# Central Tendency

Describing the dataset

Central tendency is **"the statistical measure that identifies a single value as representative of an
entire distribution"**. It aims to provide an accurate description of the entire dataset. It is the single value
that is most representative of the dataset.

The **mean**, **median** and **mode** are the three commonly used measures of central tendency.

**Mode**: is a value/range that occurred with the highest frequency.**Median**: is the number that lies in the**middle of a list of ordered numbers**. The numbers may be in the ascending or descending order. such that there is an equal probability of falling above or below it. Simply put, it is the middle value in the list of numbers.**Mean**: arithmetic average of a range of values or quantities, computed by dividing the total of all values by the number of values.

## 1. Mode

Mode is a value/range that occurred with the highest frequency.

### Mode Characteristics

a. Uniform Distribution: NO MODE.

b. Distribution can have two modes like in "Foot Size" distribution for Men and Women has TWO modes, one for Men "Foot Size" and another for women. The distribution is a bimodal distribution.

c. Modes can be used to describe both catagorical and numerical data.

d.

All scores in the dataset DOESNOT affect the mode. In the sense, if a new outliner value is added to the dataset, there is no influence on the mode value of the dataset. e. Samples from the population WILL have DIFFERENT modes or NO mode depending on the sample values.

f. There is NO equation for the mode.

## 2. Median

Number that lies in the **middle of a list of ordered numbers**. The numbers may be in
the ascending or descending order such that there is an equal probability of falling above or below it. Simply put,
it is the middle value in the list of numbers.

For **even** samples : \( \begin{equation} Median = \frac{X_\frac{n}{2}+ X_{\frac{n}{2}+1}}{2}\end{equation} \)

'X' is the sample in sample space

'n' is the number of samples in sample space

For **odd** samples : \( \begin{equation} Median = X_{\frac{n+1}{2}}\end{equation} \)

'X' is the sample in sample space

'n' is the number of samples in sample space

### Median Characteristics

a.

All scores in the dataset DOES NOT affect the median. In the sense, if a new outliner value is added to the dataset, effectively there is NO change in the medianvalue of the dataset.b.

The Medianisrobustin representing the the central/middle of the sample/population space. This is because outliners will have veryless effect on median.c. Samples from the population WILL have DIFFERENT medians as

middle of a list of ordered numbersmight vary between sample spaces of a population.

## 3. Mean

Arithmetic average of a range of values or quantities.

Sample space Mean: \( \begin{equation} \bar{x}=\frac{1}{n}\sum_{i=1}^n x_i\end{equation} \)

'n' is the number of samples in sample space

\( \begin{equation} x_i\end{equation} \) is the ith sample space value

Population Mean: \( \begin{equation} \mu=\frac{1}{N}\sum_{i=1}^N x_i\end{equation} \)

'N' is the number of samples in population space

\( \begin{equation} x_i\end{equation} \) is the ith population space value

### Mean Characteristics

a.

All scores in the dataset DOES affect the mode. In the sense, if a new outliner value is added to the dataset, effectively there is a change in the meanvalue of the dataset.b. Mean of the sample can be used to make inferences about the population it came from.

c.

The Meancan be misleading if the datasets hasoutliners. This is because outliners creates skewed distribution by pulling the mean towards outliners. This makes the mean lot less representative of middle of the data.d. Samples from the population WILL have SIMILAR means (exception to outliners as explained in

cabove).

## Summary

Central Tendency | Equation? | Changes with Sample values | Affected by bin size? | Affected by outliners? | Easy to find on histogram |
---|---|---|---|---|---|

Mean | Yes | Yes | No | Yes | No |

Median | Yes | No | No | No | No |

Mode | No | No | Yes | No | Yes |

x = No / Not much

## References:

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3127352/

https://in.udacity.com/course/intro-to-descriptive-statistics--ud827

## Share this post

Twitter

Google+

Facebook

Reddit

LinkedIn

StumbleUpon

Pinterest

Email