The normal distribution on the CG50: every button you actually need

The normal distribution on the CG50: every button you actually need

The normal distribution is a central topic in IB Maths, particularly in AI SL/HL and AA SL/HL Paper 2. My students often find the concepts straightforward, but translating that understanding into correct calculator steps can be a hurdle. This article cuts through the noise. We are focusing on the Casio CG50, and I will show you precisely which functions you need for the normal distribution questions you will face in your exams. No filler, just the buttons that matter.

My goal here is to demystify the normal distribution functions on your CG50. You do not need to memorise obscure menu trees. You need to know how to calculate probabilities, find inverse values, and sketch the curve for visualisation. These are the core skills that secure marks in the exam. I have taught this for over a decade, and I have seen students gain significant confidence once they master these specific steps.

Standard Normal vs. General Normal

Before we dive into the calculator, a quick refresher. The standard normal distribution, denoted $Z \sim N(0, 1)$, has a mean of $0$ and a standard deviation of $1$. The general normal distribution, $X \sim N(\mu, \sigma^2)$, has a mean $\mu$ and a standard deviation $\sigma$. Your CG50 handles both seamlessly. You do not need to standardise values to $Z$-scores first, unless the question explicitly asks for it or provides a $Z$-table.

All the functions we will use are found under MENU $\rightarrow$ STAT. Once in STAT mode, press F5 (DIST) to access the distribution functions. Then press F1 (NORMAL).

Calculating Probabilities: Normal CD

This is the most common task: finding the probability that a random variable falls within a certain range. For example, $P(X < x)$, $P(X > x)$, or $P(x_1 < X < x_2)$. This function is called "Normal CD" on your CG50.

From the NORMAL menu (MENU $\rightarrow$ STAT $\rightarrow$ F5 (DIST) $\rightarrow$ F1 (NORMAL)), select F2 (Ncd).

You will see a screen prompting for "Data: Variable" (which is always what you want for normal distribution problems, unless you are dealing with a frequency table, which is rare for normal distribution itself). Set "Data" to "Variable" (F2).

• Lower: This is the lower bound of your interval. For $P(X < x)$, use a very small negative number like $-10^{99}$ (you can type this as -1E99) as the lower bound.
• Upper: This is the upper bound of your interval. For $P(X < x)$, this would be $x$. For $P(X > x)$, use a very large positive number like $10^{99}$ (1E99) as the upper bound.
• $\sigma$: The standard deviation. Remember this is $\sigma$, not $\sigma^2$ (variance). If the problem gives variance, you must take the square root.
• $\mu$: The mean.

Let's say a random variable $X \sim N(50, 4^2)$. We want to find $P(45 < X < 55)$.

• Lower: $45$
• Upper: $55$
• $\sigma$: $4$
• $\mu$: $50$

After entering these values and pressing EXE, the calculator will display the probability. For $P(X < 45)$, you would set Lower to $-10^{99}$ and Upper to $45$. For $P(X > 55)$, you would set Lower to $55$ and Upper to $10^{99}$.

Finding $x$ for a Given Probability: Inverse Normal

Sometimes you are given a probability and need to find the value of $x$ corresponding to it. For example, finding the $75^{th}$ percentile. This is "Inverse Normal" on your CG50.

From the NORMAL menu, select F3 (InvN).

You will see prompts for "Data: Variable", which you keep. The key setting here is "Tail".

• Area: This is the probability (as a decimal) you are given.
• $\sigma$: The standard deviation.
• $\mu$: The mean.
• Tail: This specifies which part of the distribution the "Area" refers to.
  • LEFT (F1): The area is to the left of the unknown $x$ value. This is used for $P(X < x) = \text{Area}$. This is the most common use.
  • RIGHT (F2): The area is to the right of the unknown $x$ value. Used for $P(X > x) = \text{Area}$.
  • CENT (F3): The area is centered around the mean. This is less common in IB, but useful for confidence intervals (HL AI, Paper 3). The calculator will find $x$ such that $P(\mu - x' < X < \mu + x') = \text{Area}$. It actually gives you the $x$ values for the lower and upper bounds of this central region.

