ECON 3818

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# ECON 3818
## Chapter 2
### Kyle Butts
### 27 September 2021

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## Chapter 2: Describing Distribution with Numbers

---
## Chapter Overview
	
	
Population vs. Sample

Measures of Central Tendency
- Mean
- Median
		      
Measures of Variability
- Quartiles
- Variance and Standard Deviation

---
# Population vs Sample
	
	
.hi.purple[Population]: the entire entities under the study
- Examples: all men, all NBA players, all children under 5
		      
.hi.kelly[Sample]: subset of the population
- Can be used to draw inferences about the population
- Examples: our class, Denver Nuggets players, daycares in Colorado
		      
- Interested in parameters of the .hi.purple[population] distribution, we can estimate these parameters using data from .hi.kelly[samples] since finding population parameters is infeasible

---
# Population Distribution

.hi.kelly[Distribution of a variable]: tells us .it[what values] it takes and .it[how often] it takes these values

- We are interested in the underlying population distribution of some variable
	
- Fundamental problem of statistics is we can't collect data on every single observation

---
class: clear

---
# Population Inference

What we do instead is use a sample of the population and use that sample distribution to determine parameters of interest

.center[
<img style="width:80%;" src="sample_anim.gif"/>
]

---
# Parameters of Interest
	
	
Two primary .hi.purple[population] parameters of interest:
		      
- Measures of central tendency:
	      	      
	- Population .coral[mean], `$\mu$`
	
	- Population .cranberry[median]
	      	      
- Measures of variability:
	      	      
	- Population .alice[variance], `$\sigma^2$`
	  
	  
--

We will .it.kelly[estimate] these using the .hi.kelly[sample] distribution

---
# Measuring Center: the Mean
	
	
The most common measure of center is the arithmetic average, or .hi.coral[mean]

`$$\coral{\bar{x}} = \frac{x_1 + x_2 + .... + x_n}{n}$$`

or more compactly:

<img src="mean_annotated.png" width="45%" style="display: block; margin: auto;" />
	
	
---
# Population Inference: Mean

.center[
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]

---
# Population Inference: Mean

.center[
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]

---
# Measuring Center: the Median
	
	
The .hi.cranberry[median] is the midpoint of a distribution
		      
- Is more resistant to the influence of .hi[extreme observations]
		      
How to calculate median:
		      
- Arrange observations from smallest to largest
- If there is odd number of observations, the median is the center observation. If there are even number of observations, the median is the average of two center observations

---
# Mean vs. Median
	
	
- Although we will primarily be using the mean throughout the semester, the biggest drawback of the mean is that it is not resistant to .hi.purple[outliers]
		      
- The median, however, is resistant to .hi.purple[outliers] so it can be important to calculate for smaller samples

.center[
<img style="width: 60%;" src="meme.png"/>
]

---
# Mean vs. Median Example

.hi[Median]: 205.5 rebounds and .hi[Mean]: 250.5 rebounds

---
# Clicker Question
	
What is the sample mean of the participants's age?
	
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<table class="gt_table">
  <thead class="gt_header">
    <tr>
      <th colspan="4" class="gt_heading gt_title gt_font_normal gt_bottom_border" style><span class='hi slate' style='display: block; margin-bottom: 8px;'>Sample of individuals</span></th>
    </tr>
    
  </thead>
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">Age</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">Sex</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">BMI</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">Drinks per week</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_right">59</td>
<td class="gt_row gt_left">male</td>
<td class="gt_row gt_right">32.26</td>
<td class="gt_row gt_left">3 drinks</td></tr>
    <tr><td class="gt_row gt_right">62</td>
<td class="gt_row gt_left">male</td>
<td class="gt_row gt_right">25.09</td>
<td class="gt_row gt_left">2 drinks</td></tr>
    <tr><td class="gt_row gt_right">60</td>
<td class="gt_row gt_left">female</td>
<td class="gt_row gt_right">32.58</td>
<td class="gt_row gt_left">1 drink</td></tr>
    <tr><td class="gt_row gt_right">18</td>
<td class="gt_row gt_left">male</td>
<td class="gt_row gt_right">99.99</td>
<td class="gt_row gt_left">6 drinks</td></tr>
    <tr><td class="gt_row gt_right">57</td>
<td class="gt_row gt_left">female</td>
<td class="gt_row gt_right">31.88</td>
<td class="gt_row gt_left">2 drinks</td></tr>
    <tr><td class="gt_row gt_right">56</td>
<td class="gt_row gt_left">male</td>
<td class="gt_row gt_right">42.80</td>
<td class="gt_row gt_left">3 drinks</td></tr>
  </tbody>
  
