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 binomial's bar chart of counts humps up around its mean. Raise the number of goes and the hump grows smooth and even on both sides. Heights, reaction times and exam marks arrive with the same shape for reasons of their own, and a smooth curve fits it closely: the normal distribution. Two numbers describe the whole thing, the mean at the centre, written μ, and the standard deviation, written σ, which says how wide the hump is.
Two facts do most of the work. About 68 percent of the values sit within one σ of μ, and about 95 percent sit within two. The quiz had μ = 60 and σ = 8. So about 68 percent of members scored between 52 and 68, and about 95 percent between 44 and 76. That leaves about 5 percent outside the wider band, split evenly, so roughly 2.5 percent scored above 76.
Counting standard deviations is the useful move. The z-score of a value is z = (x - μ) / σ, which is how many standard deviations the value sits above the mean. Mia scored 76 in the quiz, so her z is (76 - 60) / 8 = 2, two standard deviations above the middle. Below the mean gives a negative z; at the mean it is zero. Inside one set each value loses the same μ and splits by the same σ, so order stays.
The point of z is that it makes different contests comparable. The plant show used a different scale, with μ = 42 and σ = 5, and Mia scored 50 there. Her z is (50 - 42) / 5 = 1.6. The raw 76 and 50 cannot be set against each other. The z-scores 2 and 1.6 can, and they say the quiz was her stronger result.
This comes straight from histograms. There, the count in a class was the area of a bar and not its height. A normal curve is that picture with infinitely thin bars, so a proportion between two marks is area, not height. Stepped values with gaps are the wrong fit for a curve that spreads over every value between. Scoring exactly 60 has chance zero: a point has no width, so no area.
A normal distribution is the smooth hump described by its mean μ and standard deviation σ. About 68 percent of values sit within one σ of the mean and about 95 percent within two. The z-score z = (x - μ) / σ counts how many standard deviations a value sits above the mean, which is negative below it, and z makes results from different scales comparable. Probability here is the area under the curve between two values, never the height at a point.
Last stop you measured spread around the mean with the standard deviation. The fair is over and Wren Lane is handing out its summer awards. Two contests to compare: the quiz night, marked with a mean of 60 and a standard deviation of 8, and the plant show, judged with a mean of 42 and a standard deviation of 5. Mia scored 76 in the quiz and 50 at the plant show.
Sam says the quiz was obviously her better result, because 76 beats 50. What is wrong with that?
Jobs that lean on this skill. Follow one to see everything it is built on.
24 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.