Normal distribution probability table

Introduction:

In this article we can study bell curve normal distribution, which figure most significantly in statistical theory and in application. Normal distribution is also called as the Normal probability distribution. Let us see how to calculate normal distribution with the help of normal distribution table. The normal distribution probability looks like a bell shaped curve. Hence it is also known as bell curve normal distribution probability.

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Definition of normal distribution probability:

A continuous random variable X is said to follows a normal distribution with parameter μ and σ (or μ and σ2) if the probability functions is

f(x) = (1/ σ 2π) e – ½ ((x – μ)/σ)2        ; −∞ < x < ∞, − ∞ < μ < ∞, and σ > 0.

This is the formula which is used to show normal distribution probability table.

Constants of Normal distribution:

Mean = μ

Variance = σ2

Standard deviation = σ

Thegraph of the normal curve is shown above. The shape of the curve is bell. These are the constants which tell how to calculate normal distribution probability by using table.

Normal distribution table:

The following are the value which resides in the normal distribution table.

This is the normal probability distribution table.

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Examples:

Examples:

If X is normally distributed with mean 3 and standard deviation 2 find. (i) P(0 ≤ X ≤ 4) (ii) P( | X − 3 | < 4).

Solution:

Given μ = 3, σ = 2

(i) P (0 ≤ X ≤ 4)

We know that Z = (X – μ) / σ

When X = 0, Z = (0 – 3) / 2 = −3 / 2 = − 1.5

When X = 4, Z = (4 – 3) / 2 = 1 / 2 =0.5

P (0 ≤ X ≤ 4) = P (−1.5 < Z < 0.5)

= P (0< Z <1.5) + P (0 < Z < 0.5)

= 0.4332 + 0.1915

= 0.6249

(ii) P (| X − 3| < 4) = P (−4 < (X − 3) < 4) ⇒ P (−1 < X < 7)

When X = −1, Z = (−1 – 3)/2 = −4/2 = − 2

When X = 7, Z = (7 – 3)/2 = 4/2 = 2

P (−1 < X < 7) = P (−2 < Z < 2)

=  P (0 < Z < 2) + P ( 0 < Z < 2)

=  2(0.4772)

= 0.9544

These are the examples of normal distribution calculated by using normal distribution table.