---
title: "Finding Probabilities and Values from a Normal Model"
description: "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."
canonical: https://lightmysky.com/learn/mathematics/finding-probabilities-and-values-from-a-normal-model-mt_naXMRySvos
source: https://lightmysky.com/learn/mathematics/finding-probabilities-and-values-from-a-normal-model-mt_naXMRySvos.md
retrieved: 2026-09-12
---

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# 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.

Subject: Mathematics · Area: Probability · Ages 17 to 18
Page: https://lightmysky.com/learn/mathematics/finding-probabilities-and-values-from-a-normal-model-mt_naXMRySvos

## Ready when they can

- 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

## Lesson: Reading the bell curve both ways

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.

**Example.** 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.

**Tip.** 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.

**Recap.** Convert values to z for probabilities, and work the table backwards from probabilities to marks.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [Cumulative Frequency Curves](https://lightmysky.com/learn/mathematics/cumulative-frequency-curves-mt_5U70qgbnST)
- [The Normal Distribution and Standardised Scores](https://lightmysky.com/learn/mathematics/the-normal-distribution-and-standardised-scores-mt_Cs32DT9KnA)

## Opens up

- [Sampling Distributions and the Central Limit Theorem](https://lightmysky.com/learn/mathematics/sampling-distributions-and-the-central-limit-theorem-mt_nugaEeUSii)
- [Confidence Intervals for Means and Proportions](https://lightmysky.com/learn/mathematics/confidence-intervals-for-means-and-proportions-mt_Pg2fPDW1sn)