A Z-score tells us:

Question

A Z-score tells us:

BlackTom AI · Accepted Answer

To begin, we restate the question and the answer choices to set the stage for analysis.
Question: A Z-score tells us:
Options:
- a. How many standard deviations a data point is from the mean
- b. The raw value of a data point
- c. The probability of a data point occurring
- d. The variance of the distribution

Now we evaluate each option in turn:

Option a: How many standard deviations a data point is from the mean. This is the essence of a Z-score. By definition, a Z-score measures the number of standard deviations away from the mean that a particular data point lies. If z = 0, the data point equals the mean; if z = 2, it is two standard deviations above the mean, and so on. This aligns with the standard normal transformation formula z = (x - μ) / σ.

Option b: The raw value of a data point. This is incorrect because the Z-score is not the original value x itself; it is a standardized version that expresses x relative to the distribution’s mean and standard deviation. The raw value is simply x, not the standardized score.

Option c: The probability of a data point occurring. A Z-score per se does not directly give the probability of that exact data point occurring; probabilities relate to ranges under the distribution, typically via the standard normal curve. A Z-score helps convert to probabilities for intervals, but it is not a probability value by itself.

Option d: The variance of the distribution. The Z-score involves the standard deviation (the square root of the variance) but it is not the variance itself. The Z-score encodes relative position using μ and σ, not the spread measure alone.

Putting these evaluations together, each incorrect option either misrepresents what the Z-score conveys (raw value, probability of a single point, or variance) or omits the defining idea. The option that correctly captures the concept is the one describing the number of standard deviations a data point lies from the mean.

Statistics and Probability

A Z-score tells us:

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