Statistical Thinking for Machine Learning: Lecture 1

# Statistical Thinking for Machine Learning: Lecture 1
### Reda Mastouri <br><span style="font-size: 50%;">UChicago MastersTrack: Coursera<br>Thank you to Gregory Bernstein for parts of these slides</span>

---

# Statistics and machine learning

- .left[Statistics and Probability Theory are the foundations for modeling and machine learning]

- .left[Statistical Modeling will be more concerned with inference. Understanding an effect we hope to observe]

- .left[Machine Learning will be more concerned with prediction. Being able to predict an unknown value based off a sample we observed. Having a representative sample and generalizable results will be crucial]

- .left[Fit the data to desired output and *learn* from the data!]

---

# Agenda

- .left[Comprehending data]
  - .left[Data in the grand scheme of things]
  - .left[Understanding your data frame: Rows and columns]
  - .left[Different types of data]

- .left[Distributions]
  - .left[Data types and distributions]
  - .left[PDF and CDF]
  - .left[Parameters and Method of Moments]
  - .left[The Normal Distribution and the Central Limit Theorem]

- .left[Sampling]
  - .left[How we can use sampling to achieve our goals]
  - .left[Descriptive statistics]

---

# Data structure and data types

---

# Where does data fit in the big picture?

- Defining a problem statement
- Collecting and storing data
  - recording data, downloading data sets, web scraping or using API, data structures and lakes
- Data
- Sampling and Modeling
- .left[Insights, inference, and visualization]

---

# Data types

.center[
|Data||Description|
|:----------||:-------------|
| Binary || a value of 0/1, true/false |
| Categorical || a discrete value with limited possibilities |
| Continuous || a numerical value that has an infinite range of possibilities |
]
--

<div class="my-footer"><span>https://www.statlearning.com</span></div>
]

---

# Rows and columns

<div class="my-footer"><span>https://r4ds.had.co.nz</span></div>
]

