PADP 7120 Data Applications in PA

class: center, middle, inverse, title-slide

.title[
# PADP 7120 Data Applications in PA
]
.subtitle[
## Data Description
]
.author[
### Alex Combs
]
.institute[
### UGA | SPIA | PADP
]
.date[
### Last updated: September 14, 2026
]

---

# Outline

- Distinguish between descriptive and inferential statistics

- Identify population, sample, parameter, and statistic

- Distributions and their description

---
# Two categories of statistics

.center[
![](lectures_files/stats_types.png)
]

---
# Population vs. Sample

- **Population:** all members of a specified group pertaining to a research question

- A population can be any size based on our research question
  
  - If we can observe the population, all we need to do is describe it to reach useful conclusions

- **Sample:** a subset of our target population

- We can describe a sample or a population

- If we cannot observe the population, we take a sample and use inferential statistics to reach useful conclusions about that population

---
# Descriptive vs. inferential

- Suppose we survey a subset of MPA graduates to learn about employment and earning outcomes.

- Are the following questions **descriptive** or **inferential**?

- What is the average income of respondents?

- What is the average income of MPA graduates?

- What percent of respondents are employed?

- Does an MPA increase income?

---
# Parameter vs. statistic

- **Parameter:** a measure pertaining to a population

- The "true" value we try to estimate if the population is unobserved

- **Statistic:** a measure pertaining to a sample

- Also referred to as an **estimate**

---
# Survey Example

Suppose our survey of UGA MPA alumni receives 100 responses and and the median income of respondents is $80,000.

- Is this a sample statistic or a population parameter?

---
# Descriptive vs. inferential

Suppose we want to know the average salary of college-educated women in Georgia. We survey a sample of 1,000 college-educated women in Georgia and record their income.

- What is the population and what is the sample?

- What is the parameter we intend to estimate?

- What is the statistic we will use to estimate the parameter?

---
class: inverse, middle, center

# Data description

---
# Distribution

- A distribution shows the (possible) values for a variable and how often they occur.

---
# Describing numerical variables

- Center
  - Mean, median, mode
- Spread
  - Variance, standard deviation, IQR, range
- Association
  - Covariance, correlation
  - Regression coefficient, coefficient of determination (will cover later)

---
# Measures of center

- Mean: the balancing point of the distribution
- Median: the middle of the distribution (50th percentile)
- Mode: the peak(s) of the distribution

---
# Mean (average)

`$$\bar{x}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}={\frac {x_{1}+x_{2}+\cdots +x_{n}}{n}}$$`

- Sum of all values and divide by the number of values

.pull-left[

<table>
 <thead>
  <tr>
   <th style="text-align:right;"> ID </th>
   <th style="text-align:right;"> variable </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 2 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 4 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 6 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 8 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 10 </td>
  </tr>
</tbody>
</table>
]

.pull-right[

``` r
(2 + 4 + 6 + 8 + 10)/5
```

```
## [1] 6
```

<img src="lectures_files/mean.png" alt="" width="1105" />
]

---
# Mean

- In R, can use the `mean(data$variable)` function

``` r
mean(data$variable)
```

```
## [1] 6
```

``` r
mean(ncbirths$weeks)
```

```
## [1] 38.4675
```

---
# Median

- Arrange values in order, find the middle value
- If no middle value because even number of values, average the two middle values

.pull-left[
<table>
 <thead>
  <tr>
   <th style="text-align:right;"> ID </th>
   <th style="text-align:right;"> variable </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 2 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 4 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 6 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 8 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 10 </td>
  </tr>
</tbody>
</table>
]

.pull-right[

``` r
median(data$variable)
```

```
## [1] 6
```

![](lectures_files/median.png)

``` r
median(ncbirths$weeks)
```

```
## [1] 39
```

]

---
# Mode

- The value that occurs most frequently

.pull-right[

- Variable has no mode. One more of any of the 5 values would make that value the mode.

]

---
# Mean vs. median vs. mode

- We use measures of center to communicate the *typical* value of a distribution.

- Which measure best conveys what is typical depends on the distribution.

