Principles & Methods of Statistics

Probability

The likelihood or chance that something will happen.

Data

A record of things that have actually happened or been observed.

Population

The entire group of individuals or objects that a researcher wants to study.

Sample

A smaller group selected from a population to collect information.

Target Population

The specific population that a researcher wants to learn about.

Representative Sample

A sample that accurately reflects the characteristics and diversity of the population.

Convenience Sample

A sample selected because it is quick and easy to access.

Random Sample

A sample selected by chance rather than personal preference.

Simple Random Sample (SRS)

A sample in which every member of the population has an equal chance of being selected and every possible sample of the same size is equally likely to be selected.

Bias

A systematic error that causes research results to be misleading or unrepresentative.

Selection Bias

Occurs when a sample systematically excludes or underrepresents certain members of a population.

Self-Selection Bias

Occurs when individuals choose whether or not to participate in a study.

Survivor Bias

Occurs when a study only includes individuals who survive or remain in a population, leaving out those who do not.

Pseudo-Random Numbers

Numbers generated by a computer algorithm that appear random but are produced by a deterministic process.

Seed

The starting number used by a computer to generate pseudo-random numbers.

Random Number Generator

A tool or algorithm used to generate random numbers for selecting samples.

Prospective Study

A study in which researchers select a sample and monitor it as time passes.

Retrospective Study

A study in which researchers select a sample and look backward at its past experiences or records.

Observational Data

Data collected by observing individuals without controlling or changing their circumstances.

Experimental Data

Data collected through experiments in which researchers deliberately change certain factors.

Controlled Experiment

An experiment in which relevant factors are held constant to isolate the effect of one variable.

Confounding Factor

An outside variable that influences results and mixes up the effects researchers are trying to study.

Multiple Regression

A statistical method used to estimate the effect of one factor while accounting for other factors.

Behavioral Economics

The study of how psychological and emotional factors influence economic decisions.

Classical Economic Theory

The theory that assumes people make rational and dispassionate economic decisions.

Sample Proportion

The proportion of individuals in a sample who have a particular characteristic.

Population Proportion

The actual proportion of individuals in the entire population who have a particular characteristic.

Parameter

A numerical characteristic of an entire population that does not change from sample to sample.

Statistic

A numerical value calculated from sample data that can vary from sample to sample.

Random Variable (X)

A variable whose value is determined by a random process.

Sampling Error

The difference between the sample mean and the population mean caused by random sampling.

Systematic Error

An error that consistently causes results to be too high or too low because of a problem with the data collection or measurement process.

Sampling Distribution

The probability distribution of a statistic, such as the sample mean, across many possible random samples.

Unbiased Estimator

A sample statistic whose sampling distribution has a mean equal to the true population parameter.

Estimator

A sample statistic used to estimate an unknown population parameter.

Sampling Variance

The amount of variation in a statistic across different random samples.

Standard Error (SE)

The estimated standard deviation of the sampling distribution of the sample mean.

Confidence Interval (CI)

A range of values calculated from sample data that is used to estimate an unknown population parameter.

Confidence Level

The percentage of confidence intervals that would contain the true population parameter over many repeated samples.

Margin of Error

The amount added to and subtracted from the sample mean to create a confidence interval.

t Distribution

A probability distribution used to calculate confidence intervals when the population standard deviation is unknown.

z Statistic

A standardized value that follows a normal distribution with a mean of 0 and standard deviation of 1 when its assumptions are met.

t Statistic

A standardized value used when estimating the population mean with an unknown population standard deviation.

Degrees of Freedom (df)

The number of independent pieces of information available when estimating a statistic; for a sample mean and sample standard deviation, it equals n minus 1.

Spurious Precision

Reporting more decimal places than the accuracy of the data or calculation justifies.

Sample Size (n)

The total number of observations or individuals included in a sample.

Finite Population

A population containing a limited, countable number of individuals or objects.

Infinite Population

A population treated as unlimited in size for statistical calculations.

Sampling Without Replacement

Selecting individuals from a population without allowing the same individual to be selected more than once.

Random Variable

Takes on numerical values that are determined by the outcomes of a random process/experiment

Sample Space

All possible outcomes

Probability Distribution (RV)

Tells us the probability of each possible value (x) the RV (X) can take on

Properties of Probability Distribution

1. 0 ≤ P(x) ≤ 1

2. 𝛴_xP(x) = 1

Covariance

A measure of linear association between two RVs

Permutations

How many ways you can arrange r objects selected from n objects when order matters

Combinations

How many ways you can arrange r objects selected from n objects when order does not matters

Binomial Probability Checks

1. Two mutually exclusive outcomes for each trial

2. Probability of success is fixed

3. Trials are independent

Central Limit Theorem

If z = sum of n independent identically distributed RVs, then the probability of z approaches normal distribution as n increases

Normal Distribution Checks

1. CLT: n ≥ 30

2. np > 5, n(1 - p) > 5

3. Independent & mutually exclusive

P(z < a) =

1 - P(z > a)

P(z < -a) =

P(z > a)

P(z > -a) =

1 - P(z > a)

P(a < z < b) =

P(z > b) - P(z > a)