Statistics for Management: Central Tendency, Dispersion, Probability Distributions, Data Collection, Sampling, Hypothesis Testing, Correlation and Regression
2. Probability distributions: binomial, Poisson, normal and exponential
| Distribution | Situation | Probability function, mean and variance | Properties and uses |
|---|---|---|---|
| Binomial (Bernoulli, 1713) | n independent trials, each a success with constant probability p; count of successes; discrete | P(x) = nCx p^x q^(n−x); mean np; variance npq; SD √(npq) | Symmetrical when p = 0.5, positively skewed when p < 0.5; mean > variance always; approximated by Poisson when n is large and p small, and by the normal when np and nq both exceed 5; quality control with a fixed sample, yes/no survey items |
| Poisson (1837) | Count of rare events in a fixed interval of time or space when events are independent and occur at a constant average rate λ; discrete | P(x) = e^(−λ) λ^x / x!; mean λ; variance λ (mean equals variance) | The limit of the binomial as n → ∞ and p → 0 with np = λ; positively skewed, approaching normal as λ grows; arrivals at a counter, defects per unit, calls per hour, accidents; the basis of queuing theory (Poisson arrivals) |
| Normal (Gauss, de Moivre) | Continuous measurements clustering symmetrically around a mean — heights, errors, sample means by the central limit theorem | Bell curve defined by μ and σ; standard normal z = (x − μ)/σ with mean 0 and SD 1; area under the curve is 1 | Mean = median = mode; symmetrical, asymptotic to the axis; 68.27% of values within ±1σ, 95.45% within ±2σ, 99.73% within ±3σ; 95% within ±1.96σ and 99% within ±2.58σ; QD = 0.6745σ and MD = 0.7979σ; points of inflexion at μ ± σ; the basis of z-tests, control limits and confidence intervals |
| Exponential | Time or distance between successive Poisson events — the waiting time until the next arrival, the life of a component with constant failure rate; continuous | f(x) = λ e^(−λx) for x ≥ 0; P(X > x) = e^(−λx); mean 1/λ; variance 1/λ²; SD equals the mean | Memoryless — the chance of waiting another t minutes does not depend on how long one has waited; positively skewed; service and inter-arrival times in queuing (M/M/1), reliability and replacement models |
3. Data collection and questionnaire design; sampling — concept, process and techniques
Primary data are collected first-hand for the problem in hand by observation (structured or unstructured, disguised or undisguised, natural or contrived; mechanical observation — eye-tracking, people meters, web analytics), experimentation (laboratory or field; test markets), and communication — the survey by personal interview (highest response and flexibility, costliest, interviewer bias), telephone, mail (cheapest, lowest response, no interviewer bias), online and mobile panels, and by focus groups, depth interviews and projective techniques for qualitative data. Secondary data — internal records, government statistics (Census, NSO surveys, RBI), trade bodies, syndicated services (Nielsen, IMRB), published research, the internet — are cheaper and faster but must be checked for relevance, accuracy, currency and the purpose for which they were collected. Questionnaire design proceeds from the information needed and the interviewing method, through question content (is it necessary, can the respondent answer, will they), question form (open-ended versus closed — dichotomous, multiple choice, Likert scale of agreement, semantic differential between bipolar adjectives, importance and rating scales, rank order, constant sum, Stapel scale), wording (simple, unambiguous, no leading, loaded or double-barrelled questions, no implicit assumptions or generalisations), sequence (funnel approach from general to specific, easy and interesting questions first, sensitive and demographic questions last, logical order, filter and branching questions), layout and length, to pretesting (pilot survey) and revision; the four measurement scales — nominal, ordinal, interval, ratio — decide which statistics may be computed.
| Item | Content |
|---|---|
| Concept | A sample is a part of the population (universe) studied to draw conclusions about the whole; sampling saves cost and time, is the only route for destructive testing or infinite populations, and can be more accurate than a badly done census; a census enumerates everyone; the sampling frame is the list from which units are drawn; a parameter describes the population, a statistic the sample; sampling error is the chance difference between them (falls with sample size) and non-sampling error (frame, non-response, measurement, processing) does not |
| Process | Define the target population; specify the sampling frame; choose the sampling unit; select the sampling technique (probability or non-probability); determine the sample size (from the desired precision, confidence level and variability — n = (zσ/E)² for a mean, n = z²pq/E² for a proportion; cost and time); execute and validate the sample |
| Probability (random) techniques | Simple random sampling (every unit an equal chance — lottery, random numbers; with or without replacement); systematic sampling (every kth unit after a random start, k = N/n — periodicity is the danger); stratified sampling (divide into homogeneous strata, sample each — proportionate or disproportionate — for precision and sub-group estimates); cluster sampling (divide into heterogeneous clusters, sample whole clusters — area sampling — cheaper, less precise); multistage sampling (clusters, then units within them); probability proportional to size |
| Non-probability techniques | Convenience sampling (whoever is at hand — mall intercepts); judgement or purposive sampling (the researcher picks typical or expert units); quota sampling (the non-random counterpart of stratified — fill quotas by characteristic); snowball sampling (respondents refer others — rare or hidden populations); self-selection; sampling error cannot be computed and results cannot be generalised statistically |
| Sampling distributions | The central limit theorem — the sampling distribution of the mean approaches normal as n grows (n ≥ 30 is the rule of thumb) whatever the population's shape, with mean μ and standard error σ/√n; the standard error of a proportion is √(pq/n); standard error falls with the square root of sample size — quadrupling n halves it; the t-distribution replaces the normal for small samples with unknown σ; finite population correction √[(N − n)/(N − 1)] |
4. Hypothesis testing — procedure and the t, Z, F and chi-square tests; correlation and regression
- State the hypotheses — the null hypothesis H0 (no difference, no effect, the status quo, always containing equality) and the alternative H1 (what the researcher suspects; two-tailed ≠, or one-tailed > or <).