![](lectures_files/center.png)

---
# Mean vs. median vs. mode

- Mean is sensitive to extreme values, median is not

- Median may be better to use when a distribution is skewed

- Mode is useful for discrete or categorical variables with few possible values

---
# Mean vs. median vs. mode

- If distribution is symmetrical, little difference between mean and median

.pull-left[
<img src="Description_files/figure-html/unnamed-chunk-13-1.png" alt="" style="display: block; margin: auto;" />
]

.pull-right[

``` r
mean(ncbirths$weeks)
```

```
## [1] 38.4675
```

``` r
median(ncbirths$weeks)
```

```
## [1] 39
```
]

---
# Mean vs. median vs. mode

- As distribution becomes skewed, mean and median begin to differ

.pull-left[
<img src="Description_files/figure-html/unnamed-chunk-15-1.png" alt="" style="display: block; margin: auto;" />
]

.pull-right[

``` r
mean(gapminder$gdpPercap)
```

```
## [1] 7215.327
```

``` r
median(gapminder$gdpPercap)
```

```
## [1] 3531.847
```
]

---
class: inverse, middle, center

# Measures of spread

---
# Measures of spread

- **Variance:** the average squared deviation from the mean

- **Standard deviation:** the average deviation from the mean

- **Interquartile range:** the difference between 75th and 25th percentiles

- **Range:** the difference between the maximum and minimum values

---
# Variance

`$$S^2={\frac {1}{n-1}}\sum _{i=1}^{n}(x_{i}-\bar{x})^2={\frac {(x_{1}-\bar{x})^2+(x_{2}-\bar{x})^2+\cdots +(x_{n}-\bar{x})^2}{n-1}}$$`

<table>
 <thead>
  <tr>
   <th style="text-align:right;"> ID </th>
   <th style="text-align:right;"> variable </th>
   <th style="text-align:right;"> xbar </th>
   <th style="text-align:right;"> deviation </th>
   <th style="text-align:right;"> devsquared </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> -4 </td>
   <td style="text-align:right;"> 16 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> -2 </td>
   <td style="text-align:right;"> 4 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 6 </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;"> 8 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 4 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 16 </td>
  </tr>
</tbody>
</table>

``` r
(16+4+0+4+16)/4
```

```
## [1] 10
```

---
# Standard deviation (SD)

- Variance does not make sense as a *descriptive* measure of spread

``` r
var(ncbirths$weeks)
```

```
## [1] 7.583423
```

- On average, weeks pregnant deviates from the average by 7.6 weeks squared.

- Standard deviation recovers the original variable

``` r
sd(ncbirths$weeks)
```

```
## [1] 2.753802
```

- On average, weeks pregnant deviates from the average by almost 3 weeks.

---
# Interquartile range (IQR)

- Divide the distribution into 4 equal parts
- Each dividing value is the 25th, 50th, and 75th percentiles
- IQR is the difference between 25th and 75th percentiles

``` r
quantile(ncbirths$weeks, c(.25, .5, .75))
```

```
## 25% 50% 75% 
##  38  39  40
```

- The IQR for weeks pregnant is 2

``` r
IQR(ncbirths$weeks)
```

```
## [1] 2
```

---
# SD vs. IQR

- We use SD and IQR to communicate the typical deviation of a distribution from its center
- SD is based on the mean; sensitive to extreme values
- IQR uses percentiles; not sensitive to extreme values

.pull-right[

``` r
sd(extreme$variable)
```

```
## [1] 42.54409
```

``` r
IQR(extreme$variable)
```

```
## [1] 4
```
]

---
# SD vs. IQR

.pull-left[
![](Description_files/figure-html/unnamed-chunk-26-1.png)
]

.pull-right[

``` r
sd(ncbirths$weeks)
```

```
## [1] 2.753802
```

``` r
IQR(ncbirths$weeks)
```

```
## [1] 2
```
]

---
# SD vs. IQR

.pull-left[
<img src="Description_files/figure-html/unnamed-chunk-28-1.png" alt="" style="display: block; margin: auto;" />
]

.pull-right[

``` r
sd(gapminder$gdpPercap)
```

```
## [1] 9857.455
```

``` r
IQR(gapminder$gdpPercap)
```

```
## [1] 8123.402
```
]

---
# Range

.pull-left[
- Describes the greatest extent to which the variable changes

- Or the possible values of the variable

- Or how different are the most different observations
]

.pull-right[

``` r
range(ncbirths$weeks)
```

```
## [1] 20 45
```

``` r
range(gapminder$gdpPercap)
```

```
## [1]    241.1659 113523.1329
```
]