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Frequency tables

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Survey results Jamilia carries out a survey to find out how many sports the students in her school do. She lists the responses in her notepad. It is not easy to see patterns or trends in the data. How could Jamilia use a table to make the results easier to read?

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**Discrete data in frequency tables**

Jamilia decides to write all the possible results in one column of a table and record how often they occur. This is called a frequency table. number of sports played frequency 20 1 17 2 15 Teacher notes Note that it can be useful to have a bottom row for the total number of data items. To ensure that students fully appreciate the links between the list of data and the frequency table, stay on this slide once everything is visible and ask them questions about the data. For example, ask them to find the mode, median and mean. Students may calculate these using the list of values or the frequency table. Students should not give 20 as the mode – the mode is 0, 20 is the number of times it occurred. The median is the middle item if the data was written in order, and 76 students were asked so the median is halfway between the 38th and 39th result. This is easy to find using the table, as we can see easily that = 37, so the 38th and 39th results are 2, so the median is 2. The method for finding the mean from a frequency table is given on the next slide. 3 10 4 9 5 3 Use the list to fill in the frequency table for Jamilia. 6 2

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**numbers of sports played**

Calculating the mean How can you find the mean number of sports? ∑(data value × frequency) Total frequency Multiply each data value by its frequency. numbers of sports played number of sports × frequency frequency 20 0 × 20 = 0 Add these values together. 1 17 1 × 17 = 17 2 15 2 × 15 = 30 Divide the sum by the total frequency. 3 10 3 × 10 = 30 Teacher notes Students may benefit from calculating the mean from the list on the previous slide first. They can then see that multiplying the number of sports by the frequency, and then adding together all the products, is the same as adding together all the responses in the list. Discuss a suitable level of accuracy for rounding off in the context of discrete data, and emphasize that no student can play sports. Mathematical Practices 4) Model with mathematics. This slide demonstrates finding the mean from grouped data in a given context. Students should use the formula and apply it to the table of data, then state their answer in the context of the question; it should be rounded to the nearest whole number because it represents the number of sports people play. 6) Attend to precision. This slide demonstrates how students should be careful about expressing numerical answers with a degree of accuracy appropriate for the problem context. They should notice that ‘number of sports’ should be a whole number, and therefore round their final answer to the nearest whole number. 4 9 4 × 9 = 36 5 3 5 × 3 = 15 140 76 6 2 6 × 2 = 12 mean = TOTAL 76 140 = 2 sports

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Continuous data Here are the race times in seconds from a downhill race event. Putting these into a frequency table as they are will not be helpful. Instead we can group the times into intervals. 5

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**Notation for class intervals**

Louise decides to create her own groups and draws a table with class intervals that she thinks fit the race data. What is wrong with this table? How should the class intervals be written down? How can your knowledge of inequalities help you to create better class intervals? Times in seconds Frequency 85 – 90 90 – 95 95 – 100 100 – 105 105 – 110 Teacher notes Students should see that some data types could have multiple entries on the table. If students do not see this, encourage them to work through the data list putting the times in the appropriate space. Discuss where 90.0, and should go. The table is ambiguous. For discrete data it would be possible to edit the table to say 86 – 90, 91 – 95, 96 – 100, 101 – 105 etc (or 85 – 89, 90 – 94 etc) but for continuous data this would not work. The students should use inequalities to clarify the boundaries of the data intervals. 6

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Intervals time in seconds frequency 85 ≤ t < 90 90 ≤ t < 95 95 ≤ t < 100 100 ≤ t < 105 105 ≤ t < 110 Use the original data from the race to complete the frequency table to show the number of times in each interval. 1 Teacher notes Note that as the data has been rounded, if a time had been recorded as 90.0 seconds then the actual time could be anything between seconds and seconds. This means that some times may be in the wrong interval. 5 28 19 7

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**Two-way frequency tables**

