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IB Mathematics AI SL The normal distribution MAI Study Notes - New Syllabus

IB Mathematics AI SL The normal distribution MAI Study Notes

LEARNING OBJECTIVE

  • The normal distribution and curve.

Key Concepts: 

  • The normal distribution and curve/ Diagrammatic representation. 
  • Properties of the normal distribution.
  • Normal probability calculations. / Inverse normal calculations

MAI HL and SL Notes – All topics

NORMAL DISTRIBUTION

◆ Normal Distribution $N(\mu, \sigma^2)$

A normal distribution is a continuous probability distribution that is symmetric about the mean $\mu$, with a bell-shaped curve.


It is fully defined by two parameters:

$\mu = \text{mean}, \quad \sigma = \text{standard deviation}$

$X \sim N(\mu, \sigma^2)$

The variable $X$ can take any value from $-\infty$ to $+\infty$, though practically most values lie within $\mu \pm 3\sigma$.

◆ Characteristics of the Curve

The total area under the curve is 1.
About 68% of data lies within $\mu \pm \sigma$.
About 95% lies within $\mu \pm 2\sigma$.
About 99.7% lies within $\mu \pm 3\sigma$. (Empirical Rule)

◆Applications

The normal distribution models real-world phenomena such as:

Human heights and weights
Product measurements (e.g., coffee packet mass)
Time durations (e.g., time spent in a store)

$\mathbf{GDC~ for ~Normal~ Distribution~Casio~ fx:}$

$\text{MENU → STAT → DIST → NORM → Ncd / InvN}$

$\text{Function}$$\text{Use Case}$
$\text{Ncd}$$\text{When finding a probability}$
$\text{InvN}$$\text{When probability is given, find the value}$

Example

Graph and examine a situation where the mean score is 46 and the standard deviation is 8.5 for a normally distributed set of data.

▶️ Answer/Explanation

Solution:

 

Examine:  

What is the probability of a value falling between the mean and the first standard deviation to the right?  Answer:  approximately 34%
Notice how this percentage supports the information found in the chart at the top of this page for the percentage of information falling within one standard deviation above the mean.

NORMAL PROBABILITY CALCULATION

 Normal Probability Calculations

In a normally distributed dataset, the probability that a value lies within a certain range can be found using a GDC or technology. The distribution is defined as \( X \sim N(\mu, \sigma^2) \), where:

  • \(\mu\) is the mean
  • \(\sigma\) is the standard deviation

Probabilities such as \( P(X < a) \), \( P(X > b) \), or \( P(a < X < b) \) are computed directly using calculator functions like normalcdf without the need for standardizing to z-scores.

Example

The weights of a certain species of birds are normally distributed with mean 250g and standard deviation 15g. Find the probability that a bird weighs less than 265g.

▶️ Answer/Explanation

Solution:

Let \( X \sim N(250, 15^2) \)

We want to find \( P(X < 265) \)

Use your GDC:
normalcdf(-1E99, 265, 250, 15)

Result: ≈ 0.8413

So, the probability that a bird weighs less than 265g is approximately 0.8413.

INVERSE NORMAL CALCULATION

Inverse Normal Calculations

Inverse normal calculations determine the value of the variable \( x \) given a probability. The mean and standard deviation are provided, and the technology gives the x-value directly from the cumulative area without needing to convert to a z-score.

This is typically done using the invNorm function in a GDC:

  • invNorm(area, μ, σ)

Note: This topic does not require transformation to the standardized variable \( z \).

Example

The heights of students in a school are normally distributed with a mean of 170 cm and standard deviation of 8 cm. Find the height corresponding to the 90th percentile.
▶️ Answer/Explanation
Solution:

Let \( X \sim N(170, 8^2) \)

We want the value of \( x \) such that \( P(X < x) = 0.90 \)

Use your GDC:
invNorm(0.90, 170, 8)

Result: ≈ 180.25 cm

So, the 90th percentile corresponds to a height of approximately 180.25 cm.
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