# Exploring Distribution mode and hypothesis testing on the fx-CG50

When we ask teachers and students to list the biggest benefits of the Casio fx-CG50 graphic calculator, one of the most common responses we receive is ‘Distribution mode’.

This dedicated app for calculating and graphing probability distributions was introduced as part of the update to OS 3.6. If you’re running an earlier operating system, you can download the latest version here.

In this article, we explore what makes Distribution mode so useful and how it can be applied to hypothesis testing, with some insights from mathematician and fx-CG50 expert, Simon May.

## What sets Distribution mode apart?

Since the fx-CG50 already has a dedicated Statistics mode with a lot of functionality related to probability distributions, you might ask what makes Distribution mode different.

One key distinction is the ease and speed of drawing probability distributions. That includes binomial vertical line graphs, which are among the various visualisation options available in Distribution mode.

There is certainly a lot you can do in Statistics mode, such as performing probability calculations, finding the values you need and displaying them on a graph. On the Distribution app, however, you’ll find the process is more seamless and intuitive, with graphs appearing automatically and highlighting relevant values, based on your input data.

“It’s a well-designed app and it can make the raw calculations a lot easier, which should make understanding stronger as well,” Simon said. “It can help with the crisis of confidence that students might experience when they’re trying to work out exactly what they should be calculating.”

Another key improvement introduced with Distribution mode is an easier method of changing the probability you want to find. If you’re doing a hypothesis test with a statistical significance level of 5%, for example, you can enter that directly using the app and view critical values that fall inside and outside your chosen range.

If you’re comparing different calculator types, particularly with A-level maths in mind, the Distribution app is also a good example of what sets the fx-CG50 apart from a scientific calculator.

The new fx-991CW, our most advanced scientific calculator, is capable of doing exact and cumulative binomial calculations. However, it isn’t as seamless as the fx-CG50’s Distribution mode when it comes to tasks such as working out right-hand probabilities or cumulative calculations between given values.

And crucially, a graphic calculator gives you the option to draw distributions for deeper analysis and understanding.

## Hypothesis testing in Distribution mode

To put the potential uses of Distribution mode into context, Simon gave us some examples of questions and scenarios involving hypothesis testing.

Students could be told that 35% of the population have brown eyes and then asked to work out the probability of exactly eight people, from a sample of 12, having brown eyes.

This is possible by entering these values in Distribution mode, with a probability of 0.35, to produce a graph showing that the probability of exactly eight people out of 12 having brown eyes is 2%.

The app’s ability to calculate right-hand probabilities means it can also show that the probability of five or more people having brown eyes is 41.7%, for instance.

Taking this a step further to involve hypothesis testing, the question might ask students to find probabilities within a critical region of 5%. Again, this data can be input directly into the Distribution app, to draw a graph showing that, in this case, the 5% critical region equates to eight or more people out of 12 having brown eyes.

Distribution mode also provides the functionality to run two-tailed hypothesis tests. This comes in useful when students are told that the probability of a certain outcome has changed, but they don’t know in what direction or by how much.

They can enter all the relevant data into the app and choose to view both left-hand and right-hand probabilities, allowing them to study the likelihood of results both less and greater than the specified value.

## Distribution mode is a hit with students and teachers

We’ve been delighted to receive a lot of positive feedback about Distribution mode from teachers and students alike.

Former secondary school teacher turned YouTube educator Seb Bicen picked out this feature as one of his favourite tools on the fx-CG50. He highlighted the benefits of being able to take a visual, intuitive approach to a topic that students often struggle to get to grips with.

James Davis, Associate Assistant Headteacher and Maths Teacher at Newstead Wood School, said both he and his students were big fans of Distribution mode, largely because of its time-saving benefits and its emphasis on graphical visualisation.

To learn more about this app and how it can be used, head to our resources centre. Here you’ll find useful guides including a video on calculating single and binomial probabilities and determining critical values.

If you’re new to the fx-CG50 and looking for an introduction to what it can do, sign up for a free training session.