Synetic College Logo
Unit 1 of 4 · M.Com Sem 1

Unit 1: Descriptive statistics

Quantitative Techniques notes · PTU syllabus (MCOP103-18)

3 min read7 topics10 exam questions
On this page
  1. Unit summary
  2. Meaning, features, importance, functions and scope
  3. Measures of central tendency
  4. Partition values
  5. Measures of dispersion
  6. Moments
  7. Skewness
  8. Kurtosis
  9. Key terms
  10. Quick revision
  11. Important questions

Unit summary

Descriptive statistics summarise data so that patterns become visible. This unit covers the meaning, features, functions, scope and limitations of statistics, measures of central tendency and partition values, measures of variation and their coefficients, moments, skewness (Karl Pearson and Bowley) and kurtosis.

After this unit you can

  • Explain the meaning, functions, scope and limitations of statistics
  • Compute measures of central tendency and partition values
  • Compute absolute and relative measures of dispersion
  • Compute moments, skewness and kurtosis and interpret them

PTU syllabus topics

  • Meaning
  • features
  • functions
  • scope and limitations of statistics
  • measures of central tendency (arithmetic/geometric/harmonic mean, mode, median, quartiles/deciles/percentiles)
  • measures of variation (range, quartile deviation, mean deviation, standard deviation and coefficients)
  • moments
  • skewness (Karl Pearson and Bowley measures) and kurtosis
Key formulasDispersion and shape

  • Standard deviation

    σ = √[Σ(x − x̄)² / n]

  • Coefficient of variation

    (σ / x̄) × 100

  • Karl Pearson skewness

    (Mean − Mode) / σ

  • Bowley skewness

    (Q3 + Q1 − 2 Median) / (Q3 − Q1)

  • Kurtosis

    β2 = μ4 / μ2²

    3 = mesokurtic

1

Topic 1

Meaning, features, importance, functions and scope

Statistics (singular) is the science of collecting, classifying, presenting, analysing and interpreting numerical data. Statistics (plural) are numerical facts collected for a purpose. Features (Secrist): aggregates of facts, numerically expressed, affected by multiple causes, collected systematically for a purpose, reasonably accurate and comparable. Importance in business: market research, demand forecasting, quality control, financial analysis, HR analytics (attrition, performance) and decision-making under uncertainty. Limitations: studies only quantitative data and groups (not individuals), results are true on average, and data can be misused.

ProcessFunctions of statistics
  1. 1

    Presents facts in definite form

  2. 2

    Simplifies mass data

  3. 3

    Facilitates comparison

  4. 4

    Helps in formulating and testing hypotheses

  5. 5

    Helps in prediction and forecasting

  6. 6

    Helps in policy formulation

2

Topic 2

Measures of central tendency

Key formulasAverages

  • Arithmetic mean

    x̄ = Σfx / Σf

  • Geometric mean

    GM = antilog (Σ log x / n)

    For growth rates and ratios

  • Harmonic mean

    HM = n / Σ(1/x)

    For rates and speeds

  • Median (grouped)

    L + [(N/2 − cf) / f] × h

  • Mode (grouped)

    L + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h

  • Relationship: for positive values, AM ≥ GM ≥ HM; for moderately skewed data, Mode = 3 Median − 2 Mean.
  • Choosing an average: mean for symmetric data; median for skewed data or open-ended classes (incomes); mode for the most common size or brand.
3

Topic 3

Partition values

Partition values divide ordered data into equal parts:

  • Quartiles (Q1, Q2, Q3) divide into 4 parts; Q2 = median. Qk = L + [(kN/4 − cf) / f] × h.
  • Deciles (D1 … D9) divide into 10 parts; percentiles (P1 … P99) into 100.

Example

If a student's score is at the 90th percentile, 90% of students scored below that student.

4

Topic 4

Measures of dispersion

MeasureFormulaRelative measure
RangeL − S(L − S)/(L + S)
Quartile deviation(Q3 − Q1)/2(Q3 − Q1)/(Q3 + Q1)
Mean deviationΣf abs(x − A)/N (A = mean or median)MD / A
Standard deviation√[Σf(x − x̄)²/N]CV = σ/x̄ × 100
Varianceσ²—

Example

Two shops have average daily sales of ₹10,000. Shop A's SD is ₹500 and Shop B's is ₹2,500, so Shop A's sales are far more stable.

