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Normal distribution

Probability distributions

Japanese school year: Math B

What you learn

This topic introduces the bell-shaped normal distribution, teaching standardization techniques and how to calculate probabilities using distribution tables. It is fundamental to statistical inference, standard scores, and experimental error analysis across sciences. Prior knowledge of continuous random variables, mean, and variance is essential.

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Key points

This formula standardizes a data value XX with mean mm and standard deviation σ\sigma into a common scale ZZ with mean 0 and standard deviation 1. It is used to compare results fairly across tests with different averages.

Z=X−mσZ = \frac{X - m}{\sigma}

This property states that the standardized variable ZZ always has an expected value E(Z)E(Z) of 0 and a variance V(Z)V(Z) of 1. It allows different datasets to be evaluated on the same unified scale.

E(Z)=0,V(Z)=1E(Z) = 0, \quad V(Z) = 1

This formula gives the height of the symmetric bell-shaped curve (normal distribution) centered at mean mm. It shows the probability density at value xx, where a larger standard deviation σ\sigma creates a flatter, wider curve.

f(x)=12πσe−(x−m)22σ2f(x) = \frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{(x-m)^2}{2\sigma^2}}

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