--- title: "The Normal Distribution and Standardised Scores" description: "A continuous symmetric model where most values sit near the mean. Converting a value to z = (x - μ) / σ says how many standard deviations from the mean it is, which makes different data sets comparabl" canonical: https://lightmysky.com/learn/mathematics/the-normal-distribution-and-standardised-scores-mt_Cs32DT9KnA source: https://lightmysky.com/learn/mathematics/the-normal-distribution-and-standardised-scores-mt_Cs32DT9KnA.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # The Normal Distribution and Standardised Scores A continuous symmetric model where most values sit near the mean. Converting a value to z = (x - μ) / σ says how many standard deviations from the mean it is, which makes different data sets comparable. Subject: Mathematics · Area: Probability · Ages 16 to 17 Page: https://lightmysky.com/learn/mathematics/the-normal-distribution-and-standardised-scores-mt_Cs32DT9KnA ## Ready when they can - Use normal-model symmetry and the 68 and 95 percent facts to describe probability regions - Convert a value to a z-score and say what it means - Explain why probability here is an area under the curve, not a height ## Lesson: One scale for two different contests A normal model is continuous, symmetric and bell-shaped. Its mean μ sets the centre and its positive standard deviation σ sets the spread. Total area is 1, so a larger σ gives a wider, lower normal curve. Measurements are not automatically normal; check the fit. A normal curve may approximate a binomial when expected success and failure counts are both sufficiently large. Binomial outcomes still remain whole counts. *(drawing: A useful approximation does not change which values the original variable can take.)* Assume a continuous normal model for quiz values with μ = 60 and σ = 8. Symmetry puts 50% of the model area on each side of μ. About 68% lie between μ - σ and μ + σ, here 52 and 68. About 95% lie between μ - 2σ and μ + 2σ, here 44 and 76. The remaining 5% splits equally between the tails, about 2.5% above 76. About 99.7% lie within three σ. These are model proportions, not guaranteed percentages in a particular recorded group. *(drawing: The marks bound model intervals: 52 to 68 contains about 68% of the normal model's area; 44 to 76 contains about 95%.)* For σ > 0, z = (x - μ) / σ counts signed standard deviations. Mia's 76 gives (76 - 60) / 8 = 2. Below μ gives negative z; at μ, z = 0. Reverse with x = μ + zσ, or x - μ = zσ. Subtracting the same μ and dividing by the same positive σ preserves order. Adding equally to x and μ, with σ unchanged, leaves z unchanged. **Example.** Mia's plant value gives (50 - 42) / 5 = 1.6, compared with z = 2 in the quiz. The quiz result is farther above its mean in σ units. Equal z-scores need not mean equal raw marks or realised ranks. Their model percentiles agree if both distributions are normal. In x = μ + zσ, different μ and σ can offset. A smaller σ makes the same positive gap a larger z. Standardising a normal model gives mean 0 and standard deviation 1. Probability in a continuous normal model is area under the curve; total area is 1. An exact model value has no width, so its probability is zero. A recorded score rounded to 60 represents 59.5 ≤ X < 60.5, an interval that can have positive probability. A continuous model may approximate discrete data. Check observed grouped frequencies for fit; discreteness alone does not decide whether the approximation is useful. **Recap.** For normal models, about 68% lie within one σ and 95% within two. With σ > 0, z counts signed standard deviations. Probability is area; exact continuous model values have probability zero. *Lesson, worked questions and flashcards authored for this learning dataset.* ## Practice 30 questions on this page, each with its working shown. ## Needs first - [Standard deviation and variance](https://lightmysky.com/learn/mathematics/standard-deviation-and-variance-mt_eNpow5tIX2) - [The Binomial Distribution](https://lightmysky.com/learn/mathematics/the-binomial-distribution-mt_X2kVGK67Md) ## Opens up - [Continuous Random Variables and Density Functions](https://lightmysky.com/learn/mathematics/continuous-random-variables-and-density-functions-mt_jlf4zDTeNm) - [Finding Probabilities and Values from a Normal Model](https://lightmysky.com/learn/mathematics/finding-probabilities-and-values-from-a-normal-model-mt_naXMRySvos) - [The Maxwell-Boltzmann Distribution of Molecular Speeds](https://lightmysky.com/learn/science/the-maxwell-boltzmann-distribution-of-molecular-speeds-mt_njr5r5YtuT)