Ideas for Extension 
The following ideas for extending the topic of constructions require the full version of Autograph. Click on the image to download the individual Autograph files. 

Note: To create each of these files, please ensure you open a new Statistics Page and you are in Advanced Mode. 

Idea 1 – Constructing a Probability Distribution 
Download 1. Probability Distributions.agg 


Autograph allows you to generate user defined probability distributions and take samples from them. Here we will construct a biased dice. 

• 
Click on Enter Raw Data 
• 
Choose Select Distribution, click on User (discrete) and click OK 
• 
Now click on Edit Distribution 
• 
Here we can specify the proability of getting a score of 0, 1, 2, 3… etc 
• 
We want a biased dice, so type in 0, 1/9, 1/9, 1/9, 1/9, 2/9, 3/9 and click OK 
• 
Now let’s generate a sample, so change the sample size to 1000, click Create Sample and then OK 
• 
What do you think the graph of 1000 rolls of this biased dice will look like? 
• 
Select Dot Plot and ensure both spacings are set to 1 
• 
Click on Default Scales if your graph goes off the grid 
• 
What do you think the mean and standard deviation of this distribution are? 
• 
Click on View Statisitics Box to find out 
• 
To change the distribution or sample size, simply doubleclick on Raw Data 1 at the bottom left of the screen 
• 
What would it look like using only 10 rolls of the dice? 
• 
Can you construct a symmetrical probability distribution for a biased dice? 
• 
We will be using this file for Idea 4 below 

Idea 2 – Binomial Distribution 
Download 2. Binomial Distribution.agg 


Here we will look at carrying out calculations and investigating the shape of the binomial distribution 

• 
Click on Enter Probability Distribution 
• 
Choose Binomial and click OK 
• 
Set n (the number of trials) to 8 and p (the probability of a success) to 0.5 
• 
What do you think this distribution will look like? 
• 
Click OK 
• 
Click on Probability Calculations and click OK 
• 
You can drag the ends of the band at the bottom of the graph over the region you want to calculate the probability of 
• 
Click View > Status Box to keep an eye on the probability calculations 
• 
Work out the proability that r is between 4 and 6 and then check it 
• 
Double clicking on the Discrete Probability Calculations allows you to change to cumulative probability measures, such as r is less than or equal to 4 
• 
What do you think would happen to the shape of the graph if we changed the probability of success to 0.8? 
• 
Click on one of the bars of the graph to select the distribution 
• 
Click on Animate and experiment by increasing and decreasing the value of p 
• 
You can also change the value of n and observe the effect on the shape of the graph and the probability calculations 

Idea 3 – Standardising the Normal Distribution 
Download 3. Standardising the Normal Distribution.agg 


A key skill students must acquire is to be able to convert other normal distributions into the standard normal distribution (mean 0 and variance 1) to be able to calculate probabilities. Autograph has a really nice way of enabling students to visualise what is going on. 

• 
Click on Enter Probability Distribution 
• 
Choose Normal and click OK 
• 
Set μ (the mean) to 10 and σ^{2} (the variance) to 4 
• 
What do you think this distribution will look like? 
• 
Click OK 
• 
Repeat the above to also produce the standard normal distribution (mean 0 and variance 1) 
• 
Adjust your axes if you need to 
• 
Why are the graphs different heights? 
• 
Click on the first normal distribution 
• 
Click on Probability Calculations and set it to between 6 and 12 
• 
What region will this be on the graph? 
• 
Click OK 
• 
Select the region and click on Text Box to display the calculation 
• 
Now, standardise the distribution and convert this into a probability calculation that you can work out using the standard normal distribution tables 
• 
When you have your answer, select the standard normal distribution 
• 
Click on Probability Calculations and set it to the appropriate values from your calculation and click OK 
• 
If you are correct, your two probabilities should be the same 
• 
Do the regions look the same size? 
• 
You can change the probability calculation by double clicking on the region, and change the parameters of the normal distribution by clicking on the curve itself, thus allowing you to test and visualise any question you may get. 

Idea 4 – Central Limit Theorem 
Download 4. Central Limit Theorem.agg 


Finally we turn our attention to one of the most important concepts in statistics – the Central Limit Theorem. Autograph can really help bring the concept to life 

• 
Open the original biased coin file that you created during Idea 1 
• 
We are going to take samples of size two from this distribution and plot their means 
• 
What do you think the distribution of these sample means will look like? 
• 
Click on Sample Means 
• 
Click on Edit Dot Plot and set the horizontal spacing to 0.1 and the vertical spacing to 1 and click OK 
• 
Change the sample size (n) to 2, the number of samples taken to 20 and click Sample 
• 
Can you describe the shape of the distribution? 
• 
Click Sample again to sample another 20 
• 
Keep taking samples unril you see the shapoe of the distirubiton emerge 
• 
What do you think would happen if instead we had a sample size of 5? 
• 
Click Clear Sample and change n to 5 
• 
Repeat the above process 
• 
Describe the shape of the distribution now. Can you explain this? 
• 
Now try n = 10. How about n = 30? 
• 
How does this relate to the Central Limit Theorem? 
• 
You can use this technique to investigate the distribution of sample means from any initial distribution. Just use the instructions from Idea 1 to create the probability distirubiotn of your choice and take it from there! 
