- Dec 2016
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mathbitsnotebook.com mathbitsnotebook.com
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You have worked with one-way tables (even though you may not have called them by that name). A one-way table is simply the data from a bar graph put into table form.
Connection: I remember starting out with one-way tables in elementary. Simple data like favorite color and then graphing that frequency.
As a teacher: prompt students to stop and make their own connections to one-way tables
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Based upon that information, if we knew the gender of a survey respondent, we could make a good prediction as to whether he/she chose a sports car or an SUV. The statistical information is strong enough to support an "association" between gender and choice of vehicle. Now, this does not mean that there is always an association between gender and choice of vehicle. It just means that such an association is evident in the data from this survey.
Prediction: I want to take a further look at the data and answer some real questions about the data.
As a teacher: Have the students analyze the data for themselves and see if they come to the same conclusion that this article comes to
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An "association" exists between two categorical variables if the row (or column) conditional relative frequencies are different for the rows (or columns) of the table. The bigger the differences in the conditional relative frequencies, the stronger the association between the variables. If the conditional relative frequencies are nearly equal for all categories, there may be no association between the variables. Such variables are said to be independent.
As a teacher: I want to make sure that the students need to be aware that association does not imply causation.
This is also an important definition for using two-way tables
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A variety of questions can be answered by examining a two-way frequency table. Let's look at some possibilities: Two-way frequency table How many people responded to the survey? 240 How many males responded to the survey? 60 How many people chose an SUV? 156 How many females chose a sports car? 45 How many males chose an SUV? 21 Two-way relative frequency table (whole table) What percentage of the survey takers was female? 75% What is the relative frequency of males choosing a sports car? Was there a higher percentage of males or females chosing an SUV? higer percentage of females
Summarizing: These are the kinds of questions that we can use the table to answer. Additionally, there are no conditional probability questions here
As the teacher: I want to ask the students about conditional probability and have them analyze the data in that way
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If you want to look for a relationship between the categorical variables, you will need to prepare a conditional relative frequency table. You will then need to decide if a "row" method or a "column" method will address the situation you wish to examine.
Summarizing: This is a how to for each kind of question that may be asked and how to answer them
As the teacher: For this, I would want to ask the students a question wherein they will need to use the tables provided to answer the question and analyze the data.
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The problem is that a column approach does not address the issue of which car men and women prefer. In the column method, we are comparing an SUV to a sports car in relation to gender. An appropriate question would be, "Were SUVs or sports cars chosen more often by females?
Summarizing: Column conditional frequency shows for one column (sports car) what percentage of men/women chose it
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If the two-way relative frequency is for columns, the entries in each column of the table are divided by the total for that column (at the bottom)
Connection: column frequency divide by column totals and row frequency divides by row totals as before.
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Do you see how this changes our previous interpretation of the data? Using a row conditional relative frequency, we can see that 65% of the 60 men responding chose Sports Car, while only 25% of the 180 women responding chose Sports Car. This method takes into account the count of men and women separately, giving us a more realistic view of the relationship between the variables.
Summarizing: These are like little checkpoints to determine understanding with the examples provided.
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"In this survey, do more men, or more women, prefer a sports car?", we need to set up a row conditional relative frequency.
Questioning: Are these the kinds of questions we will be answering with two-way tables?
As a teacher: ask the students to answer this question and then compare the result to their inferences from before
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The ratio of "1", or 100%, occurs in all right hand "total" cells.
Interesting, is this always going to be the case?
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conditional relative frequency, divide a joint frequency (count inside the table) by a marginal frequency total (outer edge) that represents the condition being investigated. You may also see this term stated as row conditional relative frequency or column conditional relative frequency. Basically, we are going to look at the women and men separately, based upon how many women were surveyed, and how many men were surveyed.
Questioning: So we are taking the joint frequency and dividing by the row total or column total depending on the variable we want the conditional frequency for? Does this make sense?
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"conditional" relative frequency
Connection/questioning: Is this similar to conditional probability?