Two-way frequency tables are used to examine the relationship between two categories or groups. For example, Rosa asked two hundred people what type of drink they had in a local coffee house. She recorded the results in this two-way frequency table. special hot drink special cold drink regular coffee total women 10 58 42 110 men 56 10 24 90 Teacher notes Rosa is comparing men and women in the table. Mathematical practices 6) Attend to precision. Students should learn the proper use of the terms “two-way frequency table” and use this technical terms in their discussions of the data. total 66 68 66 200 What two categories is Rosa comparing in the table? 8

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**Joint and marginal frequencies**

special hot drink special cold drink regular coffee total women 10 58 42 110 men 56 10 24 90 total 66 68 66 200 The numbers in the body of the table, shown in pink, are called joint frequencies. The totals, shown in blue, are called marginal frequencies. Teacher notes Students may notice many trends, for example: Most of the women preferred special hot drinks; 58 of 110. Most men preferred regular coffee; 56 of 90. Most of the women prefer special drinks in general; 100 of 110. If we look at the marginal frequencies of 66, 68, 66, we might make a false conclusion that all three drinks are almost equally preferred, however, looking more closely at the joint frequencies, it is clearly evident that most women prefer special hot drinks and most men prefer regular coffee. Most people prefer hot drinks as opposed to cold; 134 of 200. The drink that men choose the least often was special hot coffee; only 10 of 90. Mathematical practices 6) Attend to precision. Students should learn the proper use of the terms “joint frequencies” and “marginal frequencies,” and use these technical terms in their discussions of the data. Photo credit: iced coffee © Boyan Dimitrov, hot coffee © Ministr-84, both Shutterstock.com 2012 What trends do you notice from the table? List as many as you can and justify each one. 9

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**What does the number 0.12 in the table signify?**

Relative frequencies To convert a two-way frequency table to a relative frequency table divide each cell in the table by the number of participants. special hot drink special cold drink regular coffee total 10 58 42 110 women = 0.05 = 0.29 = 0.21 = 0.55 200 200 200 200 56 10 24 90 men = 0.28 = 0.05 = 0.12 = 0.45 Teacher notes Writing relative frequencies makes it easy to find the equivalent percents signifies that 12% of people asked were men who bought a special cold drink. Students may think 0.12 shows that 12% of men bought a special cold drink. However, this is false. Draw the students’ attention to the final column, which notes that 0.45 of respondents were men. Another useful type of relative frequency table is a relative frequency table for rows. This is shown on the next slide. The numbers in the body of the table (i.e. not the total row or column) are called the conditional relative frequencies. Mathematical practices 6) Attend to precision. Students should learn the proper use of the terms “relative frequency table” and “conditional relative frequencies,” and use these technical terms in their discussions of the data. 200 200 200 200 66 68 66 200 total = 0.33 = 0.34 = 1 = 0.33 200 200 200 200 What does the number 0.12 in the table signify? 10

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Rows and columns Depending on what we want to analyze we can also create relative frequencies for columns and relative frequencies for rows. for columns for rows regular coffee special hot drink special cold drink total regular coffee special hot drink special cold drink total women 0.15 0.85 0.64 0.55 women 0.09 0.53 0.38 1.00 men 0.85 0.15 0.36 0.45 men 0.62 0.11 0.27 1.00 Teacher notes All numbers have been rounded to the nearest hundredth. Lead a discussion about these two tables and what additional information may be gained from them. For instance, from the table at the left we can see that a much larger percentage of those who drink a special cold drink are women, but from the table at the right we can see that men are over twice as likely to choose regular coffee than a special cold drink. Mathematical practices 6) Attend to precision. Students should learn the proper use of the terms “relative frequencies for columns” and “relative frequencies for rows,” and use these technical terms in their discussions of the data. total 1.00 1.00 1.00 1.00 total 0.33 0.34 0.33 1.00 Every number is divided by the total for that column in the original table. Every number is divided by the total for that row in the original table. 11

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