ECON 3818

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# ECON 3818
## Chapter 6
### Kyle Butts
### 26 September 2021

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## Chapter 6: Two-Way Tables
---
# Introduction
This chapter discusses the relationship between two categorical variables

- To analyze categorical data, we use the .it[counts] or .it[percentages] of individuals that fall into various categories

---
# Two-Way Table
Typically, published data is grouped in order to save space. This chapter will talk about .hi.purple[Two-Way Tables]

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<td class="gt_row gt_center">673</td>
<td class="gt_row gt_center">1050</td>
<td class="gt_row gt_center">481</td>
<td class="gt_row gt_center">95</td></tr>
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<td class="gt_row gt_center">401</td>
<td class="gt_row gt_center">780</td>
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This two-way table describes two categorical variables. One is the sex of an individual, this is the .hi.kelly[row variable] and the other is the degree attained, which is the .hi.coral[column variable]

---
# Joint Distribution
The joint distribution is found by dividing each cell by the total count.

`$$P(x,y)=\frac{\text{number of times x and y occurs}}{\text{total number of occurrences}}$$`
Since we have 3911 people in total, we get the following joint probabilities:

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<td class="gt_row gt_center">673</td>
<td class="gt_row gt_center">1,050</td>
<td class="gt_row gt_center">481</td>
<td class="gt_row gt_center">95</td></tr>
    <tr><td class="gt_row gt_left gt_stub">Joint</td>
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<td class="gt_row gt_center">27%</td>
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<td class="gt_row gt_center">2%</td></tr>
    <tr><td class="gt_row gt_left gt_stub">Men</td>
<td class="gt_row gt_center">401</td>
<td class="gt_row gt_center">780</td>
<td class="gt_row gt_center">342</td>
<td class="gt_row gt_center">89</td></tr>
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<td class="gt_row gt_center">10%</td>
<td class="gt_row gt_center">20%</td>
<td class="gt_row gt_center">9%</td>
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This means the probability of being a woman **and** having a Master's degree is 12%

---
# Marginal Distribution

A .hi.alice[marginal distribution] is the probability distribution associated with only one of the random variables

In order to calculate, we need to look at the distribution of each variable *separately*. We do this by looking at the "Total" column and "Total" row.
 
We will have two different marginal distributions, the "row" marginal and the "column" marginal

---
# Row Marginal

In this scenario, the row marginal is the distribution of sex alone:

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}

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  font-style: italic;
  font-weight: normal;
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</style>
<table class="gt_table">
  
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1"></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Associate</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Bachelors</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Masters</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Doctorate</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Row Marginal</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_left gt_stub">Women</td>
<td class="gt_row gt_center">673</td>
<td class="gt_row gt_center">1,050</td>
<td class="gt_row gt_center">481</td>
<td class="gt_row gt_center">95</td>
<td class="gt_row gt_center" style="border-left-width: 2px; border-left-style: solid; border-left-color: #BBBBBB;">2,299</td></tr>
    <tr><td class="gt_row gt_left gt_stub">Joint</td>
<td class="gt_row gt_center">17%</td>
<td class="gt_row gt_center">27%</td>
<td class="gt_row gt_center">12%</td>
<td class="gt_row gt_center">2%</td>
<td class="gt_row gt_center" style="border-left-width: 2px; border-left-style: solid; border-left-color: #BBBBBB;">58%</td></tr>
    <tr><td class="gt_row gt_left gt_stub">Men</td>
<td class="gt_row gt_center">401</td>
<td class="gt_row gt_center">780</td>
<td class="gt_row gt_center">342</td>
<td class="gt_row gt_center">89</td>
<td class="gt_row gt_center" style="border-left-width: 2px; border-left-style: solid; border-left-color: #BBBBBB;">1,612</td></tr>
    <tr><td class="gt_row gt_left gt_stub">Joint</td>
<td class="gt_row gt_center">10%</td>
<td class="gt_row gt_center">20%</td>
<td class="gt_row gt_center">9%</td>
<td class="gt_row gt_center">2%</td>
<td class="gt_row gt_center" style="border-left-width: 2px; border-left-style: solid; border-left-color: #BBBBBB;">42%</td></tr>
  </tbody>
  
  
</table>
</div>

58.4% of individuals in this sample of degree holders are women.

