Area under Normal Distribution Revision Notes for Leaving Cert Mathematics

Area under Normal Distribution

Overview

The normal distribution is a bell-shaped probability distribution that is symmetric about the mean. It is widely used in statistics to model real-world phenomena. The area under the normal distribution curve represents probabilities.

To compute probabilities for a normal distribution, we calculate the area under the curve for a specific range of values. This area corresponds to the probability of a value falling within that range.

Standard Normal Distribution

A standard normal distribution has:

A mean ( $\mu$ ) of 0.
A standard deviation ( $\sigma$ ) of 1.

Z-Scores

To compare values from different normal distributions, we convert them into Z-scores using the formula:

Z = \frac{X - \mu}{\sigma}

Where:

$X$ : the value in the dataset.
$\mu$ : the mean of the dataset.
$\sigma$ : the standard deviation of the dataset. The Z-score indicates how many standard deviations a value is from the mean.

Finding the Area under the Curve

Convert to Z-Score: Use the formula to find the Z-score for the value(s).
Use the Z-Table or Calculator:

A Z-table provides the cumulative area (probability) to the left of a given Z-score.
For probabilities between two values, subtract the smaller Z-area from the larger Z-area.

Interpret the Area: The area represents the probability of a value falling within the specified range.

Worked Examples

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Example 1: Finding the Probability of a Value

Problem: In a distribution with $\mu = 100$ and $\sigma = 15$ ,

what is the probability that $X≤120X \leq 120$ ?

Solution:

Step 1: Calculate the Z-score:

Z = \frac{120 - 100}{15} = \frac{20}{15} = 1.33

Step 2: Use the Z-table to find the cumulative area for $Z = 1.33$

P(X \leq 120) = 0.9082

Answer: The probability is 0.9082 or 90.82%.

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Example 2: Probability between Two Values

Problem: In a standard normal distribution, what is the probability that $-1 \leq Z \leq 1.5$ ?

Solution:

Step 1: Find the area for $Z=−1$ and $Z = 1.5$ from the Z-table:

For $Z = -1$ , the cumulative area is 0.1587

For $Z = 1.5$ , the cumulative area is 0.9332

Step 2: Subtract the smaller area from the larger area:

P(-1 \leq Z \leq 1.5) = 0.9332 - 0.1587 = 0.7745

Answer: The probability is 0.7745 or 77.45%.

Summary

The area under the normal distribution curve represents probabilities.
Convert raw data to Z-scores using the formula:

Z = \frac{X - \mu}{\sigma}

Use the Z-table or calculator to find the cumulative probability for a Z-score.
To find the probability between two values, subtract the cumulative areas.
The total area under the curve is always 1, representing a 100% probability.

Area under Normal Distribution (Leaving Cert Mathematics): Revision Notes