Math · Introductory statistics · Worked example
Find a normal probability with a z-score
Adult heights in a population are approximately normal with mean 170 cm and standard deviation 8 cm. What fraction are taller than 186 cm?
Standardize
186 cm is two standard deviations above the mean.
Find the area to the left
Φ(2) ≈ 0.9772: about 97.7% of heights lie below 186 cm.
Take the complement
The area to the right is what is left.
Check with the 68–95–99.7 rule
About 95% lie within two standard deviations, so about 2.5% lie above, close to the exact 2.28%.
Result
About 0.0228, so 2.3% of adults in this population are taller than 186 cm.
Your turn
In the same population, what fraction are shorter than 158 cm?
Show the answer and explanation
About 0.0668, or 6.7%.
z = (158 − 170)/8 = −1.5, and Φ(−1.5) ≈ 0.0668.
Keep exploring
In Statistics, Distributions shows P(X ≥ 2) = 0.0227501 for the standard normal. Change the value to 1 and the right tail grows to about 0.159.
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