As a teacher: Have students determine the meaning of "conditional" for themselves
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frequency table and the relative frequency table shown above do not take into consideration how many women, and how many men, responded to this survey. There were only 60 men responding, while there were 180 women. There were three times more women responding to this survey, which presents misleading results when based upon the entire population.
Wondering: Here we cannot generalize data for the entire population because the number of women > number of men surveyed, is this always the case? We cannot make generalizations for the entire population?
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19% of the women and 16% of the men
Connection: my inference from before seems correct. Wonder: What else can this table reveal about the data?
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Each of the main body cells (blue) is telling you the percentage of people surveyed that gave that response (based upon the total number of people responding).
Wondering: Is there a way to know the percentage of only males (or females) surveyed who chose that response?
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Two-Way Relative Frequency Table: (displays "percentages")
Prediction: Ok, so now we are going to take the counts and make them into percentages, so counts/total for each joint frequency and marginal frequency square.
As a teacher: have students verify the math presented in the article
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This table shows 45 women chose Sports Car, while 39 men chose Sports Car.
Inferencing: On first sight (and thought) it appears as though more women prefer sports cars than men, but this is just a first prediction
As a teacher: encourage students to look at the data and make their own inferences and write them down to refer to later
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marginal frequencies
connection: "marginal" from the root word margin, and these frequencies happen to be in the margins of this table!
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joint frequencies.
visualize: the pictures here really help with the description provided
As a teacher: can students identify the joint frequency boxes on another two-way table?
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"If you could have a new vehicle, would you want a sport utility vehicle or a sports car?
fix-up: this is missing a " after the question mark. prediction: we will be using survey data about cars, but what will the second categorical data be??
As a teacher: prompt students to make their own predictions about what they think the results will say through this data
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The "totals" of each row appear at the right, and the "totals" of each column appear at the bottom. Note: the "sum of the row totals" equals the "sum of the column totals" (the 240 seen in the lower right corner). This value (240) is also the sum of all of the counts from the interior cells.
Summarizing: the frequency for each variable go in the respective square (column and row) and then add each row, add each column to get bottom row and right most column "totals", the sum of the row totals and the column totals should be equal and also be equal to the sum of the frequency squares (the inner four)/
As a teacher: ask the students to verify the totals by doing the math themselves
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frequency (count)
Connection: this is what I was wondering from before, so then frequency = counts
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contingency tables
Prediction: Contingency comes from the root word contingent meaning dependent on, i.e., a contingency table with show whether or not a variable depends on the other...
As a teacher: students should explore the definition of contingent and contingency to better understand the purpose of a two-way/contingency table and how we use them to analyze.
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relationships between the two categorical variables
Visualization: I visualize two variables as follows: hair color vs. gender... essentially, I see two variables being pitted against each, or compared.
As a teacher: I want the students to determine what they think "relationship between two variables" means to them? Does it mean we contrast them? Compare? Do we analyze them together or separate?
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Two-way frequency tables are a visual representation of the possible relationships between two sets of categorical data.
Summarization: Two-way tables reveal a relationship (if any) between two distinct variables, both of which are categorical. We can use this to better analyze categorical data
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displays "counts"
Wonder: I wonder what the author means by "counts". Are "counts" similar to frequency? Is there a connection?
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one categorical variable.
Connection: categorical variables are variables that are topical in nature and not numerical. ex: hair color, eye color, gender, race, age, etc.
As a teacher: ask students to provide their own examples of categorical
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- Sep 2016
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It’s easy to see why online dating has taken off. It provides you with a seemingly endless supply of people who are single and looking to date. Let’s say you’re a woman who wants a 28-year-old man who’s 5 ft. 10 in., has brown hair, lives in Brooklyn, is a member of the Baha’i faith and loves the music of Naughty by Nature. Before online dating, this would have been a fruitless quest, but now, at any time of the day, no matter where you are, you are just a few screens away from sending a message to your very specific dream man.
I agree, online dating is the ideal why to find the "dream" guy or girl. I don't think we need to settle for "good enough", but I do think this generation is obsessed with the idea of "perfection" and living the "dream". What happens when we don't find the "dream" online? What do we do then?
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