Measures of Central Tendency

class: center, middle, inverse, title-slide

.title[
# Measures of Central Tendency
]
.subtitle[
## EDP 613
]
.author[
### Week 3
]

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# Basic Idea

The mean, median and mode are measures of central tendency and attempt to summarize the typical value of a variable.

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# Why?

These may help us draw conclusions about a specific group or compare different groups using a single numerical value.

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# Recall Distributions

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# Measures of Central Tendency: The Mean

>- The *average* number.

>- There are other types of means (e.g. geometric, harmonic, etc.) but we are only using the *arithmetic* mean.

>- Essentially the **balancing point** or center of mass of a distribution

>- Found by adding all data points and dividing by the number of data points

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# Measures of Central Tendency: The Median

>- The *middle* number

>- Essentially the **point that cuts a data set in half**

>- Found by ordering data points from least to greatest or greatest to least and locating the middle number - if there are two middle data points, they are averaged

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# Measures of Central Tendency: The Mode

>- The *most frequent* number

>- Essentially the point that occurs the most
  
--

>- Found by determining the data point(s) that appear the most - if none exists, then there is no mode

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# Basic Procedure: The Mean

>- **Mean** 
  - Add the numbers up, divide by the total number of values in the set.
  - Denoted by `$\overline{Y}$`

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## Example

Compute the mean for the following sample: `$\{21.3, 31.4, 12.7, 41.6\}$`

--
### Solution

`\begin{aligned}
\overline{Y}&=\dfrac{21.3+31.4+12.7+41.6}{4}\\\\
&=\dfrac{107}{4}\\\\
&=26.75
\end{aligned}`

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## Give it a Try

Compute the mean for the following sample: `$\{2, 5, 5, 7, 7, 8, 9\}$`

### Solution

`\begin{aligned}
\overline{Y}&=\dfrac{2+5+5+7+7+8+9}{6}\\\\
&=\dfrac{43}{7}\\\\
&\approx6.14
\end{aligned}`

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# Basic Procedure: The Median

>- **Median** 
    - Put the numbers in order from least to greatest or greatest to least and find the middle number. 
    - If there are two middle numbers, average them.

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## Example

Compute the median for the following sample: `$\{2, 5, 5, 7, 7\}$`

### Solution

>- Since these data point are already in numerical order, we can use them as is without reordering.

>- `$n=5$` which is an odd number so we can locate the median by `$$\dfrac{n+1}{2}=\dfrac{5+1}{2}=\dfrac{6}{2}=3$$`
telling us to look in the *third position* from either side of the list of numbers.

>- In `$$\{2, 5, 5, 7, 7\}$$` the middle number is `$5$` so that must be the median!

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## Give it a Try

Compute the mean for the following sample: `$\{21.3, 31.4, 12.7, 41.6\}$`

### Solution

>- Since these data point are NOT already in numerical order, we must reorder them.

`$$\{12.7, 21.3, 31.4, 41.6\}$$`
--

>- `$n=4$` which is an even number so we can locate the median by taking the mean of the the numbers in

---

>+ `$\dfrac{n}{2}=\dfrac{4}{2}=2$`, or
the *second position*

>+ `$\dfrac{n}{2}+1=\dfrac{4}{2}+1=3$`, or the *third position*
 
--

>- So the median is

`$$\dfrac{21.3+31.4}{2} = 26.35$$`

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# Basic Procedure: The Mode

>- **Mode** 
  - Find the number(s) that appear the most. 
  - If none exists, then there is no mode.

---

## Example

Compute the mode for the following sample: `$\{2, 5, 5, 7, 7\}$`

### Solution

<table class="table" style="width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:center;"> Data point </th>
   <th style="text-align:center;"> Frequency </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> 2 </td>
   <td style="text-align:center;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 5 </td>
   <td style="text-align:center;"> 2 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 7 </td>
   <td style="text-align:center;"> 2 </td>
  </tr>
</tbody>
</table>

>- The data points 5 and 7 repeat twice while 2 only appears once.

>- The modes are 5 and 7.

>- Known as *bimodal*. Three modes would be *trimodal* and so on.

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## Give it a Try

Compute the mode for the following sample: `$\{21.3, 31.4, 12.7, 41.6\}$`

### Solution

>- No data point appears more than once points appear once.

>- Therefore there is no mode.

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# Something to Think About

>- A statistic is **resistant** if its value is not affected by extreme values (large or small) in the data set.

>- Which of the measures of central tendency are resistant?

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## That's it. We will work more with R next week!