Section 7.2

Applications of the Normal Distribution

Learn to standardize variables using Z-scores, calculate probabilities for any normal distribution, and find values given a probability.

1

Standardizing with Z-Scores

The Standard Normal Distribution is a special case where and .

We can transform any random variable into a standard normal variable :

Notation:

The notation represents the Z-score such that the area to its right is .

2

Find Value of Random Variable

Sometimes we know the probability (or percentile) and need to find the specific value . This is often called an "Inverse Normal" calculation.

Procedure

  1. Identify the area to the left ($p$).
  2. Find the corresponding Z-score from the table or calculator.
  3. Use the formula:
Formula
3

Find and Interpret Area

Since area under the curve represents probability, we can find the probability of any event by calculating the area to the left of .

Area to Left

Directly from table or calculator.

P(X < x)

Area to Right

Use Complement Rule.

1 - P(X < x)

Between Two Values

Subtract the smaller area from the larger.

P(X < x₂) - P(X < x₁)

Normal Probability Calculator

Use the Find Probability tab below. Enter your Mean and Std Dev, then select the region (Left, Right, or Between) to calculate the area.

Distribution Parameters

The mean (mu) of the normal distribution
The standard deviation (sigma) of the normal distribution
Region Type

Type a value between 0 and 1 to reverse-solve for x

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