---
# Column Marginal
In this scenario, the column marginal is the distribution of degrees alone:

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#spmwiymzmb .gt_super {
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#spmwiymzmb .gt_footnote_marks {
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<table class="gt_table">
  
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1"></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Associate</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Bachelors</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Masters</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Doctorate</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_left gt_stub">Women</td>
<td class="gt_row gt_center">673</td>
<td class="gt_row gt_center">1,050</td>
<td class="gt_row gt_center">481</td>
<td class="gt_row gt_center">95</td></tr>
    <tr><td class="gt_row gt_left gt_stub">Joint</td>
<td class="gt_row gt_center">17%</td>
<td class="gt_row gt_center">27%</td>
<td class="gt_row gt_center">12%</td>
<td class="gt_row gt_center">2%</td></tr>
    <tr><td class="gt_row gt_left gt_stub">Men</td>
<td class="gt_row gt_center">401</td>
<td class="gt_row gt_center">780</td>
<td class="gt_row gt_center">342</td>
<td class="gt_row gt_center">89</td></tr>
    <tr><td class="gt_row gt_left gt_stub">Joint</td>
<td class="gt_row gt_center">10%</td>
<td class="gt_row gt_center">20%</td>
<td class="gt_row gt_center">9%</td>
<td class="gt_row gt_center">2%</td></tr>
    <tr><td class="gt_row gt_left gt_stub">Column Marginal</td>
<td class="gt_row gt_center" style="border-top-width: 2px; border-top-style: solid; border-top-color: #BBBBBB;">0.273</td>
<td class="gt_row gt_center" style="border-top-width: 2px; border-top-style: solid; border-top-color: #BBBBBB;">0.47</td>
<td class="gt_row gt_center" style="border-top-width: 2px; border-top-style: solid; border-top-color: #BBBBBB;">0.21</td>
<td class="gt_row gt_center" style="border-top-width: 2px; border-top-style: solid; border-top-color: #BBBBBB;">0.047</td></tr>
  </tbody>
  
  
</table>
</div>

The probability of an individual having a Bachelor's degree is 47%.

---
# Conditional Distribution

We can use these tables to back out conditional distributions.

Remember:

`$$P(A \vert B)=\frac{P(A \cap B)}{P(B)}$$`
This is the same expression (just different notation) as:

`$$P(x\vert y)=\frac{P(x,y)}{P(y)}$$`

---
# Conditional Distribution

For example, we can calculate the probability of holding each degree, given the individual is a woman:

$$ 
P(\text{Associate's} \vert \text{Woman}) = \frac{P(\text{Associate's and Woman})}{P(\text{Woman})} = 0.17/0.584 = 29.1\%
$$

$$
P(\text{Bachelor's} \vert \text{Woman}) = \frac{P(\text{Bachelor's and Woman})}{P(\text{Woman})} = 0.27/0.584 = 46.2\%
$$

$$
P(\text{Master's} \vert \text{Woman}) = \frac{P(\text{Master's and Woman})}{P(\text{Woman})} = 0.12/0.584 = 20.5\%
$$

$$
P(\text{Doctorate} \vert \text{Woman}) = \frac{P(\text{Doctorate and Woman})}{P(\text{Woman})} = 0.024/0.584 = 4.1\%
$$

---
# Conditional Distribution
We could also calculate the probability an individual is a particular sex, based of holding an Associate's degree

$$
P(\text{Male} \vert \text{Associate's}) = \frac{P(\text{Male and Associate's})}{P(\text{Associate's})} = .103/0.273 = 37.7\%
$$

$$
P(\text{Female} \vert \text{Associate's}) = \frac{P(\text{Female and Associate's})}{P(\text{Associate's})} = .17/.273 = 62.3\%
$$

---
# Joint Probabilities vs. Conditional Probabilities

- Joint probabilities take into account the probability that each event happens on its own

- Conditional probabilities assume that one event has already happened

---
# Joint Probability vs. Conditional Probability{Example}

$$
P(\text{work in tech job} \cap \text{live in Boulder}) \text{ vs. } P(\text{work in tech job} \ \vert \  \text{live in Boulder})
$$
      
- `$P(\text{work in tech})$` = work in tech/entire US population = relatively small, let's say `$7\%$`

- `$P(\text{live in Boulder})$` = Boulder population/entire US population = also small, `$<1\%$`
      
This means the probability of BOTH happening is small, because both events are unlikely compared to the state space of the entire US population

- But the P(work in tech `$\vert$` live in Boulder) will be higher because now the state space is Boulder population, which has a greater concentration of high-tech employees

---
# Marginal and Conditional Distributions

Here are some formal definitions:

The .hi[marginal distribution] of one of the categorical variables in a two-way table of counts is the .hi[distribution of that variable among all individuals described by the table]

- distribution of sex or degrees alone

A .hi[conditional distribution] of a variable is the .hi[distribution of values of that variable among only individuals who have a given value of the other variable]. There is a separate conditional distribution for each value of the other variable.

