50% of the data (the lowest 25% and the highest 25%).
  • Not based on all observations.
  • Not suitable for further algebraic treatment.

(iii) Mean Deviation (M.D.)

• Definition: The arithmetic mean of the absolute deviations (ignoring signs) of the observations from a measure of central tendency (usually Mean or Median).
• Formulas:
  • Mean Deviation from Mean (MD_X̄):
    • Individual: MD<sub>X̄</sub> = Σ|X - X̄| / N
    • Discrete/Continuous: MD<sub>X̄</sub> = Σf|X - X̄| / Σf (or Σf|m - X̄| / Σf for continuous, where m is mid-point)
  • Mean Deviation from Median (MD_M):
    • Individual: MD<sub>M</sub> = Σ|X - M| / N
    • Discrete/Continuous: MD<sub>M</sub> = Σf|X - M| / Σf (or Σf|m - M| / Σf for continuous)
• Important Note: Mean Deviation is minimum when calculated from the Median.
• Merits:
  • Based on all observations.
  • Relatively easy to understand.
  • Less affected by extreme values compared to Standard Deviation.
• Demerits:
  • Ignoring the signs of deviations (| |) is mathematically unsound and makes it unsuitable for further algebraic treatment.
  • Can be computationally intensive, especially finding the median first for MD_M.

(iv) Standard Deviation (S.D. or σ)

• Definition: The positive square root of the arithmetic mean of the squares of the deviations of the observations from their arithmetic mean. It is the most important and widely used measure of dispersion. Also known as 'Root Mean Square Deviation'.
• Variance (σ²): The square of the Standard Deviation. It's the average of the squared deviations from the mean. Variance = σ². Sometimes variance is calculated first, then its square root gives SD.
• Formulas (using Population parameter N, for sample use n-1 sometimes in higher stats):
  • Direct Method:
    • Individual: σ = √[ Σ(X - X̄)² / N ]
    • Discrete/Continuous: σ = √[ Σf(X - X̄)² / Σf ] (or Σf(m - X̄)² / Σf for continuous)
  • Short-cut Method (Assumed Mean Method):
    • Individual: σ = √[ Σd² / N - (Σd / N)² ], where d = X - A (A=Assumed Mean)
    • Discrete/Continuous: σ = √[ Σfd² / Σf - (Σfd / Σf)² ], where d = X - A (or m - A)
  • Step-Deviation Method (for Continuous or equal-interval Discrete):
    • σ = √[ Σfd'² / Σf - (Σfd' / Σf)² ] * h, where d' = (X - A) / h (or (m - A) / h), and h = common class interval.
• Merits:
  • Based on all observations.
  • Rigidly defined and mathematically sound (doesn't ignore signs).
  • Least affected by sampling fluctuations compared to other measures.
  • Amenable to further algebraic treatment (basis for correlation, regression, etc.).
• Demerits:
  • More complex to calculate than other measures.
  • Gives more weight to extreme values because deviations are squared.
  • Difficult for a non-mathematical person to understand intuitively.

2. Relative Measures of Dispersion

These are used for comparison purposes.

(i) Coefficient of Range

• Formula: Coefficient of Range = (L - S) / (L + S)

(ii) Coefficient of Quartile Deviation

• Formula: Coefficient of Q.D. = (Q3 - Q1) / (Q3 + Q1)

(iii) Coefficient of Mean Deviation

• Formula: Coefficient of M.D. = MD / Average used
  • Coefficient of MD (Mean) = MD<sub>X̄</sub> / X̄
  • Coefficient of MD (Median) = MD<sub>M</sub> / M

(iv) Coefficient of Variation (C.V.)

• Definition: The most commonly used relative measure of dispersion. It expresses the standard deviation as a percentage of the arithmetic mean.
• Formula: C.V. = (σ / X̄) * 100
• Interpretation:
  • A lower C.V. indicates greater consistency or uniformity in the data.
  • A higher C.V. indicates greater variability or less consistency.
• Use: Extremely useful for comparing the variability of datasets with different units (e.g., height in cm vs weight in kg) or vastly different means (e.g., comparing variability in salaries of clerks vs managers).

3. Lorenz Curve

• Definition: A graphical method to study dispersion, particularly used to measure inequalities in distributions like income or wealth.
• Construction:
  1. Convert the data (items/individuals and their corresponding values like income) into cumulative frequencies and cumulative values.
  2. Calculate the percentage cumulative frequencies and percentage cumulative values.
  3. Plot the percentage cumulative frequencies (or population share) on the X-axis and the percentage cumulative values (or income/wealth share) on the Y-axis.
  4. Connect the plotted points with a smooth curve. This is the Lorenz Curve.
  5. Draw a diagonal line connecting (0%, 0%) to (100%, 100%). This is the Line of Equal Distribution.
• Interpretation:
  • If the Lorenz Curve coincides with the Line of Equal Distribution, there is perfect equality (zero dispersion).
  • The farther the Lorenz Curve is from the Line of Equal Distribution, the greater the inequality or dispersion in the distribution.
  • We can compare the inequality of two distributions by plotting their Lorenz curves on the same graph. The curve further away from the diagonal line represents greater inequality.

Summary Table of Formulas (Absolute Measures)

Summary Table of Formulas (Relative Measures)

This covers the core concepts of Measures of Dispersion. Remember to practice calculating each measure for different types of series (individual, discrete, continuous) to become proficient. Pay special attention to Standard Deviation and Coefficient of Variation, as they are very frequently tested.

Multiple Choice Questions (MCQs)

Here are 10 MCQs to test your understanding:

1. Which of the following is a measure of dispersion?
   a) Mean
   b) Median
   c) Mode
   d) Range

2. The measure of dispersion that is easiest to calculate is:
   a) Standard Deviation
   b) Quartile Deviation
   c) Range
   d) Mean Deviation

3. Which measure of dispersion ignores the signs of deviations from a central value?
   a) Standard Deviation
   b) Variance
   c) Mean Deviation
   d) Quartile Deviation

4. For distributions with open-ended classes, the most appropriate measure of dispersion is:
   a) Range
   b) Mean Deviation
   c) Standard Deviation
   d) Quartile Deviation

5. The square of the Standard Deviation is called:
   a) Coefficient of Variation
   b) Variance
   c) Mean Deviation
   d) Range

6. Which measure is considered the best and is most widely used due to its mathematical properties?
   a) Range
   b) Quartile Deviation
   c) Mean Deviation
   d) Standard Deviation

7. The Coefficient of Variation is calculated using the formula:
   a) (Mean / SD) * 100
   b) (SD / Median) * 100
   c) (SD / Mean) * 100
   d) (Range / Mean) * 100

8. The Lorenz Curve is a graphical method used to study:
   a) Central Tendency
   b) Correlation
   c) Index Numbers
   d) Dispersion (Inequality)

9. Mean Deviation is minimum when deviations are taken from the:
   a) Mean
   b) Median
   c) Mode
   d) Geometric Mean

10. Relative measures of dispersion are used primarily for:
    a) Calculating the average spread in original units
    b) Comparing the variability of different datasets
    c) Finding the central value of a dataset
    d) Calculating variance

Answer Key for MCQs:

1. d
2. c
3. c
4. d
5. b
6. d
7. c
8. d
9. b
10. b

Study these notes carefully, practice the calculations, and you'll be well-prepared for questions on dispersion in your exams. Let me know if any part needs further clarification!