##### Appliied Statistics

Pareto Diagram

is a type of chart that contains both bars and a line graph, where individual values are represented in descending order by bars, and the cumulative total is represented by the line.

Class Percentage

is the class relative frequency multiplied by 100; that is:

- class percentage = (class frequency) * 100

q

denotes the probability of failure P(F) in a binomial trial.

- q = 1 - P

s (lowercase)

represents the sample space (collection of all sample point).

Sample Standard Deviation

Standard deviation measures the spread of a data distribution. It measures the typical distance between each data point and the mean.

- if the data is a sample from a larger population, we divide by one fewer than the number of data points in the sample, n-1n−1n, minus, 1.

Population Standard Deviation

Standard deviation measures the spread of a data distribution. It measures the typical distance between each data point and the mean.

- If the data is being considered a population on its own, we divide by the number of data points, N.

Standard Deviation

Standard deviation measures the spread of a data distribution. It measures the typical distance between each data point and the mean.

P (uppercase)

the probability of success P(S) in a binomial trial.

F (uppercase)

used to denote the failure of a binomial trail.

S (uppercase).

used to denote the success of a binomial trail.

Permutation

is an arrangement of all or part of a set of objects, with regard to the order of the arrangement. For example, suppose we have a set of three letters: A, B, and C. we might ask how many ways we can arrange 2 letters from that set.

Permutation Formula

The number of permutations of n objects taken r at a time is determined by the following formula:

P(n,r) = n!(n − r)!

Example

A code have 4 digits in a specific order, the digits are between 0-9. How many different permutations are there if one digit may only be used once?

A four digit code could be anything between 0000 to 9999, hence there are 10,000 combinations if every digit could be used more than one time but since we are told in the question that one digit only may be used once it limits our number of combinations. In order to determine the correct number of permutations we simply plug in our values into our formula:

P(n,r)=10!(10−4)!=10⋅9⋅8⋅7⋅6⋅5⋅4⋅3⋅2⋅16⋅5⋅4⋅3⋅2⋅1=5040

In our
example the order of the digits were important, if the order didn't
matter we would have what is the definition of a combination. The
number of combinations of n objects taken r at a time is determined by
the following formula:

C(n,r)=n!(n−r)!r!

Combination

is a selection of all or part of a set of objects, without regard to the order in which objects are selected.

For example, suppose we have a set of three letters: A, B, and C.
We might ask how many ways we can select 2 letters from that set.
Each possible selection would be an example of a **combination**.

Combination Formula

A formula for the number of possible combinations of *r*
objects from a set of *n* objects. This is written in any of
the ways shown below.

All forms are read aloud "*n* taken *r*."

Formula:

Note:

Where _{ n}P_{ r} is the
formula for permutations of *n *objects taken *r* at a time.

Poisson Distrbution

is a statistical distribution showing the likely number of times that an event will occur within a specified period of time. It is used for independent events which occur at a constant rate within a given interval of time.

The characteristics of a Poisson Distribution

- It is a discrete distribution.
- Each occurrence is independent of the other occurrences.
- It describes discrete occurrences over an interval.
- The occurrences in each interval can range from zero to infinity.
- The mean number of occurrences must be constant throughout the experiment.

Poisson Formula

is characterized by lambda, λ, the mean number of occurrences in the interval. If a Poisson-distributed phenomenon is studied over a long period of time, λ is the long-run average of the process. The Poisson formula is used to compute the probability of occurrences over an interval for a given lambda value.

Hypergeometric Distribution

is a discrete probability distribution that describes the probability
of **
k** successes (random draws for which the
object drawn has a specified feature) in

**draws,**

*n**without*replacement, from a finite population of size {\displaystyle N} that contains exactly

**objects with that feature, wherein each draw is either a success or a failure. In contrast, the binomial distribution describes the probability of**

*K**} successes in*

**k***draws*

**n***with*replacement.

Characteristics of a Hypergeometric Distribution

Hypergeometric Experiment

is a statistical experiment that has the following properties:

- A sample of size
*n*is randomly selected without replacement from a population of*N*items. - In the
population,
*k*items can be classified as successes, and*N - k*items can be classified as failures.

**Hypergeometric Probability**

h(*x*; *N*, *n*, *k*): the probability
that an *n*-trial hypergeometric experiment results in
__exactly__ *x* successes, when the population consists of
*N* items, *k* of which are classified as successes.

Binomial Formula