Measures of central tendency
The median of 3, 7, 1, 9, 5 is: (a) 3 (b) 5 (c) 7 (d) 9
(b) 5. Sort first - 1, 3, 5, 7, 9 - then take the middle value. Forgetting to sort is the most common error in median questions, and for an even count you average the two middle values.
Which measure of central tendency is most affected by extreme values? (a) Mean (b) Median (c) Mode (d) All equally
(a) The mean, because every value enters the calculation. This is why median household income is reported rather than mean income - a few very large values would otherwise distort the picture.
The empirical relationship between mean, median and mode is: (a) Mode = 3 Median − 2 Mean (b) Mean = 3 Median − 2 Mode (c) Median = Mean + Mode (d) All are equal
(a) Mode = 3 Median − 2 Mean, an approximation for moderately skewed distributions. In a perfectly symmetric distribution all three coincide.
For a strongly right-skewed dataset, which is largest? (a) Mean (b) Median (c) Mode (d) They are equal
(a) The mean. A long right tail drags the mean above the median, which in turn sits above the mode. The ordering reverses for a left-skewed distribution.
The mean of the first 10 natural numbers is: (a) 5 (b) 5.5 (c) 6 (d) 10
(b) 5.5. Their sum is 55 by n(n+1)/2, and 55/10 = 5.5. For consecutive integers, the mean is simply the average of the first and last values.
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Dispersion: variance and standard deviation
Find the standard deviation of 2, 4, 4, 4, 5, 5, 7, 9.
2. The mean is 5; the squared deviations are 9, 1, 1, 1, 0, 0, 4, 16, summing to 32; the variance is 32/8 = 4, so the standard deviation is √4 = 2. Note that variance carries squared units, which is why the standard deviation is usually reported.
If 5 is added to every observation in a dataset, what happens to the mean and the standard deviation?
The mean increases by 5; the standard deviation is unchanged. Shifting all values moves the centre but not the spread. If instead every value were multiplied by 5, both the mean and the standard deviation would be multiplied by 5, and the variance by 25.
The coefficient of variation is defined as: (a) SD × Mean (b) (SD/Mean) × 100 (c) Variance/Mean (d) Mean/SD
(b) (SD/Mean) × 100. Being unitless, it allows spread to be compared between datasets measured in different units or with very different means.
Which measure of dispersion uses every observation and is least affected by sampling fluctuation? (a) Range (b) Quartile deviation (c) Standard deviation (d) Mode
(c) The standard deviation. The range depends on only the two extreme values, making it highly unstable; the standard deviation uses all observations and has the better mathematical properties.
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Probability
Two fair coins are tossed. What is the probability of at least one head?
3/4. The sample space is HH, HT, TH, TT and three outcomes qualify. Equivalently, 1 − P(no heads) = 1 − 1/4. The complement approach is usually faster for 'at least one' questions.
A card is drawn from a standard deck. What is the probability that it is an ace or a spade?
16/52 = 4/13. There are 4 aces and 13 spades, but the ace of spades is in both, so P(A ∪ B) = 4/52 + 13/52 − 1/52. Forgetting to subtract the overlap is the classic error.
Mutually exclusive events A and B with non-zero probabilities are: (a) Always independent (b) Never independent (c) Sometimes independent (d) Always equally likely
(b) Never independent. If they are mutually exclusive then P(A ∩ B) = 0, whereas independence would require P(A)P(B), which is non-zero. The two concepts are frequently and wrongly treated as similar.
Two dice are rolled. What is the probability that the sum is 8?
5/36. The favourable pairs are (2,6), (3,5), (4,4), (5,3) and (6,2) out of 36 equally likely outcomes. Listing beats formula-hunting for small sums.
In a binomial distribution with n trials and success probability p, the mean and variance are: (a) np and np (b) np and npq (c) p and pq (d) nq and npq
(b) Mean np and variance npq, where q = 1 − p. Since q < 1, the variance is always smaller than the mean in a binomial distribution - a useful sanity check.
