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**Two-Way Frequency Tables**

Joint, Marginal, and Conditional Frequencies

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Standards: Common Core State Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Attend to precision. MCC9-12.S.ID.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

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**By the end of this lesson, you should be able to:**

Interpret data from a two-way frequency table Identify and interpret joint, marginal, and conditional relative frequencies Create a conditional distribution table Create a relative frequency chart

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Terms Two-way Frequency table: a table listing two categorical variables who values have been paired Joint relative frequency: is found by dividing a frequency that is not in the Total row or the Total column by the grand total. Marginal relative frequency: is found by dividing a row total or a column total by the grand total. Two-Way relative frequency table: displays both the joint relative and marginal relative frequencies. Conditional relative frequency: is found by dividing a frequency that is not in the Total row or Total column by the frequency’s row total or column total.

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**Two-Way Frequency Table**

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**Conditional Distributions**

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**Creating a Relative Frequency Table:**

Old Holt 4-5/4-6; New Holt 10-1/10-2/ On-Core: 9-5 Two Way Frequency Tables Creating a Relative Frequency Table: Jenna asked 40 randomly selected students whether they preferred dogs, cats or other pets. Complete the table for both decimals and percents Preferred Pet Dog Cat Other Total Frequency 18 12 10 40 Relative Frequency Decimals Relative Frequency Percents = .45 = .30 = .25 = 1.00 45% 30% 25% 100%

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**Creating a Two-Way Frequency Table:**

Old Holt 4-5/4-6; New Holt 10-1/10-2/ On-Core: 9-5 Two Way Frequency Tables Creating a Two-Way Frequency Table: Jenna also recorded the gender of each student. The results are shown in the two-way frequency table. Find the totals and answer the questions. Preferred Pet Dog Cat Other Total Boys 10 5 9 24 Girls 8 7 1 16 Totals 18 12 10 40 1) How many total students took the survey? 40 2) How many total boys took the survey? 24 3) How many students said they like dogs as pets? 18 4) How many boys said they like dogs as pets? 10

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**Creating a Two-Way Relative Frequency Table:**

Old Holt 4-5/4-6; New Holt 10-1/10-2/ On-Core: 9-5 Two Way Frequency Tables Creating a Two-Way Relative Frequency Table: Jenna also recorded the gender of each student surveyed, complete the two-way frequency table below. Preferred Pet Dog Cat Other Total Boys 10 5 9 24 Girls 8 7 1 16 Totals 18 12 10 40 Preferred Pet Dog Cat Other Total Boys Girls Totals = .25 = .125 = .225 = .60 = .20 = .175 = .025 = .40 = .45 = .30 = .25 = 1.00

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**Creating a Two-Way Relative Frequency Table:**

Old Holt 4-5/4-6; New Holt 10-1/10-2/ On-Core: 9-5 Two Way Frequency Tables Creating a Two-Way Relative Frequency Table: Preferred Pet Dog Cat Other Total Boys Girls Totals = .25 = .125 = .225 = .60 = .20 = .175 = .025 = .40 = .45 = .30 = .25 = 1.00 State the percents for the table above. Preferred Pet Dog Cat Other Total Boys Girls Totals 25% 12.5% 22.5% 60% 20% 17.5% 2.5% 40% 45% 30% 25% 100%

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**Creating a Two-Way Relative Frequency Table:**

Old Holt 4-5/4-6; New Holt 10-1/10-2/ On-Core: 9-5 Two Way Frequency Tables Creating a Two-Way Relative Frequency Table: Preferred Pet Dog Cat Other Total Boys 25% 12.5% 22.5% 60% Girls 20% 17.5% 2.5% 40% Totals 45% 30% 100% 1) Find the joint relative frequency of students surveyed who are girls and prefer dogs as pets. 20% 2) Find the joint relative frequency of students surveyed who are boys and prefer cats as pets. 12.5% 3) Find the marginal RF of students surveyed who prefer dogs as pets and then prefer cats as pets. dogs - 45% cats - 30%

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**Calculating a Conditional Relative Frequency:**

Old Holt 4-5/4-6; New Holt 10-1/10-2/ On-Core: 9-5 Two Way Frequency Tables Calculating a Conditional Relative Frequency: Preferred Pet Dog Cat Other Total Boys 10 5 9 24 Girls 8 7 1 16 Totals 18 12 10 40 Find the conditional relative frequency that a student surveyed prefers cats as pets, given the student is a girl. Number of girls who prefer cats: 7 = .4375 = 43.75% Number of girls: 16

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**Calculating a Conditional Relative Frequency:**

Old Holt 4-5/4-6; New Holt 10-1/10-2/ On-Core: 9-5 Two Way Frequency Tables Calculating a Conditional Relative Frequency: Preferred Pet Dog Cat Other Total Boys 10 5 9 24 Girls 8 7 1 16 Totals 18 10 40 12 Find the conditional relative frequency that a student surveyed is a girl, given that the student prefers cats as pets. Number of girls who prefer cats: 7 = .583 = 58.3% Number of students who prefer cats: 12

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**Calculating a Conditional Relative Frequency:**

Old Holt 4-5/4-6; New Holt 10-1/10-2/ On-Core: 9-5 Two Way Frequency Tables Calculating a Conditional Relative Frequency: Preferred Pet Dog Cat Other Total Boys 10 5 9 24 Girls 8 7 1 16 Totals 18 12 10 40 Find the conditional relative frequency that a student surveyed prefers cats as pets, given the student is a boy. Number of boys who prefer cats: 5 = .208 = 20.8% Number of boys: 24

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**Calculating a Conditional Relative Frequency:**

Old Holt 4-5/4-6; New Holt 10-1/10-2/ On-Core: 9-5 Two Way Frequency Tables Calculating a Conditional Relative Frequency: Preferred Pet Dog Cat Other Total Boys 10 5 9 24 Girls 8 7 1 16 Totals 18 10 40 12 Find the conditional relative frequency that a student surveyed is a boy, given that the student prefers dogs as pets. Number of boys who prefer dogs: 10 = .556 = 55.6% Number of students who prefer dogs: 18

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**Finding Possible Associations and Trends**

Old Holt 4-5/4-6; New Holt 10-1/10-2/ On-Core: 9-5 Two Way Frequency Tables Finding Possible Associations and Trends Jenna survey results (numbers vs percents). Dog Cat Other Total Boys 10/25% 5/12.5% 9/22.5% 24/60% Girls 8/20% 7/17.5% 1/2.5% 16/40% Totals 18/45% 12/30% 40/100% 1) Does the table reflect gender bias towards pets? NO But, it does imply, boys like “other” pets more than girls: % to 2.5%, 2) If so, which gender, likes which pets the best? N/A 3) What are some examples of “other” pets? Hamsters, gerbils, mice, rats, snakes, rabbits, pigs, etc., 4) Girls tend not to like which types of pets? “other” types

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**Make sure you try the 10 practice problems.**

Relative Frequency Go here: Make sure you try the 10 practice problems.

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**Review: Key concepts covered in this presentation:**

How to read a two-way frequency table Marginal and joint distributions Conditional distribution tables Relative frequency

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