Statistics (Paper I — MCQ, part of 20 marks with Mathematics)¶
MCQ-level: definitions, formulas, standard properties, small mental computations. Also reused in Paper II (noise, random signals) and Financial Management (forecasting).
1. Introduction to Statistics¶
Statistics is the science of collecting, organizing, presenting, analyzing and interpreting numerical data for decision-making.
| Branch | Deals With |
|---|---|
| Descriptive | Summarizing data (tables, graphs, averages, dispersion) |
| Inferential | Drawing conclusions about a population from a sample (estimation, hypothesis testing) |
| Term | Meaning |
|---|---|
| Population | Entire group under study |
| Sample | Subset actually observed |
| Parameter | Population measure (μ, σ) |
| Statistic | Sample measure (\(\bar{x}\), s) |
| Variable | Characteristic that varies (quantitative: discrete/continuous · qualitative) |
Scales of measurement: Nominal → Ordinal → Interval → Ratio (increasing information).
2. Measures of Central Tendency¶
Mean (Arithmetic)¶
- Uses all observations; affected by extreme values (outliers).
- Combined mean: \(\bar{x}_{12} = \frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1 + n_2}\)
Median¶
Middle value of ordered data — positional average.
Grouped: \(Md = L + \frac{\frac{N}{2} - cf}{f} \times h\)
- Not affected by extreme values → best for skewed data (income!).
Mode¶
Most frequent value. Grouped: \(Mo = L + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h\)
Empirical Relation (favorite MCQ)¶
Skewness and the Three Averages¶
| Distribution | Relation |
|---|---|
| Symmetrical | Mean = Median = Mode |
| Positively skewed (right tail) | Mean > Median > Mode |
| Negatively skewed (left tail) | Mean < Median < Mode |
Other means: GM \(= \sqrt[n]{x_1x_2\cdots x_n}\) (growth rates), HM \(= \frac{n}{\sum 1/x_i}\) (rates/speeds); \(AM \ge GM \ge HM\).
3. Measures of Dispersion¶
Dispersion measures the spread/scatter of data around the center.
| Measure | Formula | Notes |
|---|---|---|
| Range | \(X_{max} - X_{min}\) | Crude, quick |
| Quartile deviation | \(\frac{Q_3 - Q_1}{2}\) | Middle 50% spread |
| Mean deviation | \(\frac{\sum\|x_i - \bar{x}\|}{n}\) | Absolute deviations |
| Variance | \(\sigma^2 = \frac{\sum(x_i - \bar{x})^2}{n} = \frac{\sum x_i^2}{n} - \bar{x}^2\) | Squared units |
| Standard deviation | \(\sigma = \sqrt{\sigma^2}\) | Same units as data; most used |
| Coefficient of variation | \(CV = \frac{\sigma}{\bar{x}} \times 100\%\) | Unit-free — compares consistency of two series; smaller CV = more consistent |
Properties of σ: unaffected by change of origin (\(x \pm a\)), scaled by change of scale (\(\sigma_{kx} = |k|\sigma_x\)). Sample variance uses \(n - 1\) (unbiased).
4. Correlation¶
Correlation measures the degree of linear relationship between two variables.
Karl Pearson's Coefficient¶
Properties (MCQ favorites):
- \(-1 \le r \le +1\)
- \(r = +1\) perfect positive, \(r = -1\) perfect negative, \(r = 0\) no linear relation
- Unit-free; unaffected by change of origin and scale
- Correlation ≠ causation
Spearman's Rank Correlation¶
(\(d_i\) = rank differences) — for ordinal/qualitative data.
5. Regression¶
Regression estimates the average value of one variable for a given value of the other (prediction line).
Regression line of y on x: \(y - \bar{y} = b_{yx}(x - \bar{x})\), where
Properties:
- \(r = \pm\sqrt{b_{yx} \cdot b_{xy}}\) (sign = sign of the b's; both b's always same sign)
- Both lines intersect at \((\bar{x}, \bar{y})\)
- If \(r = \pm 1\): the two lines coincide; if \(r = 0\): lines are perpendicular
| Correlation | Regression |
|---|---|
| Degree of relationship | Nature/equation of relationship |
| Symmetric (\(r_{xy} = r_{yx}\)) | Not symmetric (\(b_{yx} \ne b_{xy}\)) |
| No prediction | Used for prediction/forecasting |
6. Probability¶
Definitions¶
Laws¶
Addition:
(mutually exclusive: \(P(A \cap B) = 0\))
Multiplication:
(independent: \(P(A \cap B) = P(A)P(B)\))
Conditional: \(P(A|B) = \frac{P(A \cap B)}{P(B)}\)
Bayes' theorem:
Quick values: two dice sum 7 → \(\frac{6}{36} = \frac{1}{6}\); at least one head in 2 tosses → \(\frac{3}{4}\); 52-card deck: P(ace) \(= \frac{1}{13}\).