- there are two sets of conditional distributions for any two-way table, probability of having a degree based on sex, probability of being a particular sex based on degree held

---
# Clicker Question

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      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Second</th>
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    <tr><td class="gt_row gt_left gt_stub">Alive</td>
<td class="gt_row gt_center">203</td>
<td class="gt_row gt_center">118</td>
<td class="gt_row gt_center">178</td>
<td class="gt_row gt_center">212</td></tr>
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<td class="gt_row gt_center" style="border-top-width: 2px; border-top-style: solid; border-top-color: #BBBBBB;">285</td>
<td class="gt_row gt_center" style="border-top-width: 2px; border-top-style: solid; border-top-color: #BBBBBB;">706</td>
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Given this joint distribution, what is the probability of survival, given you are a first class passenger?
<ol type = "a">
  <li>9%</li>
  <li>62%</li>
  <li>1.3%</li>
  <li>30%</li>
</ol>

---
class: clear, center, middle

.center[[Source](https://ed.ted.com/lessons/how-statistics-can-be-misleading-mark-liddell)]

---
class: clear, center, middle

.center[[Source](https://www.youtube.com/watch?v=t-Ci3FosqZs)]

---
# Simpson's Paradox
Consider the survival rates for the following groups of victims who were taken to the hospital, either by helicopter or by car:

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The probability of died conditional on helicopter, is higher than the probability of died conditional on car. Does this mean that this (more costly) mode of transportation isn't helping?

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<table class="gt_table">
  
  <thead class="gt_col_headings">
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      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Car</th>
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<td class="gt_row gt_center">32%</td>
<td class="gt_row gt_center">68%</td></tr>
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<td class="gt_row gt_center">24%</td>
<td class="gt_row gt_center">76%</td></tr>
  </tbody>
  
  
</table>
</div>

---
# Simpson's Paradox

The idea is there a confounding variable, the severity of the accident, whose proportion differs between planes and car crashes

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<td class="gt_row gt_center">48</td>
<td class="gt_row gt_center">60</td></tr>
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<td class="gt_row gt_center" style="border-top-width: 2px; border-top-style: solid; border-top-color: #BBBBBB;">100</td>
<td class="gt_row gt_center" style="border-top-width: 2px; border-top-style: solid; border-top-color: #BBBBBB;">100</td></tr>
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</table>
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</div>
<div class = "pull-right">
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#lakqumsrav .gt_group_heading {
  padding: 8px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
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  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
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  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
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}

#lakqumsrav .gt_empty_group_heading {
  padding: 0.5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
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  border-bottom-color: #D3D3D3;
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}

#lakqumsrav .gt_from_md > :first-child {
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}

#lakqumsrav .gt_from_md > :last-child {
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}

#lakqumsrav .gt_row {
  padding-top: 5px;
  padding-bottom: 5px;
  padding-left: 5px;
  padding-right: 5px;
  margin: 10px;
  border-top-style: solid;
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  border-top-color: #D3D3D3;
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  border-right-style: none;
  border-right-width: 1px;
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}

#lakqumsrav .gt_stub {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
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  padding-left: 12px;
}

#lakqumsrav .gt_summary_row {
  color: #333333;
  background-color: #FFFFFF;
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  padding-right: 5px;
}

#lakqumsrav .gt_first_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
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#lakqumsrav .gt_grand_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#lakqumsrav .gt_first_grand_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-style: double;
  border-top-width: 6px;
  border-top-color: #D3D3D3;
}

#lakqumsrav .gt_striped {
  background-color: rgba(128, 128, 128, 0.05);
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#lakqumsrav .gt_table_body {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
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}

#lakqumsrav .gt_footnotes {
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  background-color: #FFFFFF;
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  border-left-color: #D3D3D3;
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  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#lakqumsrav .gt_footnote {
  margin: 0px;
  font-size: 90%;
  padding: 4px;
}

#lakqumsrav .gt_sourcenotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#lakqumsrav .gt_sourcenote {
  font-size: 12px;
  padding: 4px;
}

#lakqumsrav .gt_left {
  text-align: left;
}

#lakqumsrav .gt_center {
  text-align: center;
}

#lakqumsrav .gt_right {
  text-align: right;
  font-variant-numeric: tabular-nums;
}

