Autograph Newsletter 13 – Probability Distributions

Craig Barton
Welcome
Welcome to the thirteenth Autograph Newsletter! Each jam-packed edition looks at a specific topic in mathematics and how Autograph can help engage students and enable them to understand the key concepts better.
Contents
Introduction
Diagnostic Question
Free Online Autograph Activity
Ideas for Extension
Video Tutorials
Handy Autograph Tip
Introduction
Probability is (probably) my favourite topic in mathematics. I love the often counter-intuitive results, and how you can use knowledge of the subject to get one over on uninitiated “victims”! A key component of the topic is Probability Distributions, and I have found it frustrating in the past that students have found this particular area rather dry, abstract and uninspiring. I believe that is due to the fact that the calculations involved tend to be rather onerous, and distract from the crucial understanding behind the underlying concepts. Fortunately, Autograph can come to the rescue. The excellent statistics engine allows you to generate, sample, plot and investigate every probability distribution you could imagine, allowing your students to visualise, speculate, test and most importantly understand exactly what is going on.
Diagnostic Question
Diagnostic questions are ideal to use at the start of the lesson to enable you to get a quick and accurate picture of your students’ levels of understanding. They are designed in such a way that common misconceptions that your students may hold should steer them to one of the incorrect answers, thus allowing you to learn where the problems lie from their responses. Typically I give my class 30 seconds thinking time and then ask them to hold up their fingers: 1 for A, 2 for B, etc.
Box Plot
Free Online Autograph Activity
Standard Normal Distribution Calculator
Use this handy tool to check and visualise calculations using the standard normal distribution.
Normal Distribution Calculator
These Autograph activities do not require the full version of Autograph to run them. You just need to install the free Autograph Player (you will be guided through how to do this), which means you can use these activities in the classroom or set them for your students to do at home.
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 double-click 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 σ2 (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!
Video Tutorials
The following video takes you through a step-by-step guide as to how Autograph can be used to create Probability Distributions
Handy Autograph Tip
As you have seen, Autograph is excellent at generating data from probability distributions. But wouldn’t it be great to be able to export that data into other applications? Great news… you can!
Open Autograph in Advanced Mode and open a new Statisitics page
Click on Enter Raw Data
Choose Select Distribution, click on Binomial and click OK
Now click on Edit Distribution
Change n to 12 and p to 0.7 to simulate tossing a biased coin 12 times and counting the number of times you get heads, and click OK
To carry out the experiment 50 times, change the Sample Size (N) to 50
Click Create Sample
To export this data as a comma separate variable (.csv) file, simply click Export
Alternatively, click on one of the pieces of data and on your computer press “Ctrl” and “a” together to select all of the data
Now right-click and select Copy
You can now paste this data into Excel, Word, PowerPoint, or any application you should choose!

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