The normal distribution is the symmetric, single-peaked "bell curve" that describes many natural measurements — heights, test scores, measurement errors. Two facts make it easy to work with on the GRE. First, it is perfectly symmetric about its mean, so the mean, median and mode all coincide and exactly 50% of the data lies on each side. Second, its entire shape is fixed by just two numbers: the mean (where the peak sits) and the standard deviation (how wide the bell is). A larger standard deviation gives a flatter, wider curve; a smaller one gives a tall, narrow spike. The GRE never asks you to integrate the curve or use a z-table — it expects you to reason with symmetry and the 68-95-99.7 bands. A value’s z-score, (value − mean)/SD, just says how many standard deviations from the mean it lands, which turns any normal question into a bands question.
✅ Solved examples
1. Adult male heights are normally distributed with mean 175 cm. What percent of men are taller than 175 cm?
The mean splits a normal curve exactly in half by symmetry. So 50% are above 175 cm.
2. Scores are normal with mean 500 and standard deviation 100. What is the z-score of a score of 680?
z = (value − mean)/SD = (680 − 500)/100 = 180/100 = 1.8. The score is 1.8 standard deviations above the mean.
3. Two normal curves have the same mean; curve X has SD 4 and curve Y has SD 10. Which curve is taller and narrower at the peak?
A smaller standard deviation concentrates the data near the mean, giving a taller, narrower peak. So curve X (SD 4) is taller and narrower.
4. Quantitative Comparison. A dataset is normally distributed. Quantity A: the median. Quantity B: the mean. Which is greater?
In any normal distribution the curve is symmetric, so the mean and median are equal. The two quantities are equal.
✏️ Practice — try these, take hints as needed
1. Weights are normal with mean 60 kg. What percent weigh less than 60 kg?
The mean bisects a normal curve.
Symmetry gives equal halves.
Half of 100%.
50%
2. A normal set has mean 200 and SD 25. Find the z-score of 250.
z = (value − mean)/SD.
(250 − 200)/25.
50/25.
z = 2
3. A value has z-score −1.5 in a normal set with mean 80 and SD 6. What is the value?
value = mean + z × SD.
80 + (−1.5)(6).
80 − 9.
71
4. Curve P has SD 3, curve Q has SD 8, same mean. Which is more spread out?
Spread is measured by the SD.
Bigger SD, wider bell.
Compare 3 and 8.
Curve Q
5. In a normal distribution, what is the relationship between the mode and the mean?
Consider the symmetry of the peak.
The peak sits at the centre.
All three centres coincide.
They are equal
📝 Topic test — 8 questions
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