#lakqumsrav .gt_font_normal {
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}

#lakqumsrav .gt_font_bold {
  font-weight: bold;
}

#lakqumsrav .gt_font_italic {
  font-style: italic;
}

#lakqumsrav .gt_super {
  font-size: 65%;
}

#lakqumsrav .gt_footnote_marks {
  font-style: italic;
  font-weight: normal;
  font-size: 65%;
}
</style>
<table class="gt_table">
  <thead class="gt_header">
    <tr>
      <th colspan="3" class="gt_heading gt_title gt_font_normal gt_bottom_border" style><span class='hi slate' style='display: block; margin-bottom: 8px;'>Less Serious Accidents</span></th>
    </tr>
    
  </thead>
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1"></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Helicopter</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Car</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_left gt_stub">Victim Died</td>
<td class="gt_row gt_center">16</td>
<td class="gt_row gt_center">200</td></tr>
    <tr><td class="gt_row gt_left gt_stub">Victim Survived</td>
<td class="gt_row gt_center">84</td>
<td class="gt_row gt_center">800</td></tr>
    <tr><td class="gt_row gt_left gt_stub">Total</td>
<td class="gt_row gt_center" style="border-top-width: 2px; border-top-style: solid; border-top-color: #BBBBBB;">100</td>
<td class="gt_row gt_center" style="border-top-width: 2px; border-top-style: solid; border-top-color: #BBBBBB;">1000</td></tr>
  </tbody>
  
  
</table>
</div>
</div>

<br/>

<div class = "pull-left">
<div id="wshtxiyhdg" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
<style>@import url("https://fonts.googleapis.com/css2?family=Fira+Code:ital,wght@0,100;0,200;0,300;0,400;0,500;0,600;0,700;0,800;0,900;1,100;1,200;1,300;1,400;1,500;1,600;1,700;1,800;1,900&display=swap");
html {
  font-family: 'Fira Code', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen, Ubuntu, Cantarell, 'Helvetica Neue', 'Fira Sans', 'Droid Sans', Arial, sans-serif;
}

#wshtxiyhdg .gt_table {
  display: table;
  border-collapse: collapse;
  margin-left: auto;
  margin-right: auto;
  color: #333333;
  font-size: 16px;
  font-weight: normal;
  font-style: normal;
  background-color: #FFFFFF;
  width: auto;
  border-top-style: solid;
  border-top-width: 3px;
  border-top-color: transparent;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 3px;
  border-bottom-color: transparent;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
}

#wshtxiyhdg .gt_heading {
  background-color: #FFFFFF;
  text-align: left;
  border-bottom-color: #FFFFFF;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#wshtxiyhdg .gt_title {
  color: #333333;
  font-size: 125%;
  font-weight: initial;
  padding-top: 4px;
  padding-bottom: 4px;
  border-bottom-color: #FFFFFF;
  border-bottom-width: 0;
}

#wshtxiyhdg .gt_subtitle {
  color: #333333;
  font-size: 85%;
  font-weight: initial;
  padding-top: 0;
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  border-top-color: #FFFFFF;
  border-top-width: 0;
}

#wshtxiyhdg .gt_bottom_border {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#wshtxiyhdg .gt_col_headings {
  border-top-style: solid;
  border-top-width: 3px;
  border-top-color: transparent;
  border-bottom-style: solid;
  border-bottom-width: 3px;
  border-bottom-color: black;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#wshtxiyhdg .gt_col_heading {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  text-transform: uppercase;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 6px;
  padding-left: 5px;
  padding-right: 5px;
  overflow-x: hidden;
}

#wshtxiyhdg .gt_column_spanner_outer {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  text-transform: uppercase;
  padding-top: 0;
  padding-bottom: 0;
  padding-left: 4px;
  padding-right: 4px;
}

#wshtxiyhdg .gt_column_spanner_outer:first-child {
  padding-left: 0;
}

#wshtxiyhdg .gt_column_spanner_outer:last-child {
  padding-right: 0;
}

#wshtxiyhdg .gt_column_spanner {
  border-bottom-style: solid;
  border-bottom-width: 3px;
  border-bottom-color: black;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 6px;
  overflow-x: hidden;
  display: inline-block;
  width: 100%;
}

#wshtxiyhdg .gt_group_heading {
  padding: 8px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  text-transform: uppercase;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
}

#wshtxiyhdg .gt_empty_group_heading {
  padding: 0.5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: middle;
}

