Descriptive Statistics

# Descriptive Statistics
## What to do with lots of numbers
### Matthew Crump
### 2018/07/20 (updated: 2021-11-28)

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# What do lots of number look like?

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# Lots of Numbers look like this

Like this

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# What can we say about them?

We can see they aren't all the same. Not much else really. Looking at a bunch of numbers is hard work.

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# Summary numbers

It would be nice to reduce the big set of numbers down to a few numbers that we can look at and make sense of.

## Sameness (Central Tendency)

- What are all the numbers close to?

## Differentness (Variance)

- How different are the numbers?

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# Descriptive Statistics

- Give us summaries of big sets of numbers

- Useful single numbers to look at

- They tell us about patterns of sameness and differentness

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# Graph the numbers to get a better look

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# Dot plot (unordered)

Graphing the numbers gives a quick and dirty sense of what they are like. Here's 200 numbers presented as dots

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# Dot plot (ordered)

Sorting the numbers from smallest to largest

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# Histograms

Histograms count up the numbers inside specific ranges

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# Histograms

Bars show you which bins have more or less numbers in the range

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# So what are these numbers like?

What single number would you say best describes most of these numbers?

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# Question

Is the red or blue value a better summary of all the numbers?

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# Measures of Central Tendency

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# Central Tendency

1. **Central tendency** should describe what most of the data is like

2. We want our summary number to be most like the other numbers. We want it to be a **representative value**

3. There are **multiple measures** of central tendency

4. They have **different properties**

5. Some work better than others depending on the data

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# Mode

The mode is the single most frequently occuring number

> 1 1 2 2 3 4 5 6 7 7 7 7 7

- The mode is 7 because 7 happens the most

- Find the mode by counting the occurence of each number, the mode is the most frequently occuring number

- If there is a tie, then you have two or three or more modes (depends on how many differen numbers tie)

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# Finding the Mode in R

We make 25 numbers, how do we get R to find the mode?

```r
#make some numbers
a <- round(rnorm(n=25, mean=24, sd=5)) 
```

|   |   |   |   |   |
|--:|--:|--:|--:|--:|
| 29| 30| 16| 28| 22|
| 17| 29| 26| 21| 29|
| 27| 31| 22| 19| 22|
| 27| 28| 30| 29| 24|
| 23| 33| 27| 26| 28|

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# Finding the Mode in R

`table` function automatically counts the occurence of each number

```r
table(a)
```

```
## a
## 16 17 19 21 22 23 24 26 27 28 29 30 31 33 
##  1  1  1  1  3  1  1  2  3  3  4  2  1  1
```

We can see that 29 occurs the most

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# Custom function for the mode in R

You can always write your own function for the mode. This one is called `my_mode`

```r
my_mode <- function(x) {
     ux <- unique(x)
     ux[which.max(tabulate(match(x, ux)))]
 }
```

```r
my_mode(a)
```

```
## [1] 29
```

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# Thinking about the mode

1. Tells us the most frequent number(s)

2. Is it representative of all the numbers?

3. When would the mode be a good thing to know?

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