Questions: Standard Normal Distribution and Z-Score Standardization

Question 1

A student scores 78 on Exam A (mean = 70, SD = 10) and 88 on Exam B (mean = 85, SD = 5). Which score was stronger relative to its distribution?

• A. Exam B, because 88 is a higher raw score than 78
• B. Exam A, because its Z-score (0.8) is higher than Exam B's Z-score (0.6)
• C. They are equivalent, since both scores are above their respective means by similar amounts
• D. Exam B, because the smaller standard deviation means less competition at the top

Answer: (B)

Comparing raw scores across distributions with different means and spreads is misleading. Z-scores standardize both: Z_A = (78−70)/10 = 0.8 and Z_B = (88−85)/5 = 0.6. The student was 0.8 standard deviations above the mean on Exam A but only 0.6 on Exam B. Despite the higher raw score on B, relative performance was stronger on A. Option A commits exactly the error Z-scores are designed to correct: treating raw scores as comparable when they come from distributions with different parameters.

Question 2

Why can a single Z-table (the standard normal CDF) be used to find probabilities for any normal distribution, regardless of its mean and variance?

• A. All normal distributions assign the same probability to the same raw values because they share the same bell shape
• B. The transformation Z = (X−μ)/σ converts any N(μ,σ²) probability question into an equivalent N(0,1) question, where probabilities are tabulated
• C. Z-tables approximate all continuous distributions, not just normal ones, which is why they work universally
• D. Normal distributions with different parameters are literally the same distribution, so their probability tables are identical

Answer: (B)

A single table works because the transformation Z = (X−μ)/σ converts 'what is P(X ≤ x) for N(μ,σ²)?' into 'what is P(Z ≤ z) for N(0,1)?' where z = (x−μ)/σ. Every normal distribution has the same shape — just centered and scaled differently — so once standardized, you read probabilities from the universal N(0,1) curve. The table doesn't need a separate entry for every (μ,σ²) pair; the transformation does that work. Option A is subtly wrong: different normal distributions assign different probabilities to the same raw value; they only assign the same probability to the same Z-score.

Question 3

A Z-score of −2 means the observation is 2 standard deviations below the mean of its distribution.

• T. True
• F. False

Answer: True

The Z-score formula Z = (X−μ)/σ encodes signed distance from the mean in standard deviation units. When Z is negative, X < μ, so the observation is below the mean. A Z of −2 means X = μ − 2σ, exactly 2 standard deviations below the mean. The sign indicates direction (above or below) and the magnitude indicates how many standard deviations away. This signed distance interpretation is the intuitive core of the Z-score concept.

Question 4

The standard normal distribution N(0,1) is a fundamentally different type of distribution from N(5,4) and requires separate mathematical tools to analyze.

• T. True
• F. False

Answer: False

N(0,1) and N(5,4) are the same type of distribution — both are normal (Gaussian) distributions with the same bell-shaped form and identical mathematical structure, just different parameters. N(0,1) is N(5,4) after applying Z = (X−5)/2 (subtracting the mean, dividing by the standard deviation). Standardization is a change of variable, not a change of distribution family. This is exactly what makes the Z-table work: there is only one shape of normal distribution, and all instances are rescaled and recentered versions of each other.

Question 5

Explain why Z-scores enable meaningful comparison of values from two different normal distributions, and what goes wrong if raw scores are compared instead.

Model answer: Z-scores measure how many standard deviations a value sits above or below its distribution's mean — a scale-free measure of relative position. Two values from distributions with different means and spreads can only be compared meaningfully by their position within each distribution. Raw scores conflate the level (mean) and spread (SD) of the distribution with the individual's relative standing. A score of 85 might be mediocre in one distribution (mean 90, SD 5) and excellent in another (mean 70, SD 10).

The deeper point is that Z-scores remove the influence of the distribution's location and scale, placing both values on the universal N(0,1) scale. This is why standardization underlies hypothesis testing: a test statistic of the form Z = (X̄−μ₀)/(σ/√n) converts a raw difference into a Z-score — measuring how many 'sampling standard deviations' the observed mean sits from the hypothesized value — making it directly comparable to the universal null distribution regardless of the original units or scale.