#wshtxiyhdg .gt_from_md > :first-child {
  margin-top: 0;
}

#wshtxiyhdg .gt_from_md > :last-child {
  margin-bottom: 0;
}

#wshtxiyhdg .gt_row {
  padding-top: 5px;
  padding-bottom: 5px;
  padding-left: 5px;
  padding-right: 5px;
  margin: 10px;
  border-top-style: solid;
  border-top-width: 1px;
  border-top-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  overflow-x: hidden;
}

#wshtxiyhdg .gt_stub {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  text-transform: uppercase;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 12px;
}

#wshtxiyhdg .gt_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#wshtxiyhdg .gt_first_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
}

#wshtxiyhdg .gt_grand_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#wshtxiyhdg .gt_first_grand_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-style: double;
  border-top-width: 6px;
  border-top-color: #D3D3D3;
}

#wshtxiyhdg .gt_striped {
  background-color: rgba(128, 128, 128, 0.05);
}

#wshtxiyhdg .gt_table_body {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#wshtxiyhdg .gt_footnotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#wshtxiyhdg .gt_footnote {
  margin: 0px;
  font-size: 90%;
  padding: 4px;
}

#wshtxiyhdg .gt_sourcenotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#wshtxiyhdg .gt_sourcenote {
  font-size: 12px;
  padding: 4px;
}

#wshtxiyhdg .gt_left {
  text-align: left;
}

#wshtxiyhdg .gt_center {
  text-align: center;
}

#wshtxiyhdg .gt_right {
  text-align: right;
  font-variant-numeric: tabular-nums;
}

#wshtxiyhdg .gt_font_normal {
  font-weight: normal;
}

#wshtxiyhdg .gt_font_bold {
  font-weight: bold;
}

#wshtxiyhdg .gt_font_italic {
  font-style: italic;
}

#wshtxiyhdg .gt_super {
  font-size: 65%;
}

#wshtxiyhdg .gt_footnote_marks {
  font-style: italic;
  font-weight: normal;
  font-size: 65%;
}
</style>
<table class="gt_table">
  <thead class="gt_header">
    <tr>
      <th colspan="3" class="gt_heading gt_title gt_font_normal gt_bottom_border" style><span class='hi slate' style='display: block; margin-bottom: 8px;'>Serious Accidents</span></th>
    </tr>
    
  </thead>
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1"></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Helicopter</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Car</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_left gt_stub">Victim Died</td>
<td class="gt_row gt_center">48%</td>
<td class="gt_row gt_center">52%</td></tr>
    <tr><td class="gt_row gt_left gt_stub">Victim Survived</td>
<td class="gt_row gt_center">60%</td>
<td class="gt_row gt_center">40%</td></tr>
  </tbody>
  
  
</table>
</div>

</div>

<div class = "pull-right">
<div id="kbxmjcuroa" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
<style>@import url("https://fonts.googleapis.com/css2?family=Fira+Code:ital,wght@0,100;0,200;0,300;0,400;0,500;0,600;0,700;0,800;0,900;1,100;1,200;1,300;1,400;1,500;1,600;1,700;1,800;1,900&display=swap");
html {
  font-family: 'Fira Code', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen, Ubuntu, Cantarell, 'Helvetica Neue', 'Fira Sans', 'Droid Sans', Arial, sans-serif;
}

#kbxmjcuroa .gt_table {
  display: table;
  border-collapse: collapse;
  margin-left: auto;
  margin-right: auto;
  color: #333333;
  font-size: 16px;
  font-weight: normal;
  font-style: normal;
  background-color: #FFFFFF;
  width: auto;
  border-top-style: solid;
  border-top-width: 3px;
  border-top-color: transparent;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 3px;
  border-bottom-color: transparent;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
}

#kbxmjcuroa .gt_heading {
  background-color: #FFFFFF;
  text-align: left;
  border-bottom-color: #FFFFFF;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#kbxmjcuroa .gt_title {
  color: #333333;
  font-size: 125%;
  font-weight: initial;
  padding-top: 4px;
  padding-bottom: 4px;
  border-bottom-color: #FFFFFF;
  border-bottom-width: 0;
}

#kbxmjcuroa .gt_subtitle {
  color: #333333;
  font-size: 85%;
  font-weight: initial;
  padding-top: 0;
  padding-bottom: 4px;
  border-top-color: #FFFFFF;
  border-top-width: 0;
}

#kbxmjcuroa .gt_bottom_border {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#kbxmjcuroa .gt_col_headings {
  border-top-style: solid;
  border-top-width: 3px;
  border-top-color: transparent;
  border-bottom-style: solid;
  border-bottom-width: 3px;
  border-bottom-color: black;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
}