- Choose the significance level α — the probability of rejecting a true H0 (Type I error), conventionally 5% or 1%; β is the probability of accepting a false H0 (Type II error) and 1 − β the power of the test; the two errors trade off for a given sample size.
- Select the test and compute the test statistic — Z, t, F or chi-square according to the parameter, the sample size, whether σ is known, and the number of groups; its sampling distribution under H0 is known.
- Find the critical value or the p-value — the critical region is the tail area α (split between two tails for a two-tailed test); the p-value is the probability of a result at least as extreme as the one observed if H0 were true.
- Decide and interpret — reject H0 if the statistic falls in the critical region or p < α; otherwise fail to reject (never 'prove') H0; report the conclusion in the problem's terms, remembering statistical significance is not practical importance.
| Test | Used for | Statistic and degrees of freedom | Conditions |
|---|---|---|---|
| Z-test | A mean or proportion, or the difference of two means or proportions, in large samples or when σ is known | z = (x̄ − μ)/(σ/√n); for a proportion z = (p − P)/√(PQ/n); critical values ±1.96 at 5% and ±2.58 at 1% two-tailed, 1.645 and 2.33 one-tailed; no degrees of freedom | n ≥ 30, or normal population with known σ |
| t-test (Student — W. S. Gosset, 1908) | A mean or the difference of two means in small samples with unknown σ: one-sample, two independent samples, paired (matched) samples; also the significance of a correlation or regression coefficient | t = (x̄ − μ)/(s/√n), df = n − 1; two independent samples df = n1 + n2 − 2 with a pooled variance; paired t on the differences, df = n − 1; the t-curve is flatter than the normal and approaches it as df grows | n < 30, σ unknown, population approximately normal; for two samples, equal variances (Welch's version otherwise) |
| F-test (Fisher, Snedecor) | Equality of two variances; and, as analysis of variance (ANOVA), equality of three or more means — one-way (one factor) and two-way (two factors, with or without interaction); the significance of a regression as a whole | F = larger variance/smaller variance, df (n1 − 1, n2 − 1); in ANOVA F = mean square between groups/mean square within groups, df (k − 1, N − k); always right-tailed | Normal populations, independent samples, homogeneity of variance; a significant ANOVA is followed by post-hoc tests (Tukey, Scheffé, Duncan) |
| Chi-square test (Karl Pearson, 1900) | Categorical (nominal) data: goodness of fit of observed frequencies to an expected distribution; independence of two attributes in a contingency table; homogeneity of several populations; a population variance | χ² = Σ(O − E)²/E; goodness of fit df = k − 1 (less parameters estimated); independence df = (r − 1)(c − 1); a 2 × 2 table has df = 1 and takes Yates's continuity correction; the distribution is positively skewed and right-tailed | Frequencies, not percentages; expected frequency in each cell at least 5 (pool cells otherwise); random sample; a non-parametric (distribution-free) test |
Correlation measures the degree and direction of the linear relationship between two variables without implying causation. Karl Pearson's coefficient r = Σ(x − x̄)(y − ȳ)/√[Σ(x − x̄)² Σ(y − ȳ)²] = Cov(x, y)/(σx σy) lies between −1 and +1: +1 is perfect positive, −1 perfect negative, 0 no linear correlation; it is independent of origin and scale, symmetric in x and y, and is the geometric mean of the two regression coefficients. Its square, the coefficient of determination r², is the proportion of variation in y explained by x — r = 0.8 means 64% explained; 1 − r² is the coefficient of non-determination. The probable error 0.6745(1 − r²)/√n judges significance — r is significant if it exceeds six times its probable error — and the t-test with df = n − 2 is the exact version. Spearman's rank correlation ρ = 1 − 6Σd²/[n(n² − 1)] serves ordinal data and small samples (with a correction for tied ranks); Kendall's tau is the alternative; the scatter diagram gives the picture, and concurrent deviations the quick sign test. Spurious correlation from a common cause and the ecological fallacy are the traps. Regression estimates the value of the dependent variable from the independent one: the line of y on x, y = a + bxy·x, minimises the sum of squared vertical deviations (least squares — Gauss, Legendre), with slope byx = r σy/σx = Σ(x − x̄)(y − ȳ)/Σ(x − x̄)² and intercept a = ȳ − byx x̄; the line of x on y has bxy = r σx/σy. The two lines intersect at (x̄, ȳ), coincide when r = ±1 and are perpendicular when r = 0; both regression coefficients carry the sign of r, their product is r², and their geometric mean is r (so both cannot exceed 1 together). The standard error of estimate σy√(1 − r²) measures the scatter about the line; multiple regression y = a + b1x1 + b2x2 + … adds predictors, with R², adjusted R², partial coefficients, the F-test for overall fit, t-tests for each coefficient, and the assumptions of linearity, independence, homoscedasticity, normal errors and no multicollinearity; the Durbin-Watson statistic tests autocorrelation.
