How to Find Probability Using the Mean and Variance in Normal Distribution? (original) (raw)

Last Updated : 23 Jul, 2025

You can calculate the probability in a normal distribution using the z-score formula: _P (_X < _x) = Φ(_x – __μ_​)/_σ, where Φ is the cumulative distribution function, _x is the value, _μ is the mean, and _σ is the standard deviation.

To calculate the probability in a normal distribution given the mean (_μ) and variance (__σ_2), you can use the z-score formula along with the standard normal distribution. The formula is:

**P (X<x) = Φ (x–μ​)/σ

Here's a detailed explanation of the steps involved:

  1. **Understand the Components:
    • _P(_X < _x): This represents the probability that a random variable _X in a normal distribution is less than a specific value _x.
    • **Φ: This symbolizes the cumulative distribution function (CDF) of the standard normal distribution.
    • _****(x–μ​)/σ**_​: This part calculates the z-score, representing how many standard deviations a particular value _x is from the mean _μ in terms of the standard deviation _σ.
  2. **Calculate the Z-Score:
    • Subtract the mean (_μ) from the specific value _x.
    • Divide the result by the standard deviation (_σ).
    • The z-score (****(x–μ​)/σ**) tells you how many standard deviations the specific value _x is from the mean.
  3. **Use the Standard Normal Distribution Table:
    • Once you have the z-score, you can use a standard normal distribution table to find the cumulative probability.
    • The cumulative probability (Φ) gives the probability that a standard normal random variable is less than or equal to the calculated z-score.
  4. **Interpret the Result:
    • The final result _P(_X < _x) is the probability that a random variable _X in the normal distribution is less than the specific value _x.

**Example: Suppose you have a normal distribution with a mean (_μ) of 50 and a variance (__σ_2) of 25. You want to find the probability that _X is less than 55.

Calculate the z-score: _(x–μ​)/σ = (55-50)/√25 = 5/5 = 1

Use the standard normal distribution table or calculator to find Φ (1). For z = 1, Φ (1) is approximately 0.8413.

Interpret the result: The probability _P(_X < 55) is approximately 0.8413, meaning there's an 84.13% chance that a randomly selected value from the distribution is less than 55.