---

# Rows and columns: Diamond Data Set

- Two use-cases: supervised and unsupervised learning

<table>
 <thead>
  <tr>
   <th style="text-align:right;"> carat </th>
   <th style="text-align:left;"> cut </th>
   <th style="text-align:left;"> color </th>
   <th style="text-align:left;"> clarity </th>
   <th style="text-align:right;"> depth </th>
   <th style="text-align:right;"> table </th>
   <th style="text-align:right;"> price </th>
   <th style="text-align:right;"> x </th>
   <th style="text-align:right;"> y </th>
   <th style="text-align:right;"> z </th>
   <th style="text-align:right;"> id </th>
   <th style="text-align:left;"> purchased </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 1.54 </td>
   <td style="text-align:left;"> Ideal </td>
   <td style="text-align:left;"> J </td>
   <td style="text-align:left;"> VS1 </td>
   <td style="text-align:right;"> 62.2 </td>
   <td style="text-align:right;"> 59 </td>
   <td style="text-align:right;"> 8848 </td>
   <td style="text-align:right;"> 7.34 </td>
   <td style="text-align:right;"> 7.38 </td>
   <td style="text-align:right;"> 4.58 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:left;"> TRUE </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.91 </td>
   <td style="text-align:left;"> Ideal </td>
   <td style="text-align:left;"> E </td>
   <td style="text-align:left;"> SI2 </td>
   <td style="text-align:right;"> 61.5 </td>
   <td style="text-align:right;"> 56 </td>
   <td style="text-align:right;"> 3968 </td>
   <td style="text-align:right;"> 6.20 </td>
   <td style="text-align:right;"> 6.23 </td>
   <td style="text-align:right;"> 3.82 </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:left;"> FALSE </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2.01 </td>
   <td style="text-align:left;"> Good </td>
   <td style="text-align:left;"> H </td>
   <td style="text-align:left;"> SI2 </td>
   <td style="text-align:right;"> 63.1 </td>
   <td style="text-align:right;"> 55 </td>
   <td style="text-align:right;"> 13849 </td>
   <td style="text-align:right;"> 7.99 </td>
   <td style="text-align:right;"> 8.09 </td>
   <td style="text-align:right;"> 5.07 </td>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:left;"> FALSE </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 1.01 </td>
   <td style="text-align:left;"> Good </td>
   <td style="text-align:left;"> E </td>
   <td style="text-align:left;"> SI1 </td>
   <td style="text-align:right;"> 64.1 </td>
   <td style="text-align:right;"> 62 </td>
   <td style="text-align:right;"> 4480 </td>
   <td style="text-align:right;"> 6.26 </td>
   <td style="text-align:right;"> 6.19 </td>
   <td style="text-align:right;"> 3.99 </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:left;"> TRUE </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 1.52 </td>
   <td style="text-align:left;"> Good </td>
   <td style="text-align:left;"> F </td>
   <td style="text-align:left;"> VS2 </td>
   <td style="text-align:right;"> 57.8 </td>
   <td style="text-align:right;"> 59 </td>
   <td style="text-align:right;"> 12283 </td>
   <td style="text-align:right;"> 7.58 </td>
   <td style="text-align:right;"> 7.50 </td>
   <td style="text-align:right;"> 4.36 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:left;"> TRUE </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 1.61 </td>
   <td style="text-align:left;"> Premium </td>
   <td style="text-align:left;"> D </td>
   <td style="text-align:left;"> SI1 </td>
   <td style="text-align:right;"> 61.4 </td>
   <td style="text-align:right;"> 60 </td>
   <td style="text-align:right;"> 13582 </td>
   <td style="text-align:right;"> 7.56 </td>
   <td style="text-align:right;"> 7.51 </td>
   <td style="text-align:right;"> 4.63 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:left;"> FALSE </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.33 </td>
   <td style="text-align:left;"> Ideal </td>
   <td style="text-align:left;"> G </td>
   <td style="text-align:left;"> VS1 </td>
   <td style="text-align:right;"> 61.5 </td>
   <td style="text-align:right;"> 56 </td>
   <td style="text-align:right;"> 699 </td>
   <td style="text-align:right;"> 4.45 </td>
   <td style="text-align:right;"> 4.48 </td>
   <td style="text-align:right;"> 2.74 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:left;"> FALSE </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 1.17 </td>
   <td style="text-align:left;"> Ideal </td>
   <td style="text-align:left;"> F </td>
   <td style="text-align:left;"> VS2 </td>
   <td style="text-align:right;"> 61.8 </td>
   <td style="text-align:right;"> 55 </td>
   <td style="text-align:right;"> 8072 </td>
   <td style="text-align:right;"> 6.81 </td>
   <td style="text-align:right;"> 6.74 </td>
   <td style="text-align:right;"> 4.19 </td>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:left;"> FALSE </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.30 </td>
   <td style="text-align:left;"> Premium </td>
   <td style="text-align:left;"> H </td>
   <td style="text-align:left;"> VS2 </td>
   <td style="text-align:right;"> 62.6 </td>
   <td style="text-align:right;"> 58 </td>
   <td style="text-align:right;"> 608 </td>
   <td style="text-align:right;"> 4.28 </td>
   <td style="text-align:right;"> 4.22 </td>
   <td style="text-align:right;"> 2.66 </td>
   <td style="text-align:right;"> 9 </td>
   <td style="text-align:left;"> TRUE </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0.70 </td>
   <td style="text-align:left;"> Ideal </td>
   <td style="text-align:left;"> G </td>
   <td style="text-align:left;"> VS2 </td>
   <td style="text-align:right;"> 61.8 </td>
   <td style="text-align:right;"> 57 </td>
   <td style="text-align:right;"> 2929 </td>
   <td style="text-align:right;"> 5.68 </td>
   <td style="text-align:right;"> 5.71 </td>
   <td style="text-align:right;"> 3.52 </td>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:left;"> TRUE </td>
  </tr>
</tbody>
</table>

- **Independent variables** can be used for unsupervised modeling
- **Dependent variables** and independent variables can be used together for supervised modeling

---

# Rows and columns

- Rows are linked — some rows belong to the same group
- Are the group means the same?

