07: Measures of Centrality

Review: Summation Notation

The Fibonnaci Sequence $\{1,1,2,3,5,8,13,...\}$ is an example of a sequence where

$F_1=1,F_2=1,F_n=F_{n-1}+F_{n-2}$

Compute

• ${\displaystyle \sum _{i=1}^1{a}_i}$
• ${\displaystyle \sum _{i=1}^2{a}_i}$
• ${\displaystyle \sum _{i=1}^3{a}_i}$
• ${\displaystyle \sum _{i=1}^4{a}_i}$

Populations versus Samples

We tend to study a relatively small sample to understand the behavior of a much larger population.

Mean (Average)

• “mean” and “average” are synonymous and will be used interchangably

• The mean of $\{x_1,x_2,...,x_n\}$ is denoted by

  • Greek letter $\mu$ (“mu”) for a population mean (where we know all of the elements)
  • Anglicized $\overline{x}$ (“x bar”) for a sample mean (where we are working with a sample of data)

• To calculate the mean

  • Add up all of the numbers
  • Divide by the amount of numbers

$$\mu ={\displaystyle \frac{1}{N}{\displaystyle \sum _{i=1}^N{x}_i\text{or}\overline{x}={\displaystyle \frac{1}{n}{\displaystyle \sum _{i=1}^n{x}_i}}}}$$

Example: One Die

• Find the mean of the roll of one six-sided die.
• Find the mean of the sample $\{21,22,23,32\}$.

Median

The median of an ordered, discrete set of numbers is the number in the middle. If there are an even amount of data, then the median is the average of the middle two numbers in the ordered data set.

• Compute the median of $\{1,2,1,5,3\}$
• Compute the median of $\{1,1,2,3,5,8\}$

Weighted Mean

$$\overline{x}={\displaystyle \frac{{\displaystyle \sum _{i=1}^n{w}_i\cdot x_i}}{{\displaystyle \sum _{i=1}^n{w}_i}}}$$

Example: Cal Kulas

Setting

Going into the final exam for a Statistics course, Cal Kulas had earned the following marks in the other categories.

Category	Weight	Cal Kulas
attendance	10%	95%
quizzes	20%	75%
midterms	25%	60%
project	20%	90%

Tasks

• What is his current grade in the course?
• What does Cal Kulas need on the final exam so that he earns at least 80% in the course?

Review: Summation Notation

The Fibonnaci Sequence $\{1,1,2,3,5,8,13,...\}$ is an example of a sequence where

$F_1=1,F_2=1,F_n=F_{n-1}+F_{n-2}$

Compute

• ${\displaystyle \prod _{i=1}^1{a}_i}$
• ${\displaystyle \prod _{i=1}^2{a}_i}$
• ${\displaystyle \prod _{i=1}^3{a}_i}$
• ${\displaystyle \prod _{i=1}^4{a}_i}$

For Formulas

In particular, sigma and product notation will allow us to express probability formulas more efficiently. For example, the independence formula for two events $A$ and $B$,

$$P(AB)=P(A)\cdot P(B)$$ becomes the following for $n$ independent events:

$$P(\{X_i{\}}_{i=1}^n)={\displaystyle \prod _{i=1}^{n}P(X_i)}$$

Looking Ahead

• due Fri., Feb. 3:
  • WHW3
  • JHW1
  • Identity Statement (short essay)
• and the before-lecture quizzes

Exam 1 will be on Wed., Mar. 1

Discussion section 24D (Wed., 530 PM) now meets in GRAN 135

Even NASA makes mistakes!

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