  
</table>
</div>

---
# Clicker Question
	
Which measure of central tendency best describes the age of participants?

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    </tr>
    
  </thead>
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      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">Age</th>
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    <tr><td class="gt_row gt_right">59</td>
<td class="gt_row gt_left">male</td>
<td class="gt_row gt_right">32.26</td>
<td class="gt_row gt_left">3 drinks</td></tr>
    <tr><td class="gt_row gt_right">62</td>
<td class="gt_row gt_left">male</td>
<td class="gt_row gt_right">25.09</td>
<td class="gt_row gt_left">2 drinks</td></tr>
    <tr><td class="gt_row gt_right">60</td>
<td class="gt_row gt_left">female</td>
<td class="gt_row gt_right">32.58</td>
<td class="gt_row gt_left">1 drink</td></tr>
    <tr><td class="gt_row gt_right">18</td>
<td class="gt_row gt_left">male</td>
<td class="gt_row gt_right">99.99</td>
<td class="gt_row gt_left">6 drinks</td></tr>
    <tr><td class="gt_row gt_right">57</td>
<td class="gt_row gt_left">female</td>
<td class="gt_row gt_right">31.88</td>
<td class="gt_row gt_left">2 drinks</td></tr>
    <tr><td class="gt_row gt_right">56</td>
<td class="gt_row gt_left">male</td>
<td class="gt_row gt_right">42.80</td>
<td class="gt_row gt_left">3 drinks</td></tr>
  </tbody>
  
  
</table>
</div>

<ol type="a">
	<li>Median</li>
	<li>Mean</li>
</ol>

---
# Measuring Variability
	
	
Measures of central tendency do not tell the whole story. To further characterize the distribution, we need to know how the data is spread out
		      
- Quartiles
- Variance

---
# Variability: Quartiles
	
	
Measure of center alone can be misleading. One way to measure variability is to use quartiles.

How to calculate quartiles:
		      
- Arrange observations in increasing order and locate .hi.cranberry[median] 
- The .hi.kelly[first quartile] is the median of the observations located to the left of the median
- The .hi.kelly[third quartile] is the median of observations located to the right of the median
              
<img src="quartiles.png" width="50%" style="display: block; margin: auto;" />

---
# Boxplots
	
.hi.alice[Five-number summary]: smallest observation (minimum), the first quartile, the median, the third quartile, and the largest observation (maximum)

We can use the .hi.purple[boxplot] using this five number summary to display quantitative data

How to make a boxplot:
		      
- A central box spans the first and third quartiles

- A line in the box marks the median

- Line extends from the box out to the smallest and largest observations

---
# Boxplots

---
# Interquartile Range
	
	
The .hi.ruby[interquartile range], IQR, is the distance between the first and third quartiles
		      
- IQR = `$Q_3 - Q_1$`
- The IQR measures the spread of the data and it also helps to identify outliers
	      
Rule for outliers:
		      
- An observation is an outlier if it falls more than `$1.5 \times IQR$` above the third quartile or below the first

---
# Variability: Variance
	
	
.hi.purple[Variance]: denoted, `$s^2$`, measures how "spread out" the data are on average
		     
`$$s^2 = \frac{(x_1-\coral{\bar{x}})^2 + (x_2-\coral{\bar{x}})^2 + .... + (x_n - \coral{\bar{x}})^2}{n-1},$$`
or more compactly

---
# Visualizing Variance
	
	
<img src="ch2_files/figure-html/multiple-vars-1.svg" width="90%" style="display: block; margin: auto;" />

