The normal curve, the three benchmarks worth knowing, and working backwards from a probability to a value.
A test is scaled so scores are normal with mean 100 and standard deviation 15. Drag the boundaries. The number underneath is the shaded area, and the shaded area is the probability.
The z-value tells you how many standard deviations out you are. It is not the probability.
| Within | Range here | Probability |
|---|---|---|
| μ ± σ | 85 to 115 | about 68% |
| μ ± 2σ | 70 to 130 | about 95% |
| μ ± 3σ | 55 to 145 | about 99.7% |
These are worth knowing by heart. They give you a sanity check on every answer: if a calculator says P(85 < X < 115) is 0.23, something has gone in wrong, because one standard deviation either side is always about two thirds.
The inverse normal is the one people set up wrongly. Calculators usually want the area to the LEFT. If a question asks for the top 10%, feed in 0.9, not 0.1. Getting 80.8 instead of 119.2 is the standard symptom, and the symmetry of the curve makes it look plausible.
1. With μ = 100 and σ = 15, find P(X < 115) to 4 decimal places.
2. Find P(X > 130) to 4 decimal places.
3. The top 10% of scores start at what value, to 1 decimal place?
A sketch with the region shaded. It is quick, it is often a mark, and it catches a tail on the wrong side before the calculator does.
Stating X ~ N(μ, σ²) with the numbers. Note the notation uses the variance, so N(100, 225) here, not N(100, 15).
On an inverse normal, saying which area you fed in. Showing 0.9 for the top 10% is what distinguishes a method slip from a wrong answer.
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