Business Statistics I: Description, Association and Probability
2. Dispersion and skewness
Two distributions with the same average can be utterly different in spread, so an average alone is half a description. The measures of dispersion differ in what they are built from and, critically, in whether they carry the unit of the data or are free of it.
- Range — largest minus smallest. Uses two values only, so a single outlier decides it.
- Quartile deviation — half the interquartile range, (Q₃ − Q₁)/2. Ignores the extreme quarters entirely, which is its virtue with skewed data and its defect otherwise.
- Mean deviation — the average of the absolute deviations from the mean or the median. It is least from the median, but the absolute value makes it awkward algebraically.
- Standard deviation — the square root of the mean of squared deviations from the mean. Uses every value, is algebraically tractable, and is the measure every later technique is built on. Its square is the variance.
- Coefficient of variation — the standard deviation as a percentage of the mean, and therefore UNIT-FREE. It is the only measure here that can compare the variability of two series measured in different units or with very different means, and a lower value means greater consistency.
Skewness is the departure from symmetry. In a symmetrical distribution mean, median and mode coincide; in a positively skewed one the long tail is to the right and the mean is dragged above the median, which is above the mode; in a negatively skewed one the order reverses. Karl Pearson's coefficient is (mean − mode) divided by the standard deviation, or, using the empirical relation mode ≈ 3 median − 2 mean, 3(mean − median) divided by the standard deviation. Bowley's coefficient uses the quartiles and is the one to reach for with open-ended classes.
3. Correlation and regression of two variables
Correlation measures the degree to which two variables move together; regression estimates one from the other. Karl Pearson's coefficient r lies between −1 and +1, is unit-free, and is unaffected by changing the origin or the scale of either variable — so measuring price in paise instead of rupees changes nothing. Its sign gives the direction, its magnitude the closeness, and r = 0 means no LINEAR relation, not no relation.
- Two regression lines, not one — Y on X for estimating Y, X on Y for estimating X. They intersect at the point of means, and they coincide only when r is ±1.
- The regression coefficients b_yx and b_xy carry the same sign as r, and r is the geometric mean of the two: r² = b_yx × b_xy. So if both are given, r follows at once — and if their product exceeds one, the figures are wrong.
- The coefficient of determination r² is the proportion of the variance in one variable explained by the other. An r of 0.6 explains 36% of the variance, not 60% — which is the arithmetic the paper likes to set.
- Spearman's rank correlation applies the same idea to ranks: ρ = 1 − 6Σd²/[n(n² − 1)]. It suits qualities that can be ordered but not measured, and it is the rank correlation test the hypothesis-testing list names later in the unit.
4. Probability: three approaches and Bayes' theorem
The syllabus names the approaches to probability, and they are approaches to one question — where does the number come from.
| Approach | Where the number comes from | Its limit |
|---|---|---|
| Classical, or a priori | Favourable outcomes over equally likely total outcomes, before any trial | Needs the outcomes to be equally likely, which assumes what it sets out to find |
| Relative frequency, or empirical | The limit of the proportion of times an event occurs as trials increase | Needs repeatable trials; says nothing about a one-off event |
| Subjective | A degree of belief, revised as evidence arrives | Two people may assign different numbers to the same event |
Two rules do most of the work. The addition rule gives the probability of A or B as P(A) + P(B) − P(A and B), the subtraction being what candidates forget when the events can both happen; for mutually exclusive events it vanishes. The multiplication rule gives the probability of A and B as P(A) × P(B|A), which reduces to P(A) × P(B) only when the events are independent. Mutually exclusive and independent are not the same thing — mutually exclusive events are strongly dependent, since one occurring rules the other out.
5. Binomial, Poisson and normal
The three distributions are told apart by what they count and by one property each. The binomial counts successes in a fixed number of independent trials with a constant probability; the Poisson counts rare events in a continuous interval of time or space; the normal describes a continuous measurement clustering symmetrically about a mean.
| Distribution | Parameters | Mean and variance | The property that is asked |
|---|---|---|---|
| Binomial | n and p | np and npq | Variance is always less than the mean, since q < 1; symmetrical only when p = 0.5 |
| Poisson | λ alone | λ and λ | Mean equals variance — the one-line identifier; the limit of the binomial as n grows and p shrinks with np fixed |
| Normal | μ and σ | μ and σ² | Symmetrical and bell-shaped; mean = median = mode; about 68%, 95% and 99.7% of values within 1, 2 and 3 standard deviations |
The normal distribution matters beyond its own questions because of the standard normal variate z = (x − μ)/σ, which converts any normal measurement into standard deviations from its mean and lets one table serve every normal distribution. It is also the distribution to which sample means tend, whatever the population looks like, by the central limit theorem — which is why it reappears throughout the next chapter as the basis of estimation and testing.
Key takeaways
- The mean is dragged by extreme values and the median is not; for any positive data AM ≥ GM ≥ HM, and the geometric mean is the right average of growth rates.
- Every 'which is more consistent' question wants the coefficient of variation, because it is the only unit-free measure of dispersion and the only one that compares series of different scale.
- r is the geometric mean of the two regression coefficients, r² is the proportion of variance explained, and r = 0 rules out only a linear relation.
- Mutually exclusive is not independent: the addition rule drops its subtraction for the first, the multiplication rule simplifies for the second, and Bayes' theorem revises a prior by evidence.
- Binomial variance is always below its mean, Poisson mean equals variance, and the normal puts 68%, 95% and 99.7% of values within 1, 2 and 3 standard deviations.
