Finding Probabilities and Values from a Normal Model
Get the probability of a range of values from a normal model, and go the other way from a stated probability back to the cut-off value, sketching the region every time.
What a learner can do afterwards
- Find P(X < 70) for a stated mean and standard deviation
- Find the mark that only the top 10 percent beat
- Find an unknown mean or standard deviation from a given probability
1 · Read
Mia beat almost everyone in the quiz, and you want to say how rare a mark like hers is. A z-score counts how many standard deviations a value sits from the centre: subtract the mean, divide by the spread. For mean 60 and spread 5, the value 70 gives (70 - 60)/5 = 2. The empirical rule says about 68% sit within 1, 95% within 2, and 99.7% within 3 deviations.
Going backwards from a probability to a mark means working the table in reverse. The top 10% cutoff has z about 1.28, so with mean 60 and spread 8 the mark is 60 + 1.28 times 8 = 70.24, about 70. Sketch the curve and shade the region every time.
Unknown means and deviations fall out the same way once z is known. If P(X < 70) is about 0.975 with mean 60, about 2 deviations sit below 70, so the gap of 10 covers 2 deviations and the spread is 5. Always mark what you know on the sketch before converting.
The pattern never changes: sketch, mark, convert to z, finish with the table. Below 2.5% sits about 2 deviations down: for IQ style scores with mean 100 and spread 15 that is 100 - 30 = 70. Copy the sketch habit and your error rate will drop.
Convert values to z for probabilities, and work the table backwards from probabilities to marks.
2 · Watch
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Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.