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Distributions
In a normal distribution, approximately what percentage of values lie within one standard deviation of the mean? (a) 50% (b) 68% (c) 95% (d) 99.7%
(b) About 68%. The empirical rule gives roughly 68% within one SD, 95% within two and 99.7% within three - worth knowing as a set.
Which is true of a normal distribution? (a) It is skewed right (b) Mean, median and mode coincide (c) It is discrete (d) It has no tails
(b) Mean, median and mode all coincide at the centre. It is continuous, symmetric and bell-shaped, with tails extending indefinitely in both directions without touching the axis.
A distribution is used to model the number of successes in a fixed number of independent trials. It is the: (a) Normal (b) Binomial (c) Poisson (d) Uniform
(b) Binomial. The Poisson distribution models counts of events in a fixed interval when they are rare, and the normal is continuous rather than a count.
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Correlation and regression
This is where interpretation questions concentrate, and where confident wrong answers are most common.
The correlation coefficient r always lies between: (a) 0 and 1 (b) −1 and 1 (c) −∞ and ∞ (d) 0 and 100
(b) −1 and 1. A value near ±1 indicates a strong linear relationship, and the sign gives its direction. Values outside this range signal a computational error.
If r = 0 between two variables, it means: (a) There is no relationship (b) There is no linear relationship (c) The variables are independent (d) One causes the other
(b) There is no linear relationship. A perfect parabola can produce r ≈ 0 while the variables are entirely dependent on each other. This distinction is the single most-tested idea in the topic.
A strong correlation between ice-cream sales and drowning incidents implies: (a) Ice cream causes drowning (b) Drowning causes ice-cream sales (c) A likely common cause (d) The correlation must be spurious
(c) A likely common cause - hot weather increases both. Correlation does not establish causation, and the standard explanation is a lurking third variable rather than any direct link.
The regression line of y on x always passes through: (a) The origin (b) The point (x̄, ȳ) (c) The largest observation (d) No fixed point
(b) The point of means (x̄, ȳ). This is a defining property of least-squares regression and a frequent one-mark question.
If the correlation between x and y is 0.8, what does the coefficient of determination tell you?
r² = 0.64, so about 64% of the variation in y is explained by the linear relationship with x. Note that squaring reduces the figure substantially - a strong-sounding correlation of 0.8 still leaves over a third of the variation unexplained.
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Sampling and inference
The standard error of the mean is: (a) σ (b) σ/n (c) σ/√n (d) σ²/n
(c) σ/√n. Because the divisor is the square root of the sample size, quadrupling the sample only halves the standard error - which is why very precise estimates get expensive quickly.
A Type I error occurs when you: (a) Reject a true null hypothesis (b) Accept a false null hypothesis (c) Choose the wrong test (d) Use too small a sample
(a) Reject a true null hypothesis - a false positive. A Type II error is failing to reject a false null hypothesis, a false negative. Reducing one generally increases the other for a fixed sample size.
Which sampling method gives every member of the population an equal chance of selection? (a) Convenience sampling (b) Simple random sampling (c) Quota sampling (d) Snowball sampling
(b) Simple random sampling, which is what makes the results generalisable. Convenience sampling is the cheapest and most biased - the reason many online polls are unrepresentative.
A p-value of 0.03 means: (a) There is a 3% chance the null hypothesis is true (b) The result is 97% certain (c) Data this extreme would occur 3% of the time if the null hypothesis were true (d) The effect is large
(c). A p-value is a probability computed under the assumption that the null hypothesis holds - it is not the probability that the hypothesis is true, and it says nothing about the size or importance of the effect.
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How to prepare statistics MCQs
- ✓Split your revision into compute questions and interpret questions - they fail for different reasons and need different practice.
- ✓Learn the effect of transformations as a rule: adding a constant shifts the mean only; multiplying scales the mean and SD, and the variance by the square.
- ✓Memorise the interpretation traps: r = 0 means no linear relationship; correlation is not causation; a p-value is conditional on the null hypothesis.
- ✓For probability, work the complement for 'at least one' questions and always subtract the overlap in 'A or B'.
- ✓Check computations for plausibility - r outside [−1, 1], a variance larger than the mean in a binomial, or an SD bigger than the range all signal an error.