Practice questions (9)
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.
Which measure of dispersion is appropriate for comparing the variability of two series measured in different units?
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Answer: A — Coefficient of variation
Standard deviation, range and quartile deviation are all absolute measures expressed in the unit of the data, so rupees cannot be set against tonnes and even two rupee figures mislead when the means differ greatly. The coefficient of variation divides the standard deviation by the mean and expresses the result as a percentage, which strips the unit out and makes the two series directly comparable; the lower value is the more consistent series. Every question asking which series is more consistent or more variable is asking for this measure, and answering from the standard deviation alone is the standard error.The two regression coefficients are b_yx = 0.8 and b_xy = 0.45. The coefficient of correlation is:
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Answer: A — 0.6
The correlation coefficient is the geometric mean of the two regression coefficients, so r² = b_yx × b_xy = 0.8 × 0.45 = 0.36 and r = 0.6, taking the positive root because both coefficients are positive and r carries their sign. The value 0.36 is r² itself, the coefficient of determination, left unrooted. 0.625 is the arithmetic mean of the two coefficients, which is the wrong average — the relation is multiplicative, not additive. 1.25 is their sum, which exceeds one and could not be a correlation coefficient at all; a product of regression coefficients greater than one would itself signal an error in the data.A discrete distribution has mean equal to its variance. It is most likely:
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Answer: A — Poisson
Equality of mean and variance is the defining signature of the Poisson distribution, whose single parameter λ is both. The binomial cannot have this property because its variance npq is always less than its mean np, since q is below one. The normal is a continuous distribution, so it is ruled out by the word discrete before its moments are even considered, and its mean and variance are independent parameters in any case. A discrete uniform distribution has a variance determined by its range that in general bears no fixed relation to its mean. A question stating mean and variance equal is identifying the Poisson without naming it.Two machines produce 60% and 40% of a firm's output with defect rates of 2% and 5% respectively. A randomly chosen item is defective. The probability that it came from the first machine is:
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Answer: A — 0.375
Bayes' theorem reverses the conditioning. The joint probability of coming from the first machine and being defective is 0.60 × 0.02 = 0.012; from the second, 0.40 × 0.05 = 0.020; so the total probability of a defect is 0.032 and the posterior probability for the first machine is 0.012 ÷ 0.032 = 0.375. The value 0.6 is the prior, which ignores the evidence of the defect entirely. 0.012 is the joint probability left undivided by the total, and 0.625 is the posterior for the second machine, which is the likelier source once a defect has been seen because defects are more characteristic of it.In a positively skewed distribution:
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Answer: A — Mean > median > mode
Positive skewness means the long tail extends to the right, towards high values. The mode sits at the peak, the median is pulled a little towards the tail because half the observations must lie beyond it, and the mean is pulled furthest because it alone is weighted by the size of every value, so the few very large observations drag it right. The order is therefore mean, median, mode from largest to smallest. The reverse order describes negative skewness, and equality of all three describes a symmetrical distribution such as the normal. Income and wealth are the standard examples of positive skew.An investment doubles in the first year and halves in the second. Its average annual growth rate is correctly given by the:
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Answer: A — Geometric mean, which gives zero growth
Growth compounds multiplicatively, so the correct average of growth factors is their geometric mean: the factors are 2 and 0.5, their product is 1, and the square root of 1 is 1 — a growth rate of zero, which is exactly right because the investment ends where it started. The arithmetic mean of +100% and −50% is +25%, which is the standard wrong answer and would describe an investment that had grown by more than half over the two years. The harmonic mean is the average for rates measured per unit such as speed over equal distances, not for compound growth, and the median of two values is simply their arithmetic mean here.Two events are mutually exclusive. It follows that they are:
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Answer: A — Dependent, since the occurrence of one rules out the other
Mutually exclusive events cannot occur together, so knowing that one has occurred tells you with certainty that the other has not — which is the strongest possible form of dependence, not independence. Independence would require P(A and B) to equal P(A) × P(B), and for mutually exclusive events with non-zero probabilities the left side is zero while the right side is not. Confusing the two is the standard error, and it matters for the rules: the addition rule loses its subtracted term for mutually exclusive events, while the multiplication rule simplifies for independent ones. Neither equal likelihood nor exhaustiveness follows from mutual exclusivity.The coefficient of correlation between two variables is 0.6. The proportion of variation in one explained by the other is:
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Answer: A — 36%
The proportion of variance explained is the coefficient of determination, which is the square of the correlation coefficient: 0.6² = 0.36, or 36%. Reading the correlation coefficient itself as the proportion explained gives 60% and is the standard error — a correlation of 0.6 is considerably weaker as an explanation than the raw figure suggests. 40% is the complement of 60% and describes nothing, and 77% is roughly the square root of 0.6 rather than its square, the operation applied in the wrong direction. The unexplained proportion, 1 − r², is 64% here.For a normal distribution, the proportion of values lying within two standard deviations of the mean is approximately:
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Answer: A — 95%
The empirical rule for the normal distribution places about 68% of values within one standard deviation of the mean, about 95% within two and about 99.7% within three. The three figures are routinely tested against one another, so 68% and 99.7% are the neighbours a candidate is expected to distinguish from the two-sigma value. 50% is the proportion on either side of the mean, since the distribution is symmetrical, and describes no interval around it. The rule is what makes the standard normal variate useful: any normal measurement expressed as z = (x − μ)/σ can be read against these same proportions.