Engineering Data Analysis

STATISTICS

• It involves the methods of collecting, processing. analyzing, and summarizing data in order to provide answers or solutions to an inquiry.

• It is defined as a science that studies data to be able to make a decision.

DESCRIPTIVE STATISTICS

Describe what is there in our data

DESCRIPTIVE STATISTICS

Make inferences from our data to more general conditions

DATA

It is a collection of facts from experiments, observations, sample surveys and censuses, and administrative reporting systems

QUALITATIVE

• categorical data

• labels

• cannot be measured

QUANTITATIVE

• numerical data

• can be measured

Nominal Data

unordered categories

Ordinal Data

ordered categories

Discrete Data

• whole numerical value (countable)

• cannot be broken down to smaller parts

• ex. population

Continuous Data

• can take any value within range

• can be broken down to smaller parts

• ex. time

UNIVERSE

It is the collection or set of units or entities from whom we got the data.

VARIABLE

It is a characteristic that is observable or measurable in every unit of the universe.

POPULATION

The set of all possible values of a variable

SAMPLE

A subgroup of a universe or of a population is a sample.

Slovin’s Formula

To solve how many samples needed

err (margin of error)

= 1 - confidence level

If not stated: 5% = 0.05

• probability for committing mistakes

• probability for the research to be in erroe

Confidence level

• If not stated: 95% = 0.95

• Confidence that the research is correct

the margin of error decreases

As the sample increases

location

Location of Median

Median

The middle value when the data is arranged in ascending or descending order.

Mode

The value(s) that occur most frequently

Mean

The average-sum of all values divided by the number of values

MEASURE OF CENTRAL TENDENCY

Values that describe the center or typical value of a dataset.

MEASURE OF LOCATION

Values that describe the relative position of an observation within the dataset.

PERCENTILE

Values that divide the ordered dataset into 100 equal parts.

QUARTILES

Values that divide the ordered dataset into 4 equal parts.

DECILES

Values that divide the ordered dataset into 10 equal parts.

Population Mean

Sample Mean

MEASURE OF VARIATION

Values that describe the variability or spread of a dataset.

RANGE

• Difference between the maximum and minimum values,

• Sensitive to extreme values.

INTERQUARTILE RANGE (IQR) / midspread

• Measure the middle 50% of data

• calculated by subtracting the first quartile from the third quartile, making it robust against outliers.

VARIANCE

• Average of the squared deviations from the mean

• measures how far values are spread.

STANDARD DEVIATION / Effective Variance

• The square root of variance

• Typical distance of values from the mean

Population Variance

Population Standard Deviation

Sample Variance

Sample Standard Deviation

Caltech for mean, population SD, sample SD

Coefficient of variation (CV)

Tells how much the data varies relative to its mean.

P75

3rd Quartile (Q3) in Percentile

P25

1st Quartile (Q1) in Percentile

1st FUNDAMENTAL PRINCIPLE OF COUNTING

• If a thing can be done in m different ways and another thing can be done in n different ways, then the two things can be done one after the other in m times n different ways.

• “AND”

2nd FUNDAMENTAL PRINCIPLE OF COUNTING

• If a thing can be done in m different ways and another thing can be done in n different ways, then either of these two things can be done in m plus n different ways.

• “OR”

PERMUTATION

An ordered arrangement of a finite number of elements, either all of the available n element or of a part of them

Permutation of n distinct objects taken n at a time:

Permutation of n distinct objects taken r at a time

Permutation of repeated objects

• When some objects are identical, swapping identical objects does not create a new arrangement.

• To avoid counting duplicates, divide by the factorial of the number of identical objects.

Permutations with grouped objects or permutations with objects required to stay together

*like chain rule

• get the # of ways of the groups then multiply with # of ways within a group

Cyclic Permutation

denominator is the repeated objects

Circular arrangement with identical objects

Treat each group as one block, use (number of blocks−1)!, then multiply by the internal arrangements of each block

Circular arrangement with grouped objects

Necklace/bracelet (rotations and reflections considered the same)

COMBINATION

• A grouping arrangement of all or of any elements of a set regardless of the order.

