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Unit 2 of 4 · B.Sc IT Sem 2

Unit 2: Statistics and applications of logarithms

Mathematics-II notes · PTU syllabus (BSIT202/BSBC202)

3 min read6 topics10 exam questions
On this page
  1. Unit summary
  2. Mean, median and mode
  3. Mean deviation, standard deviation and coefficient of variation
  4. Logarithms: basics
  5. Compound interest
  6. Depreciation
  7. Annuities
  8. Key terms
  9. Quick revision
  10. Important questions

Unit summary

Statistics summarises data, and logarithms simplify growth and decay calculations in finance. This unit covers measures of central tendency — mean, median and mode — measures of dispersion — mean deviation, standard deviation and coefficient of variation — and applications of logarithms to compound interest, depreciation and annuities.

After this unit you can

  • Compute mean, median and mode
  • Compute mean deviation, standard deviation and coefficient of variation
  • Use logarithms in compound interest and depreciation
  • Use logarithms to compute annuities

PTU syllabus topics

  • Measures of central tendency (mean, median, mode)
  • measures of dispersion (mean deviation, standard deviation, coefficient of variation)
  • applications of logarithms to compound interest
  • depreciation and annuities
Key formulasStatistics and logarithm applications

  • Mean

    x̄ = Σx / n

  • Standard deviation

    σ = √[Σ(x − x̄)² / n]

  • Coefficient of variation

    (σ / x̄) × 100

  • Compound interest

    A = P (1 + r/100)^n

  • Depreciation (reducing balance)

    V = P (1 − r/100)^n

1

Topic 1

Mean, median and mode

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.
2

Topic 2

Mean deviation, standard deviation and coefficient of variation

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.

3

Topic 3

Logarithms: basics

Key formulasLaws of logarithms

  • Product

    log(mn) = log m + log n

  • Quotient

    log(m ÷ n) = log m − log n

  • Power

    log(mⁿ) = n log m

  • Change of base

    loga b = log b ÷ log a

4

Topic 4

Compound interest

Key formulasCompound interest

  • Amount

    A = P (1 + r/100)ⁿ

  • Using logs

    log A = log P + n log(1 + r/100)

  • Compounded k times a year

    A = P (1 + r/(100k))^(nk)

Example

P = ₹10,000, r = 8%, n = 5: log A = 4 + 5 × log 1.08 = 4 + 5 × 0.0334 = 4.1671; A = antilog 4.1671 ≈ ₹14,693.

5

Topic 5

Depreciation

Key formulasDepreciation (reducing balance)

  • Value after n years

    V = P (1 − r/100)ⁿ

  • Using logs

    log V = log P + n log(1 − r/100)

Example

A machine costing ₹2,00,000 depreciates at 10% a year. After 3 years: V = 2,00,000 × 0.9³ = ₹1,45,800.

6

Topic 6

Annuities

Key formulasAnnuities

  • Amount of an annuity (future value)

    A = (a ÷ i) × [(1 + i)ⁿ − 1]

  • Present value of an annuity

    P = (a ÷ i) × [1 − (1 + i)^(−n)]

  • Notation

    a = periodic payment, i = rate per period, n = number of periods

Example

Saving ₹5,000 a year for 4 years at 10%: A = (5,000 ÷ 0.1) × (1.4641 − 1) = ₹23,205.

  • Applications: recurring deposits, loan EMIs, sinking funds, pensions.

Key terms

Arithmetic mean
Sum of values divided by their number
Median
Middle value of ordered data
Standard deviation
Root mean square deviation from the mean
Coefficient of variation
SD as a percentage of the mean
Annuity
Series of equal periodic payments

Quick revision

  • Mean (direct, shortcut, step deviation), median, mode for grouped data.
  • Range, QD, mean deviation, SD, CV.
  • Laws of logarithms.
  • Compound interest and depreciation with logs.
  • Future and present value of annuities.

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.Define median.
  2. Q2.State the formula for coefficient of variation.
  3. Q3.State the product law of logarithms.
  4. Q4.Find the amount on ₹1,000 at 10% for 2 years compounded annually.
  5. Q5.What is reducing-balance depreciation?
  6. Q6.Define an annuity.

Long-answer questions

  1. Q1.Compute mean, median and mode for a grouped frequency distribution (numerical).
  2. Q2.Compute standard deviation and coefficient of variation and compare two series (numerical).
  3. Q3.Solve compound interest and depreciation problems using logarithms (numerical).
  4. Q4.Find the amount and present value of an annuity (numerical).

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