Central Tendency

ABSTRACT

Central tendency refers to the statistical measures used to identify the “center” or “typical” value of a dataset. These metrics provide a summary that represents the entire distribution with a single point.

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1. Primary Measures

Mean

• The man of a set of quantitative variables is given by:

$$\bar{x}=\boxed{\frac{1}{n}\sum _{i=1}^n{x}_i=\frac{x_1+x_2+\cdots x_n}{n}}$$

The mean of $(1,2,3,4,10,x)$ is $3.3333$. What is $x$?

Answer $x$ we will go backwards on the operations in finding the mean $\bar{x}$

To find

$$\begin{array}{rl}\bar{x}& =3.3333\\ n& =6\\ x& =\bar{x}\cdot n-(\sum _{i=1}^{n-1}{x}_i)\\ x& =3.3333\cdot 6-\sum _{i=1}^5{x}_i\\ & =\mathrm{19.999...}-(1+2+3+4+10)\\ & =20-20\\ x& =\boxed{0}\end{array}$$

Median

• Let the data points be $x_1,x_2,...,x_n$ arranged in non-decreasing order ($x_i\le x_{i+1}$ for all $i$). Then the median, $M$, is:

$$M=\{\begin{array}{ll}{X}_{\frac{n+1}{2}}& \text{if }n\text{ is odd}\\ \frac{X_{\frac{n}{2}}+X_{(\frac{n}{2}+1)}}{2}& \text{if }n\text{ is even}\end{array}$$

The median of $(1,2,3,4,10,x)$ is $2.5$. What is $x$?

Answer $(2,3,4)$ to have the set that we look for, which the $\frac{2+3}{2}=2.5$, therefore $2$ and $3$ should be the middle 2 values. With that in mind, it is required that $x\le 2$ to satisfy this requirement

Since we find the median by taking the middle 2 values and take their average, we see

Mode

• The mode of data points $x_1,x_2,\cdots ,x_n$ is the value which appears most frequently

The mode of $(1,2,3,4,10,x)$ is $1$. What is $x$?

Answer $1$ to be considered the most, the value $1$ must appear more than once.

Each known value in the data points are only themselves. In order for

With that in mind, it is clear that $x=1$ to satisfy the condition

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2. Comparing Measures by Distribution Shape

The relationship between the mean, median, and mode changes based on the Shape of the distribution.

Distribution Shape	Relationship
Symmetric (Normal)	Mean $\approx$ Median $\approx$ Mode
Right-Skewed (Positive)	Mode $<$ Median $<$ Mean
Left-Skewed (Negative)	Mean $<$ Median $<$ Mode

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3. When to Use Which?

• Use Mean when the data is symmetric and you need to account for every value in the set.
• Use Median when you want to describe the “typical” experience in a skewed dataset (e.g., Household Income).
• Use Mode when you are dealing with non-numerical categories (e.g., “What is the most popular car color?”).