• “Choose”

Combination of n distinct objects taken r at a time:

SETS

• Any collection of distinct objects.

• Ordering of elements are unnecessary.

VENN DIAGRAM

Pictorial representation of sets.

Union

Contains all elements that are in either set or in both sets.

Intersection

Contains only the elements common to both sets

Sample space, S

• Universal set

• The set of all possible outcome of a random variable

Outcome

• Element

• The result of a single trial.

Event, E

• Set

• Collection of 1 or more outcomes

The probability of an event happening is:

P(S) = 1

Probability of sure event

P (null) = 0

Probability of impossible event

0 < P(A) < 1

Probability of always non-negative event

COMPLEMENT OF AN EVENT

MUTUALLY EXCLUSIVE

[ADDITION RULE “OR”]
Two or more events that can never happen in the same trial
P (A∩B) = 0

NON-MUTUALLY EXCLUSIVE

[ADDITION RULE “OR”]
Two or more events that can happen in the same trial

INDEPENDENT

[MULTIPLICATION RULE “AND”]
An event is independent if the outcome of one trial has no effect on the outcome of any other trial.

DEPENDENT

[MULTIPLICATION RULE “AND”]
One event has an effect on the outcome of the next event.

Non-mutually exclusive and independent

CONDITIONAL PROBABILITY

• P(B | A) is the probability of B given that A has occurred.

• Since A is known to have occurred, it becomes the new sample space replacing the original.

Bayes’ Theorem

• a mathematical formula used to calculate the probability of an event based on new evidence or prior knowledge.

• It allows you to update your belief in a hypothesis (the "cause") when a related event (the "evidence") occurs.

DISCRETE PROBABILITY DISTRIBUTION

• a probability distribution that depicts the occurrence of discrete (individually countable) outcomes, such as 1,2,3… or zero

Mean / average (Expected value) in Discrete Probability

Variance in Discrete Probability

Standard Deviation in Discrete Probability

BINOMIAL OR REPEATED TRIALS

There are two possible outcomes of an event, and the possibilities of the outcome are independent and constant.

∴2 outcomes, constant probability for each outcome, trials are independent

for conditions with “at most” or “at least”

[Binomial or repeated trials]
Use summation

HYPERGEOMETRIC DISTRIBUTION

• Describes the probability of getting a specific number of successes in a fixed number of draws, without replacement, from a finite population containing a certain number of successes and failures.

• Probability is not given, only number of trials

POISSON'S DISTRIBUTION

• A limiting case of a Binomial distribution when the number of trials, n, gets very large and p, the probability of success, is small.

• ↑n = ↓p

• Average / mean

λ in formula for Poisson’s Distribution

=np = (number of trials)(probability)

= mean/average value

=variance

n ≥ 20, number of trials in the mean (Poisson’s Distribution)

n ≥ 100, number of trials in the mean (Poisson’s Distribution)

for conditions with “more than”

[Poisson’s distribution]
Use 1 - summation of less than the specified value

Binomial

Type of Experiment for Repeated Trials with replacement

Hypergeometric

Type of Experiment for drawing without replacement from a finite population

Poisson’s

Type of Experiment for Counts of events in a continuous Interval (e g. time)

Typically large

Population Size for Binomial

Finite and known

Population Size for Hypergeometric

Infinite or undefined

Population Size for Poisson’s

n; number of trials
p; probability of success

Key Parameters for Binomial

N: population size
K: successes in population
n: number drawn

Key parameters for Hypergeometric

What is the probability of k successes in n independent trials, each with the same chance of success?

Quick Intuition for Binomial

What is the probability of k success in n dependent draws from a known group without replacements?

Quick Intuition for Hypergeometric