Number's Between 0 and 100

A random variable with a normal distribution is called normally distributed. For a normally distributed random variable X with mean 𝜇 and standard deviation 𝜎:

P(𝜇–𝜎<X<𝜇+𝜎) ≈ 0.68

P(𝜇–2𝜎<X<𝜇+2𝜎) ≈ 0.95

P(𝜇–3𝜎<X<𝜇+3𝜎) ≈ 0.997

In other words: The probability that X is within one standard deviation of the mean is P≈0.68. The probability that X is within two standard deviations of the mean is P≈0.95. The probability that X is within three standard deviations of the mean is P≈0.997.

The probability that X is between 63 and 71 can be written as P(63<X<71).

Write 63 and 71 as the mean, 𝜇=67, plus or minus a multiple k of the standard deviation, 𝜎=2: 63 = 𝜇+𝜎k 63 = 67+2k-4 = 2k-2 = k

So, 63=𝜇–2𝜎. By a similar calculation, 71=𝜇+2𝜎. This means P(63<X<71)=P(𝜇–2𝜎<X<𝜇+2𝜎). Since X is normally distributed:

P(𝜇–2𝜎<X<𝜇+2𝜎)≈0.95 So, P(63<X<71)=P(𝜇–2𝜎<X<𝜇+2𝜎)≈0.95. Got it