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.
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.
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.
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.
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.
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.
Readings and extra questions authored for this learning dataset. Video references credit Khan Academy.
You have measured spread with standard deviation. Mia scored 76 in the quiz, where μ = 60 and σ = 8, and 50 at the plant show, where μ = 42 and σ = 5.
Sam says the raw marks prove which result is farther above its mean in standard-deviation units. What is missing?
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