#kbxmjcuroa .gt_col_heading {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  text-transform: uppercase;
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  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
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  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 6px;
  padding-left: 5px;
  padding-right: 5px;
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}

#kbxmjcuroa .gt_column_spanner_outer {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  text-transform: uppercase;
  padding-top: 0;
  padding-bottom: 0;
  padding-left: 4px;
  padding-right: 4px;
}

#kbxmjcuroa .gt_column_spanner_outer:first-child {
  padding-left: 0;
}

#kbxmjcuroa .gt_column_spanner_outer:last-child {
  padding-right: 0;
}

#kbxmjcuroa .gt_column_spanner {
  border-bottom-style: solid;
  border-bottom-width: 3px;
  border-bottom-color: black;
  vertical-align: bottom;
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  padding-bottom: 6px;
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}

#kbxmjcuroa .gt_group_heading {
  padding: 8px;
  color: #333333;
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  font-size: 80%;
  font-weight: bolder;
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}

#kbxmjcuroa .gt_empty_group_heading {
  padding: 0.5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
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  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: middle;
}

#kbxmjcuroa .gt_from_md > :first-child {
  margin-top: 0;
}

#kbxmjcuroa .gt_from_md > :last-child {
  margin-bottom: 0;
}

#kbxmjcuroa .gt_row {
  padding-top: 5px;
  padding-bottom: 5px;
  padding-left: 5px;
  padding-right: 5px;
  margin: 10px;
  border-top-style: solid;
  border-top-width: 1px;
  border-top-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  overflow-x: hidden;
}

#kbxmjcuroa .gt_stub {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 80%;
  font-weight: bolder;
  text-transform: uppercase;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 12px;
}

#kbxmjcuroa .gt_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
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  padding-right: 5px;
}

#kbxmjcuroa .gt_first_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
}

#kbxmjcuroa .gt_grand_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#kbxmjcuroa .gt_first_grand_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-style: double;
  border-top-width: 6px;
  border-top-color: #D3D3D3;
}

#kbxmjcuroa .gt_striped {
  background-color: rgba(128, 128, 128, 0.05);
}

#kbxmjcuroa .gt_table_body {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#kbxmjcuroa .gt_footnotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#kbxmjcuroa .gt_footnote {
  margin: 0px;
  font-size: 90%;
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}

#kbxmjcuroa .gt_sourcenotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
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  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#kbxmjcuroa .gt_sourcenote {
  font-size: 12px;
  padding: 4px;
}

#kbxmjcuroa .gt_left {
  text-align: left;
}

#kbxmjcuroa .gt_center {
  text-align: center;
}

#kbxmjcuroa .gt_right {
  text-align: right;
  font-variant-numeric: tabular-nums;
}

#kbxmjcuroa .gt_font_normal {
  font-weight: normal;
}

#kbxmjcuroa .gt_font_bold {
  font-weight: bold;
}

#kbxmjcuroa .gt_font_italic {
  font-style: italic;
}

#kbxmjcuroa .gt_super {
  font-size: 65%;
}

#kbxmjcuroa .gt_footnote_marks {
  font-style: italic;
  font-weight: normal;
  font-size: 65%;
}
</style>
<table class="gt_table">
  <thead class="gt_header">
    <tr>
      <th colspan="3" class="gt_heading gt_title gt_font_normal gt_bottom_border" style><span class='hi slate' style='display: block; margin-bottom: 8px;'>Less Serious Accidents</span></th>
    </tr>
    
  </thead>
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1"></th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Helicopter</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">Car</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_left gt_stub">Victim Died</td>
<td class="gt_row gt_center">16%</td>
<td class="gt_row gt_center">84%</td></tr>
    <tr><td class="gt_row gt_left gt_stub">Victim Survived</td>
<td class="gt_row gt_center">20%</td>
<td class="gt_row gt_center">80%</td></tr>
  </tbody>
  
  
</table>
</div>

</div>

---
# Simpson's Paradox

---
# Clicker Question

From 2010-2013 the US median wage increased 1%, however over the same time period the median wage has decreased within each education subgroup (high school drop outs, high school graduates, some college, bachelor's or more).

Which of the following explanations is consistent with Simpson's paradox?
<ol type = "a">
  <li>The BLS didn't control for inflation</li>
  <li>There are more people with bachelor's degrees (high income people)</li>
  <li>The wage of the highest income earner went up even more</li>
  <li>There are less unemployed people now</li>
</ol>