Unit 2: Central tendency, variation & sampling distribution
Business Statistics notes · PTU syllabus (BCOMGE 201-18)
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Unit summary
Two summary measures describe any data set: where its centre lies (central tendency) and how spread out it is (dispersion). This unit covers the arithmetic, geometric and harmonic means, median and mode, partition values, and the range, quartile deviation, mean deviation, standard deviation and variance, plus the idea of a sampling distribution.
After this unit you can
- Calculate the arithmetic, geometric and harmonic means, median and mode
- Find quartiles, deciles and percentiles
- Calculate range, quartile deviation, mean deviation and standard deviation
- Explain the sampling distribution and standard error
PTU syllabus topics
- Arithmetic
- geometric and harmonic mean
- mode and median
- quartiles/deciles/percentiles
- range
- quartile deviation
- mean deviation
- standard deviation and variance
- concept of sampling distribution
Arithmetic mean
Σfx / Σf
Geometric mean
Antilog (Σ log x / n)
Quartile deviation
(Q3 − Q1) / 2
Variance
σ² = Σ(x − x̄)² / n
Coefficient of variation
(σ / x̄) × 100
Topic 1
Measures of central tendency
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.
Topic 2
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.
Topic 3
Measures of dispersion
| Measure | Formula | Relative measure |
|---|---|---|
| Range | L − S | (L − S)/(L + S) |
| Quartile deviation | (Q3 − Q1)/2 | (Q3 − Q1)/(Q3 + Q1) |
| Mean deviation | Σf abs(x − A)/N (A = mean or median) | MD / A |
| Standard deviation | √[Σf(x − x̄)²/N] | CV = σ/x̄ × 100 |
| Variance | σ² | — |
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.
Topic 4
Sampling distribution
If many samples of the same size are drawn from a population, the sample means form a sampling distribution. Its standard deviation is the standard error (SE): SE = σ / √n. Larger samples give smaller standard errors and more precise estimates. By the central limit theorem, the sampling distribution of the mean is approximately normal for large n.
Key terms
- Median
- The middle value of ordered data
- Quartile
- A value dividing data into four equal parts
- Standard deviation
- Square root of the mean squared deviation
- Coefficient of variation
- SD as a percentage of the mean
- Standard error
- Standard deviation of a sampling distribution
Quick revision
- AM ≥ GM ≥ HM; Mode = 3 Median − 2 Mean.
- Q2 = D5 = P50 = median.
- QD = (Q3 − Q1)/2; CV = σ/x̄ × 100.
- SE = σ/√n.
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
- Q1.When is the geometric mean used?
- Q2.What are partition values?
- Q3.Find the range and coefficient of range of 15, 25, 40, 10, 30.
- Q4.Define coefficient of variation.
- Q5.What is a standard error?
Long-answer questions
- Q1.Calculate the mean, median and mode of a grouped frequency distribution.
- Q2.Calculate the quartiles and quartile deviation of a frequency distribution.
- Q3.Calculate the standard deviation and coefficient of variation of two series and compare their consistency.
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