**Definition**

In statistics, when we talk about distributions we usually mean

probability distributions.

Definition

(informal): A distribution is a function

that shows the possible

values for a variable and how often they occur.

Definition

(Wikipedia): In probability theory and statistics,

a probability distribution is a

mathematical function that, stated in

simple terms, can be thought of as providing the probabilities of occurrence of different possible outcomes in an experiment.

Examples:

Normal distribution, Student’s T distribution, Poisson distribution, Uniform distribution, Binomial distribution

Graphical representations

It is a common mistake to believe

that the distribution is the graph.

In

fact, the distribution is the ‘rule’

that determines how values are

positioned in relation to each other.

Very often, we use a graph to

visualize the data. Since

different distributions have a particular graphical representation, statisticians like to plot them.

The Normal distribution is also known as Gaussian distribution or the Bell

curve. It is one of the most common distributions due to the following reasons:

•

It approximates a wide variety

of random variables

•

Distributions of sample means

with large

enough samples

sizes could

be approximated to normal

•

All computable statistics are elegant

•

Heavily used in regression

analysis

. Good track record

Examples:

•

Biology. Most biological measures are normally

distributed, such as: height; length

of arms, legs, nails; blood pressure; thickness of tree

barks, etc.

•

IQ tests

•

Stock market

information

__Controlling for the standard deviation__

Keeping the standard deviation constant, the graph of a normal distribution with:

• a smaller mean would look in the same way, but be situated to the left (in gray)

• a larger mean would look in the same way, but be situated to the right (in red)

**Controlling for the mean**

Keeping the mean constant,

a normal distribution with:

• a smaller

standard deviation would

be situated in the same spot, but have a higher

peak and thinner tails (in red)

• a larger standard deviation

would be situated

in the same spot, but have a lower

peak and fatter tails (in gray)

The Standard Normal

distribution is a particular case

of the Normal

distribution. It has a mean of 0

and a standard deviation of 1.

Every Normal distribution

can be ‘standardized’ using

the standardization formula:

A variable following the Standard Normal distribution is denoted with the letter z.

Why standardize?

Standardization allows us to:

• compare different

normally distributed

datasets

• detect normality

• detect outliers

• create confidence intervals

• test hypotheses

• perform regression analysis

Rationale of the formula for standardization:

We want to transform a random variable from N~ μ, σ² to N~(0,1).

Subtracting the mean from all observations would cause a transformation from N~ μ,σ²

to N~ 0, σ² , moving the graph to the origin.

Subsequently, dividing all observations by the standard deviation would

cause a transformation from N~ 0, σ² to N~ 0,1, standardizing the peak and

the tails of the graph.

1- Types of data and level of measurement
2- Graphs and Tables that Represent Categorical Variables
3- Excel formulas
4- Graphs and tables that represent numerical variables
5- Graphs and Tables for Relationships Between Variables.
6- Mean, Median, Mode
7- Variance and Standard Deviation
8- Covariance and Correlation
9- Distributions
10- The Central Limit Theorem
11- Estimators and Estimates
12- Confidence Intervals and the Margin of Error
13- Student’s T Distribution
14- Formulas for Confidence Intervals
15- Scientific method
16- Hypotheses
17- Decisions You Can Take
18- Statistical Errors (Type I Error and Type II Error)
19- P-Value
20- Formulae for Hypothesis Testing
21- Basics
22- Linear regression equation
23- How to do linear regression in Excel with Analysis ToolPak
24- Interpret regression analysis output
25- How to make a linear regression graph in Excel
26- How to do regression in Excel using formulas