.center[
|city_type  |population_mil    |rainfall_inches|
|:----------|-------------:|-------------:|
| urban    | 1.2 | 38 |
| urban | .75 | 6 |
| suburban | .5 | 14 |
| suburban | .5 | 18 |
| rural | .5 | 32 |
| rural | .5 | 12|
]

- Label and categorical independent variable that can be used in a model: `city_type`
- potential dependent variables: `population_mil`, `rainfall_inches`

---

# Categorical variables and dummy encoding

.left-column[
<table>
 <thead>
  <tr>
   <th style="text-align:right;"> id </th>
   <th style="text-align:right;"> price </th>
   <th style="text-align:left;"> clarity </th>
   <th style="text-align:left;"> cut </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 8848 </td>
   <td style="text-align:left;"> VS1 </td>
   <td style="text-align:left;"> Ideal </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 3968 </td>
   <td style="text-align:left;"> SI2 </td>
   <td style="text-align:left;"> Ideal </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 13849 </td>
   <td style="text-align:left;"> SI2 </td>
   <td style="text-align:left;"> Good </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 4480 </td>
   <td style="text-align:left;"> SI1 </td>
   <td style="text-align:left;"> Good </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 12283 </td>
   <td style="text-align:left;"> VS2 </td>
   <td style="text-align:left;"> Good </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 13582 </td>
   <td style="text-align:left;"> SI1 </td>
   <td style="text-align:left;"> Premium </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 699 </td>
   <td style="text-align:left;"> VS1 </td>
   <td style="text-align:left;"> Ideal </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:right;"> 8072 </td>
   <td style="text-align:left;"> VS2 </td>
   <td style="text-align:left;"> Ideal </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 9 </td>
   <td style="text-align:right;"> 608 </td>
   <td style="text-align:left;"> VS2 </td>
   <td style="text-align:left;"> Premium </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:right;"> 2929 </td>
   <td style="text-align:left;"> VS2 </td>
   <td style="text-align:left;"> Ideal </td>
  </tr>
</tbody>
</table>
]

<table>
 <thead>
  <tr>
   <th style="text-align:right;"> id </th>
   <th style="text-align:right;"> price </th>
   <th style="text-align:right;"> clarity_1 </th>
   <th style="text-align:right;"> clarity_2 </th>
   <th style="text-align:right;"> clarity_3 </th>
   <th style="text-align:right;"> clarity_4 </th>
   <th style="text-align:right;"> clarity_5 </th>
   <th style="text-align:right;"> clarity_6 </th>
   <th style="text-align:right;"> clarity_7 </th>
   <th style="text-align:right;"> clarity_8 </th>
   <th style="text-align:right;"> cut_1 </th>
   <th style="text-align:right;"> cut_2 </th>
   <th style="text-align:right;"> cut_3 </th>
   <th style="text-align:right;"> cut_4 </th>
   <th style="text-align:right;"> cut_5 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 8848 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 3968 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 13849 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 4480 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 12283 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 13582 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 699 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:right;"> 8072 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 9 </td>
   <td style="text-align:right;"> 608 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:right;"> 2929 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
</tbody>
</table>

---

# Distributions

---

# Distributions depend on the data

- A normal distribution describes continuous, numeric data

- Bernoulli and Binomial distributions describes random variables that take binary values

- A uniform distribution can handle discrete or continuous data
]

---

# Terminology of distributions

- **Moments:** Values from our sample that can help us understand our distribution

- **Parameters:** The values that define our distribution]
  - Normal: `$\mu$`, `$\sigma$`
  - Binomial: `$n$`, `$p$`
  - Uniform: `$a$`, `$b$`

- **Probability Density (Mass) Function (PDF)/(PMF):** A description of how likely certain outcomes are in a distribution

- **Cumulative Distribution Function (CDF):** A description of how much of a distribution is contained up to a certain point

---

# Moments of random variables

- Moments describe a distribution with a set of attributes

- `$\text{E}[X^n]$`: general raw moment

- `$\text{E}[X]$`: first moment, `$\mu=\frac{\sum{x}}{n}$`

- `$\text{E}[X^2]$`: second raw moment (unrefined, so contains first moment information)

- `$\text{E}[X^2]$` - `$\text{E}[X]^2$`: second central moment, `$\sigma^2=\frac{\sum{(x-\mu})^2}{n}$`

- For continuous data, we can mathematically represent the second central moment:

`$$\int_{\infty}^{-\infty}f(x)x^2dx - \text{E}[X]^2$$`
where `$f(x)=PDF$`

---

# Uniform PMF/PDF, CDF

<img src="index_files/figure-html/unnamed-chunk-9-1.png" width="504" />
]