7. Random Variables and Distributions¶
Random Variable¶
A random variable assigns a numerical value to each outcome of a random experiment. Discrete (countable values, PMF) or continuous (interval values, PDF).
Properties: \(E(aX + b) = aE(X) + b\); \(Var(aX + b) = a^2Var(X)\).
Standard Distributions¶
| Distribution | Type | PMF/PDF | Mean | Variance | Models |
|---|---|---|---|---|---|
| Binomial \(B(n,p)\) | Discrete | \(\binom{n}{x}p^xq^{n-x}\) | \(np\) | \(npq\) | n independent success/fail trials (bit errors in a frame) |
| Poisson \(P(\lambda)\) | Discrete | \(\frac{e^{-\lambda}\lambda^x}{x!}\) | \(\lambda\) | \(\lambda\) | Rare events per interval (call arrivals — links to Erlang) |
| Normal \(N(\mu,\sigma^2)\) | Continuous | bell curve | \(\mu\) | \(\sigma^2\) | Noise, measurement errors |
| Uniform | Continuous | \(\frac{1}{b-a}\) | \(\frac{a+b}{2}\) | \(\frac{(b-a)^2}{12}\) | Quantization error |
| Exponential | Continuous | \(\lambda e^{-\lambda x}\) | \(\frac{1}{\lambda}\) | \(\frac{1}{\lambda^2}\) | Holding/service times |
Facts: mean = variance ⇒ Poisson; binomial → Poisson as \(n\to\infty, p\to 0\), \(np = \lambda\); binomial → normal for large \(n\).
Normal Distribution Properties (MCQ favorites)¶
- Symmetric, bell-shaped; mean = median = mode
- Standard normal: \(Z = \frac{X - \mu}{\sigma}\), \(N(0, 1)\)
- 68–95–99.7 rule: \(\mu \pm 1\sigma\) → 68.3%, \(\mu \pm 2\sigma\) → 95.4%, \(\mu \pm 3\sigma\) → 99.7%
- Total area = 1; points of inflection at \(\mu \pm \sigma\)
8. Samples and Sampling¶
Concepts¶
Sampling is selecting a representative subset to estimate population characteristics — cheaper, faster, sometimes the only option (destructive testing).
| Random (probability) Methods | Non-random Methods |
|---|---|
| Simple random (lottery/random numbers) | Convenience |
| Systematic (every \(k^{th}\), \(k = N/n\)) | Judgment/purposive |
| Stratified (divide into homogeneous strata, sample each) | Quota |
| Cluster (sample whole groups) | Snowball |
Errors¶
- Sampling error — chance difference between sample and population (decreases as \(n\) increases)
- Non-sampling error — bias, bad questionnaire, non-response (can occur even in census)
Key Results¶
- Standard error of mean: \(SE = \frac{\sigma}{\sqrt{n}}\) — halving SE needs 4× sample
- Central Limit Theorem: for large \(n\) (≥ 30), the sampling distribution of \(\bar{x}\) is approximately normal regardless of population shape
- Census vs sample survey: complete coverage/costly vs partial/economical
9. Speed Facts for MCQs¶
| Fact | Value |
|---|---|
| Mode (empirical) | \(3Md - 2\bar{x}\) |
| \(AM \ge GM \ge HM\) | Equality iff all values equal |
| CV | \(\frac{\sigma}{\bar{x}} \times 100\), unit-free |
| \(r\) range | \([-1, +1]\); \(r = \pm\sqrt{b_{yx}b_{xy}}\) |
| Regression lines meet at | \((\bar{x}, \bar{y})\) |
| Binomial mean/var | \(np\), \(npq\) (var < mean) |
| Poisson mean = var | \(\lambda\) |
| Normal areas | 68 – 95 – 99.7 |
| SE of mean | \(\sigma/\sqrt{n}\) |
| Two dice sum 7 | 1/6 |
Key Exam Points — Statistics
- Median resists outliers; mean uses all data; empirical relation Mode = 3Median − 2Mean.
- CV (not σ) compares consistency between two data sets.
- Correlation: strength of linear relation; regression: prediction equation; \(r = \pm\sqrt{b_{yx}b_{xy}}\).
- Binomial (np, npq) vs Poisson (λ, λ) vs Normal (68-95-99.7) — the distribution trio covers most MCQs.
- CLT and \(SE = \sigma/\sqrt{n}\) are the two sampling facts asked repeatedly.