---
# Example

.footnote[Figure from [Teacups, Giraffes, & Statistics](https://tinystats.github.io/teacups-giraffes-and-statistics/04_variance.html)]

1. Calculate the mean height in sample

---
# Example

.footnote[Figure from [Teacups, Giraffes, & Statistics](https://tinystats.github.io/teacups-giraffes-and-statistics/04_variance.html)]

<ol start = "2">
<li>Calculate deviations from mean</li>
<li>Square and sum</li>
</ol>

---
# Variability: Standard Deviation
.hi.purple[Standard deviation]: looks at how far each observation is from the mean; square root of the variance

`$$s=\sqrt{\frac{1}{n-1}\sum_{i=1}^n(x_i-\bar{x})^2} = \sqrt{s^2}$$`
	
- `$n-1$` is referred to as the degrees of freedom
- `$s$` measures variability about the mean
	- More variable `$\implies$` larger `$s$`
- `$s$` is always greater than or equal to zero, but usually `$> 0$`
  - When would it be `$=0$`?
- `$s$` is not resistant to outliers.

---
# Practice Question

Calculate the standard deviation of age?

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<table class="gt_table">
  <thead class="gt_header">
    <tr>
      <th colspan="4" class="gt_heading gt_title gt_font_normal gt_bottom_border" style><span class='hi slate' style='display: block; margin-bottom: 8px;'>Sample of individuals</span></th>
    </tr>
    
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      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">Age</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">Sex</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">BMI</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">Drinks per week</th>
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    <tr><td class="gt_row gt_right">59</td>
<td class="gt_row gt_left">male</td>
<td class="gt_row gt_right">32.26</td>
<td class="gt_row gt_left">3 drinks</td></tr>
    <tr><td class="gt_row gt_right">62</td>
<td class="gt_row gt_left">male</td>
<td class="gt_row gt_right">25.09</td>
<td class="gt_row gt_left">2 drinks</td></tr>
    <tr><td class="gt_row gt_right">60</td>
<td class="gt_row gt_left">female</td>
<td class="gt_row gt_right">32.58</td>
<td class="gt_row gt_left">1 drink</td></tr>
    <tr><td class="gt_row gt_right">18</td>
<td class="gt_row gt_left">male</td>
<td class="gt_row gt_right">99.99</td>
<td class="gt_row gt_left">6 drinks</td></tr>
    <tr><td class="gt_row gt_right">57</td>
<td class="gt_row gt_left">female</td>
<td class="gt_row gt_right">31.88</td>
<td class="gt_row gt_left">2 drinks</td></tr>
    <tr><td class="gt_row gt_right">56</td>
<td class="gt_row gt_left">male</td>
<td class="gt_row gt_right">42.80</td>
<td class="gt_row gt_left">3 drinks</td></tr>
  </tbody>
  
  
</table>
</div>

---
# Summary of Summary Statistics

Two basic ways to summarize the center and spread of a distribution
	
- Mean and standard deviation (or variance)

- The five-number summary

.hi.slate[When to Use Which]
	
Use `$\bar{x}$` and `$s$` when the distribution is reasonably symmetric and free of outliers
		      
Use five-number summary if distribution is skewed, or has outliers

---
# Greek Letters and Statistics

.pull-left[
.hi.kelly[Latin Letters]

- Latin letters like `$\bar{x}$` and `$s^2$` are calculations that represent guesses (estimates) at the population values.
]
.pull-right[
.hi.purple[Greek Letters]

- Greek letters like `$\mu$` and `$\sigma^2$` represent the truth about the population.
]

The goal for the class is for the latin letters to be good guesses for the greek letters:

$$
	\kelly{\text{Data}} \longrightarrow \kelly{\text{Calculation}} \longrightarrow \kelly{\text{Estimates}} \longrightarrow^{hopefully!} \purple{\text{Truth}}
$$

For example, 
$$
	\kelly{X} \longrightarrow \kelly{\frac{1}{n} \sum_{i=1}^n X_i} \longrightarrow \kelly{\bar{x}} \longrightarrow^{hopefullly!} \purple{\mu}
$$