5933 /www/nrich/html/content/id/5933/ 1 2011-02-01T00:00:01 <mdoxml version="1.0"><br></br> Four different, unknown, probability distributions have each been used four times to generate 16 sets of data. By searching for patterns in the data, can you work out which groups of 4 are most likely to have come from the same distributions? <br></br> <br></br> <mdo:image width="572" height="825" alt="" src="dataSet1.JPG"></mdo:image><br></br> If you like, you may download a <a href="/content/id/5933/5933.xls">spreadsheet</a> containing these numbers and an additional data set.<br></br></mdoxml> <?xml version="1.0" encoding="ISO-8859-1"?> <mdoxml version="1.0"><br></br> Patrick from Woodbridge school sent in the following solution:<br></br> <br></br> &quot;I made frequency graphs of the data sets and I tried to find similar features.<br></br> <br></br> 1. A G K M have no odd numbers so they are very likely to be one set.<br></br> 2. D E L N have no early numbers or late numbers (they are all grouped around the middle)<br></br> 3. C F H P have the same general shape - a rise up to 17 ish, then a very steep drop.<br></br> 4. B I J O each have almost all the numbers centred in two peaks at around 7 and 13.<br></br> <br></br> E could possibly be in group 4. C and H are not quite as well defined as the others. I would say that the best example of my definitions in each group is:<br></br> <br></br> 1. (none needed)<br></br> 2. D<br></br> 3. F<br></br> 4. B<br></br> <br></br></mdoxml> <mdoxml version="1.0"><br></br> <h3>Why use this problem</h3> <p>This problem will help students to understand the concept of a probability distribution: various results are possible, and each result occurs with a certain probable frequency. It will help students to understand that there is some structure in random processes, even though individual parts of the problem are random.<br></br>  </p> <h3>Possible Approach</h3> <p>Cut out the 16 grids and ask students to work in pairs to group the numbers into four sets of four.<br></br> <br></br> Once the students have grouped the pairs ask them to explain in turns to the rest of the group the reasoining behind their grouping. Encourage them to present their arguments as clearly as possible. Do the others agree or disagree? Can the others refine their argument?<br></br> <br></br> Once the cards are sorted, can students suggest the probability distributions from which the cards were drawn? They can use the <a href="http://nrich.maths.org/public/viewer.php?obj_id=5932&amp;part=">distribution maker</a> interactivity to help.<br></br>  </p> <h3>Key Questions</h3> <ul> <li>Describe the cards in words.</li> <li>How might you start to quantify the data on each card more precisely? How would you represent this?</li> <li>Can you spot any patterns occurring in some of the cards? Does this help you to give a grouping?</li> </ul> <ul> <li>Can you know for sure that you have correctly grouped the cards? Can you think of any other sensible ways in which the cards may have been grouped?</li> </ul> <h3>Possible Extension</h3> <p>Can you suggest possible distributions from which the numbers on each card were drawn? Can you suggest distributions from which the cards were unlikely or impossible to be drawn?<br></br>  </p> <h3>Possible Support</h3> <div>Students struggling to start should be asked to make frequency and cumulative frequency tables. Can they spot any patterns in these? If your students are having trouble getting started, you might like to try the similar Stage 4 problem <a href="http://nrich.maths.org/7731">Which list is which?</a></div> <br></br></mdoxml> <mdoxml version="1.0"><p>Try constructing frequency tables.</p> <p> </p> <p>If you are having trouble getting started, you might like to try the similar Stage 4 problem <a href="http://nrich.maths.org/7731">Which list is which?</a></p></mdoxml> <mdoxml version="1.0"><br></br> The four distributions used are Spiky, Stepped, Twin and Normal. The cards are grouped as follows<br></br> <br></br> <br></br> <mdo:image width="325" height="275" alt="" src="solution.JPG"></mdo:image><br></br> The distributions used look like:<br></br> <mdo:image width="352" height="259" alt="" src="spiky.JPG"></mdo:image><br></br> <mdo:image width="351" height="224" alt="" src="stepped.JPG"></mdo:image><br></br> <mdo:image width="353" height="196" alt="" src="twinTowers.JPG"></mdo:image><mdo:image width="352" height="253" alt="" src="normal.JPG"></mdo:image><br></br> <br></br> <br></br></mdoxml> 2 3 0 0 0 0 1 Use your skill and judgement to match the sets of random data. Handling, Processing and Representing Data Handling data Handling, Processing and Representing Data Processing and representing data Handling, Processing and Representing Data Interpreting data Handling, Processing and Representing Data Frequency distribution Advanced Probability and Statistics Probability distributions, expectation and variance Applications Maths in STEM Admin Discussion