Let's say $X \sim N(50, 4^2)$ and we want to find $x$ such that $P(X < x) = 0.75$.

• Area: $0.75$
• $\sigma$: $4$
• $\mu$: $50$
• Tail: LEFT (F1)

If you wanted $P(X > x) = 0.25$, you could use Area $0.25$ with Tail RIGHT, or equivalently, Area $0.75$ with Tail LEFT, as these describe the same $x$ value due to the symmetry of the distribution. Always sketch to be sure.

The "CENT" option is particularly useful in HL AI when discussing confidence intervals, where you might need to find the $z$-scores that define the central $95\%$ of the distribution. My HL AI Paper 3 notes cover this in detail.

Plotting the Normal Distribution Curve

Visualising the normal distribution is incredibly helpful. Your CG50 can plot the curve and shade regions for you. This is less about getting a numerical answer and more about understanding the shape and the probability's location.

To plot a normal distribution:

1. Go to MENU $\rightarrow$ GRAPH.
2. Delete any existing graphs (F2 $\rightarrow$ F1 $\rightarrow$ EXIT).
3. Press F3 (TYPE) and select F6 ($\rightarrow$) $\rightarrow$ F1 (DIST).
4. Select F1 (NORM) $\rightarrow$ F2 (Ncd) or F1 (Npd) if you want the PDF directly. Ncd (Normal Cumulative Distribution) is more common for probability regions.
5. Enter your $\sigma$ and $\mu$ values.
6. Set your X-range (SHIFT $\rightarrow$ F3 (V-Window)). A good rule of thumb is $\mu \pm 3\sigma$ for your Xmin/Xmax. So for $N(50, 4^2)$, set Xmin to $50 - 3 \times 4 = 38$ and Xmax to $50 + 3 \times 4 = 62$. Ymin should be $0$ and Ymax something like $1 / (\sigma \sqrt{2\pi})$ (the peak of the PDF), or just $0.1$ or $0.2$ to start with if you are unsure.
7. Press F6 (DRAW). You will see the normal curve.
8. To shade an area, press SHIFT $\rightarrow$ F5 (G-SOLV) $\rightarrow$ F1 (GRPH) $\rightarrow$ F3 (Integral) $\rightarrow$ F2 (Normal CD).
9. Enter your Lower and Upper bounds, $\sigma$ and $\mu$. The calculator will shade the region and give you the probability. This is a great way to double-check your Ncd calculations and to build intuition.

Plotting can really help you grasp the meaning of $P(X < x)$, especially when the value $x$ is far from the mean. I often encourage my students to use this function when they are first learning the topic, as it makes the abstract concept of probability density tangible. You can find more comprehensive calculator guidance on our CG50 Guide page.

Normal PD (Probability Density) - Less Common, But Good to Know

Within the NORMAL menu, you will also see F1 (Npd). This calculates the probability density function (PDF) value $f(x)$ for a given $x$, $\mu$, and $\sigma$. That is, it calculates $f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$.

This is rarely asked for directly in IB exams, as $f(x)$ itself is not a probability. Probabilities are found by integrating the PDF (which Ncd does for you). However, if you are asked to sketch the PDF by plotting points or to verify the maximum height of the curve (which occurs at $x=\mu$), this function can be useful.

For example, if $X \sim N(50, 4^2)$, Npd with $x=50$, $\sigma=4$, $\mu=50$ would give you the peak value of the curve. It is a good way to understand the shape, but Ncd and InvN are your primary tools for answering IB questions related to the normal distribution.

Final Thoughts and Practice

Mastering these three functions—Normal CD, Inverse Normal, and the ability to plot and shade—will equip you for almost any normal distribution question in IB Maths. The key is consistent practice. Do not just read this; open your CG50 and follow along with example problems from your textbook or past papers.

Normal distribution is a recurring topic in Paper 2 for both AA and AI students. The concepts are relatively contained, making it a reliable source of marks if you know your calculator inside out. Regularly reviewing these steps will make your exam experience smoother. Check out our study notes for more worked examples across all topics.