<img src="index_files/figure-html/unnamed-chunk-11-1.png" width="504" />
]

---

# Normal PDF, CDF

---

# Method of Moments Calculation — Normal, Uniform

---

# Simulated data: Normal

```python
import random, numpy, math
import statistics as stats, matplotlib.pyplot as plt
import seaborn as sns, pandas as pd

random.seed(50390) # reproducibility
sample_df = numpy.random.normal(
  loc=0, scale = 2, size = 1000
  ) # Normal
sample_df[0:3]
```

```
## array([-0.42501843, -0.93878938, -2.88702722])
```

```python
stats.mean(sample_df)
```

```
## -0.051744188640231566
```

```python
# Solving for sigma using sample variance
math.sqrt(stats.variance(sample_df))
```

```
## 1.9919425634418428
```
]

---

# Mystery data: Uniform

```python
my_nums = pd.read_csv("my_sample.csv") #sample is uniform
my_nums = my_nums["x"].tolist()
my_nums[0:5]
```

```
## [8.74912820779718, 9.16765952832066, 6.329065448371691, 4.0551836041268, 8.59842579229735]
```

```
## Data length: 10000
```

```python
stats.mean(my_nums) #Can we use sample moments to find parameters
```

```
## 5.513906104243407
```

```python
stats.variance(my_nums)
```

```
## 6.673169412967319
```

---

# Calculating Method of Moments: Uniform

`\begin{align}
\text{E}[\mu] & =\int_{\infty}^{-\infty}\frac{1}{b-a}xdx \\
 & =  \frac{X^2}{2(b-a)} \\
 \frac{X^2}{2(b-a)} ]_a^b& = \frac{b^2}{2(b-a)}-\frac{a^2}{2(b-a)} \\
 & = \frac{b^2-a^2}{2(b-a)} \\
 \frac{b^2-a^2}{2(b-a)} & = \frac{{(b-a)}(b+a)}{2(b-a)} \\
 \text{E}[\mu] & = \frac{a+b}{2} \\
\end{align}`

---

# Calculating Method of Moments: Uniform

`$\text{1) } \text{E}[\mu]=5.51=\frac{a+b}{2}$`
`$\text{2) } \text{V}=6.67=\frac{(b-a)^2}{12}$`
]

.right-column[
`\begin{align}
\text{Solve eq. 1 for b: } a+b & =11.02 \\
b & =11.02 - a \\
\end{align}`

`\begin{align}
\text{Plug b into eq. 2:  }
\frac{(11.02-a-a)^2}{12} & =6.67 \\
11.02-2a & =6.67\sqrt{12} \\
-2a& =6.67\sqrt{12}-11.02 \\
a& =\frac{6.67\sqrt{12}-11.02}{-2} \\
a& =1.04 \\
\end{align}`

`\begin{align}
\text{Complete eq. 1 for b: } b & =11.02 - 1.04 \\
b & =9.98 \\
\end{align}`
]

---

# Checking Method of Moments with the data

```python
print(f"a = {min(my_nums)}, max = {max(my_nums)}")
```

```
## a = 1.0011236150749, max = 9.999872184358539
```

---

# Sampling and descriptive statistics

---

# Choosing a representative sample

- Our goal when modeling is to use a sample to draw conclusions about a greater population

- It is important that our sample is representative of our population; Ensuring we have a sufficient sample size is helpful to accomplish this goal

- Sample Moments = Theoretical Moments! If the sample is not representative, this equation will mislead us

- Methods of repeated sampling are also good to utilize, Central Limit Theorem

---

# Central Limit Theorem

```python
dice = [1, 2, 3, 4, 5, 6] #uniform
```

```python
# Central limit
# Draws from Uniform Distribution becoming normal distribution
rolls = 6
dice_means = [0] * rolls
for i in range(0,rolls):
    this_roll = random.choices(dice,k=rolls)
    dice_means[i] = stats.mean(this_roll)
```

---

# Using descriptive statistics from a sample

- Measure of central tendency:
  - Mean — arithmetic average
  - Median — middle quantile (based on data sorted)
  - Mode — most common value
- Variance
- Range, IQ range
- Correlation — the foundation of linear regression