Exam tip

Standard deviation is the best measure of dispersion because it uses all values and allows further algebraic treatment.

5

Topic 5

Moments

Moments are averages of the powers of deviations — they describe the shape of a distribution.

Key formulasMoments

  • First central moment

    μ1 = Σf(x − x̄) ÷ N = 0

  • Second central moment

    μ2 = Σf(x − x̄)² ÷ N = variance

  • Third central moment

    μ3 = Σf(x − x̄)³ ÷ N — skewness

  • Fourth central moment

    μ4 = Σf(x − x̄)⁴ ÷ N — kurtosis

  • Conversion from raw moments

    μ2 = μ2′ − μ1′²; μ3 = μ3′ − 3μ2′μ1′ + 2μ1′³; μ4 = μ4′ − 4μ3′μ1′ + 6μ2′μ1′² − 3μ1′⁴

6

Topic 6

Skewness

Skewness measures the lack of symmetry. In a positively skewed distribution, Mean > Median > Mode (long right tail); in a negatively skewed one, Mean < Median < Mode.

Key formulasMeasures of skewness

  • Karl Pearson's coefficient

    Sk = (Mean − Mode) ÷ σ, or 3 (Mean − Median) ÷ σ

  • Bowley's coefficient (quartiles)

    Sk = (Q3 + Q1 − 2Median) ÷ (Q3 − Q1)

  • Moment coefficient

    β1 = μ3² ÷ μ2³; γ1 = √β1 with the sign of μ3

Example

Mean 52, median 50, σ 8: Karl Pearson's Sk = 3 × (52 − 50) ÷ 8 = 0.75 — moderately positively skewed. Q1 = 40, Q3 = 66, median 50: Bowley's Sk = (66 + 40 − 100) ÷ 26 = 0.23.

  • Karl Pearson's measure lies (practically) between −3 and +3; Bowley's between −1 and +1.
7

Topic 7

Kurtosis

Kurtosis measures the peakedness (or tail-heaviness) of a distribution relative to the normal curve.

Key formulasKurtosis

  • Coefficient

    β2 = μ4 ÷ μ2²

  • Excess kurtosis

    γ2 = β2 − 3

ClassificationTypes of kurtosis
Kurtosis

  • Mesokurtic

    β2 = 3 — normal curve

  • Leptokurtic

    β2 > 3 — more peaked, heavy tails (stock returns)

  • Platykurtic

    β2 < 3 — flatter than normal

Exam tip

Skewness tells the direction of asymmetry; kurtosis tells the peakedness — a four-measure summary (mean, SD, skewness, kurtosis) describes any distribution well.

Key terms

Central tendency
A single value representing the centre of data
Coefficient of variation
SD as a percentage of the mean
Moment
Average of powered deviations from the mean
Skewness
Lack of symmetry in a distribution
Kurtosis
Degree of peakedness of a distribution

Quick revision

  • Averages: AM, GM, HM, median, mode; partition values Q, D, P.
  • Dispersion: range, QD, MD, SD; CV for comparison.
  • Moments μ1 to μ4; μ2 = variance.
  • Skewness: Karl Pearson, Bowley, moments.
  • Kurtosis: β2 = 3 meso, > 3 lepto, < 3 platy.

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.State four limitations of statistics.
  2. Q2.What are partition values?
  3. Q3.Define coefficient of variation.
  4. Q4.What are central moments?
  5. Q5.Distinguish Karl Pearson's and Bowley's coefficients of skewness.
  6. Q6.What is a leptokurtic curve?

Long-answer questions

  1. Q1.Explain the meaning, functions, scope and limitations of statistics.
  2. Q2.Explain measures of central tendency and their merits.
  3. Q3.Explain measures of dispersion with a numerical example.
  4. Q4.Explain moments, skewness and kurtosis with their measures.

Stuck on this unit?

Message SBS on WhatsApp for help with Quantitative Techniques, or to ask about studying M.Com at Synetic.

WhatsApp us