Key takeaways
- The mean uses all values and is pulled by extremes, the median is positional and the mode is the only average for nominal data; mode = 3 median − 2 mean in moderate skew; AM ≥ GM ≥ HM; the coefficient of variation compares consistency and the lower is more consistent.
- Binomial has mean np and variance npq (mean greater); Poisson has mean equal to variance λ and models rare events; the normal has 68-95-99.7 within one, two and three SDs and 95% within 1.96; the exponential has mean 1/λ, SD equal to mean, and is memoryless.
- Questionnaires funnel from general to specific with demographics last and no leading or double-barrelled questions; probability sampling is simple random, systematic, stratified, cluster and multistage, non-probability is convenience, judgement, quota and snowball; standard error is σ/√n and the central limit theorem makes sample means normal.
- H0 contains equality; α is the Type I error rate and 1 − β the power; Z for large samples or known σ, t for small samples with df n − 1, F for variances and ANOVA, chi-square for categorical frequencies with df (r − 1)(c − 1); reject when p < α.
- r lies between −1 and +1, r² is the explained proportion, Spearman's ρ serves ranks; the two regression lines meet at the means, both coefficients share r's sign, their product is r², and byx = r σy/σx.
Practice questions (10)
Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.
For a moderately skewed distribution with mean 50 and median 48, Karl Pearson's empirical relation gives a mode of:
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Answer: A — 44
Mode = 3 median − 2 mean = 144 − 100 = 44. Since mean > median > mode the distribution is positively skewed, with its long tail to the right.Series A has mean 40 and standard deviation 8; series B has mean 100 and standard deviation 15. Which is more consistent?
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Answer: A — B, because its coefficient of variation (15%) is lower than A's (20%)
Consistency is judged by relative dispersion, CV = SD/mean × 100: A is 8/40 = 20%, B is 15/100 = 15%. The absolute SD misleads when means differ.Which probability distribution has its mean equal to its variance?
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Answer: A — Poisson
Poisson has mean and variance both λ. Binomial has mean np greater than variance npq; the normal's mean and variance are independent parameters; the exponential has mean 1/λ and variance 1/λ², so its SD, not its variance, equals the mean.In a normal distribution, the proportion of observations lying within one standard deviation on either side of the mean is approximately:
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Answer: A — 68%
The empirical rule: 68.27% within ±1σ, 95.45% within ±2σ, 99.73% within ±3σ; 95% lies within ±1.96σ, the figure behind the 5% significance level.A researcher divides a city's households into income strata and then draws a random sample from each stratum in proportion to its size. This is:
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Answer: A — Proportionate stratified random sampling
Strata are internally homogeneous and every stratum is sampled randomly, which raises precision. Cluster sampling samples whole heterogeneous groups; quota sampling fills the same proportions non-randomly; systematic sampling takes every kth unit.If the sample size is increased four times, the standard error of the sample mean:
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Answer: A — Is halved
Standard error = σ/√n; multiplying n by 4 multiplies √n by 2, so the error halves. Precision improves only with the square root of sample size, which is why doubling accuracy costs four times the sample.Rejecting a null hypothesis that is actually true is:
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Answer: A — A Type I error, whose probability is the significance level α
Type I is the false alarm and α its probability; Type II is failing to reject a false null and β its probability; power is 1 − β, the chance of detecting a real effect. For a fixed sample size, lowering α raises β.To test whether brand preference is independent of gender using a 3 × 4 contingency table, the appropriate test and its degrees of freedom are:
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Answer: A — Chi-square test with 6 degrees of freedom
Categorical frequencies in a contingency table call for the chi-square test of independence with df = (r − 1)(c − 1) = 2 × 3 = 6. F compares variances or several means, t compares one or two means.Comparing the mean sales of four regions to see whether they differ significantly requires:
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Answer: A — One-way analysis of variance (F-test)
Three or more means need ANOVA, which partitions total variation into between-group and within-group components and tests their ratio as F with (k − 1, N − k) degrees of freedom. A t-test handles at most two means; chi-square handles frequencies.If the two regression coefficients are byx = 0.8 and bxy = 0.45, the coefficient of correlation is:
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Answer: A — 0.6
r is the geometric mean of the regression coefficients: √(0.8 × 0.45) = √0.36 = 0.6, positive because both coefficients are positive. 0.36 is r², the coefficient of determination — 36% of the variation explained.