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Unit 4 of 4 · BCA Sem 2

Unit 4: Measures of dispersion

Fundamental of Statistics notes · PTU syllabus (UGCC2504)

3 min read5 topics8 exam questions
On this page
  1. Unit summary
  2. Meaning and objectives of dispersion
  3. Range
  4. Mean deviation
  5. Standard deviation
  6. Coefficient of variation
  7. Key terms
  8. Quick revision
  9. Important questions

Unit summary

Averages tell us the centre of the data, but not how spread out it is. Two classes can have the same average marks while one is consistent and the other varies wildly. Measures of dispersion measure that spread.

After this unit you can

  • Explain the objectives and properties of a good measure of dispersion
  • Calculate range, mean deviation and standard deviation
  • Calculate the coefficient of variation and use it to compare consistency
  • State the merits and demerits of each measure

PTU syllabus topics

  • Objectives and properties of a good measure of dispersion
  • range
  • mean deviation
  • standard deviation
  • coefficient of variation — calculation
  • merits and demerits of each
Key formulasMeasures of dispersion

  • Range

    Largest − Smallest

  • Mean deviation

    Σ abs(x − x̄) / n

    Average absolute distance from the mean

  • Standard deviation

    σ = √[Σ(x − x̄)² / n]

  • Coefficient of variation

    CV = (σ / x̄) × 100

    Lower CV means more consistent data

1

Topic 1

Meaning and objectives of dispersion

Dispersion is the degree to which values are spread around the average. Objectives: to judge how reliable an average is, to compare the consistency of series, and to support further analysis (correlation, quality control).

  • Absolute measures are in the units of the data (range, mean deviation, standard deviation).
  • Relative measures are ratios without units (coefficient of range, coefficient of variation), used to compare series in different units.

A good measure of dispersion should be easy, rigidly defined, based on all items, not unduly affected by extremes, and capable of algebraic treatment.

2

Topic 2

Range

Range = Largest value (L) − Smallest value (S). Coefficient of range = (L − S) / (L + S).

Example

Daily temperatures 22, 25, 30, 27, 24: range = 30 − 22 = 8°C.

Merits: simplest. Demerits: uses only two values and is heavily affected by extremes. It is used in quality control and weather reports.

3

Topic 3

Mean deviation

Mean deviation is the average of the absolute deviations from the mean (or median), ignoring signs.

Key formulasMean deviation

  • Individual series

    MD = Σ abs(x − x̄) / n

  • Frequency series

    MD = Σf abs(x − x̄) / Σf

  • Coefficient of MD

    MD / mean (or MD / median)

Merits: based on all values, less affected by extremes than SD. Demerits: ignoring signs is mathematically illogical, so it cannot be used for further algebra.

4

Topic 4

Standard deviation

Standard deviation (σ) is the square root of the mean of squared deviations from the mean. It is the most important and widely used measure.

Key formulasStandard deviation

  • Individual series

    σ = √[Σ(x − x̄)² / n]

  • Frequency series

    σ = √[Σf(x − x̄)² / Σf]

  • Short-cut

    σ = √[Σfd²/Σf − (Σfd/Σf)²]

  • Variance

    σ²

Example

Values 2, 4, 6, 8, 10: mean = 6; squared deviations 16, 4, 0, 4, 16 sum to 40; σ = √(40 / 5) = √8 ≈ 2.83.

Merits: based on all values, algebraic treatment possible, used in advanced statistics. Demerits: harder to calculate, gives more weight to extreme values.

5

Topic 5

Coefficient of variation

CV = (σ / x̄) × 100. It expresses SD as a percentage of the mean. The series with the lower CV is more consistent.

Example

Batsman A: mean 50, σ 10 → CV 20%. Batsman B: mean 40, σ 12 → CV 30%. A is more consistent.

ComparisonComparing measures of dispersion
Uses all values?
Main weakness

Range

No

Only two values

Mean deviation

Yes

Ignores signs

Standard deviation

Yes

Harder to compute

Coefficient of variation

Yes

Needs a non-zero mean

Key terms

Dispersion
The spread of values around the average
Range
Highest value minus lowest value
Standard deviation
Square root of the average squared deviation from the mean
Variance
The square of the standard deviation
Coefficient of variation
SD as a percentage of the mean

Quick revision

  • Range = L − S; coefficient = (L − S)/(L + S).
  • MD uses absolute deviations; SD squares deviations.
  • SD is the best and most used measure.
  • CV = σ/x̄ × 100; lower CV = more consistent.

Important exam questions

Practice questions written to the PTU exam pattern for this unit's syllabus: short answers (Section A style) and long answers (Sections B and C style).

Short-answer questions

  1. Q1.What is dispersion? Why is it needed?
  2. Q2.Differentiate between absolute and relative measures of dispersion.
  3. Q3.Find the range and coefficient of range of 12, 18, 7, 25, 15.
  4. Q4.Define standard deviation.
  5. Q5.What does a lower coefficient of variation indicate?

Long-answer questions

  1. Q1.Explain the different measures of dispersion with their merits and demerits.
  2. Q2.Calculate the standard deviation and coefficient of variation of a frequency distribution of your choice.
  3. Q3.Two players' scores are given. Using the coefficient of variation, find who is more consistent, and explain why CV is used for such comparisons.

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