Question 1

What does a z-score of 2 mean?

Accepted Answer

A z-score of 2 means the value sits exactly 2 standard deviations above the mean. In a normal distribution that puts it at roughly the 97.7th percentile — higher than about 97.7% of all values. For example, if test scores average 70 with a standard deviation of 10, a score of 90 has z = 2 and beats nearly 98% of the class.

Question 2

What is a good z-score?

Accepted Answer

It depends entirely on what is being measured. For things where higher is better (test scores, income), a positive z-score is above average and z above 1 is clearly strong (84th percentile or better). For things where closer to the mean is better (manufacturing tolerances, lab values), a z-score near 0 is good and values beyond ±2 are unusual — only about 5% of a normal distribution falls outside that range.

Question 3

What is the difference between a z-score and a percentile?

Accepted Answer

A z-score measures distance from the mean in standard deviations and can be any number, negative or positive. A percentile tells you what share of values fall below yours, from 0 to 100. They map to each other through the normal curve: z = 0 is the 50th percentile, z = 1 is about the 84th, z = 2 about the 97.7th. The z-score is linear in the raw value; the percentile is not — one extra standard deviation matters much more near the middle than in the tails.

Question 4

When is the normal-distribution assumption wrong?

Accepted Answer

The percentile conversion is only accurate if the data is roughly bell-shaped. It breaks down for skewed data (income, house prices, waiting times), bounded data with many values at the edge (percent scores near 100), small samples, or data with heavy tails or multiple peaks. The z-score itself — distance from the mean in standard deviations — can still be computed, but reading a percentile from it would be misleading in those cases.

Question 5

How do I calculate a z-score by hand?

Accepted Answer

Subtract the mean from your value, then divide by the standard deviation: z = (x − μ) / σ. With x = 85, μ = 70, σ = 10: z = (85 − 70) / 10 = 1.5. To go the other way, multiply the z-score by the standard deviation and add the mean: x = μ + zσ = 70 + 1.5 × 10 = 85.

Z-score	Percentile	Higher than
−3.0	0.13	0.13% of values
−2.5	0.62	0.62% of values
−2.0	2.28	2.28% of values
−1.5	6.68	6.68% of values
−1.0	15.87	15.87% of values
−0.5	30.85	30.85% of values
0.0	50.00	50.00% of values
+0.5	69.15	69.15% of values
+1.0	84.13	84.13% of values
+1.5	93.32	93.32% of values
+2.0	97.72	97.72% of values
+2.5	99.38	99.38% of values
+3.0	99.87	99.87% of values

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