Statistics Cheatsheet

Mean () (population)

where:

• = total number of data points
• = each data point

Mean () (sample)

where:

• = number of data points in the sample

Median

• Ifis odd:

• Ifis even:

Mode

• Value that appears most frequently in the dataset.

Variance () (population)

Standard Deviation () (population)

Variance () (sample)

Standard Deviation () (sample)

Range

Probability of an event

Complementary Probability

Addition Rule (for mutually exclusive events)

Addition Rule (for non-mutually exclusive events)

Multiplication Rule (for independent events)

Multiplication Rule (for dependent events)

whereis the probability ofgiven thathas occurred.

As the size of a sample increases, the sample mean will approach the expected mean of the population.

Weak Form

For a sequence of independent and identically distributed (i.i.d.) random variables with a meanand variance, the sample mean will converge in probability to the population mean as the sample size increases

for any.

This means that the sample mean tends to get closer to the population mean, but there is no guarantee that this will happen in every case.

Strong Form

States that the sample mean converges almost surely (with probability 1) to the population mean, as the sample size approaches infinity

This means that the sample mean will definitely equal the population mean when the number of trials is infinite.

Example

We roll a die 1000 times and calculate the mean of the results. As the number of rolls (n) increases, the mean of the results will approach the expected value ()

If we roll the die once, we can get any number from 1 to 6, but as we increase the number of rolls, the mean of those rolls will approach 3.5.

Fundamental Counting Principle

If one task can be performed inways and another task can be performed inways, then the two tasks can be performed inways.

Permutations ofelements

Permutations ofelements with repetitions

Combinations ofelements takenat a time

Combinations with repetition

Binomial expansion

A Bernoulli trial is a random experiment that has the following characteristics:

• Discrete Outcomes: It has only two possible outcomes, typically called success (usually represented by 1) and failure (represented by 0).

• Constant Probability: The probability of successis constant in each trial. Consequently, the probability of failure is.

• Independence: The trials are independent, meaning the outcome of one trial does not affect the outcome of another.

Examples

• Tossing a coin, where “heads” can be considered a success and “tails” a failure
• Taking an exam, where “passing” is considered a success and “failing” a failure
• Measuring the effectiveness of a medical treatment in which the outcome can be “effective” (success) or “ineffective” (failure)

Discrete distributions are those that describe probabilities of variables that can only take specific and finite values, such as integers.

Binomial Distribution

Models the number of successes in a sequence of independent trials, each with the same probability of success.

where:

• = number of trials
• = number of successes
• = probability of success in a single trial
• = binomial coefficient

Negative Binomial Distribution

Models the number of failures before achieving a fixed number of successes in Bernoulli trials.

where:

• is the number of successes required
• is the probability of success
• is the number of failures

Geometric Distribution

Models the number of trials until the first success in a sequence of Bernoulli trials.

where:

• is the probability of success
• is the number of trials until the first success

Hypergeometric Distribution

Models the number of successes in a fixed-size sample drawn without replacement from a finite population.

where:

• is the population size
• is the number of successes in the population
• is the sample size

Poisson Distribution

Models the number of events that occur in a fixed interval of time or space when events occur with a constant average rate.

where:

• is the average rate of occurrence of the events
• is the number of events

Normal (Gaussian) Distribution

Models many variables in nature and society. It is symmetric and bell-shaped.

where:

• is the mean
• is the variance

Cumulative Distribution Function of the Normal (CDF)

Exponential Distribution

Models the time between events in a Poisson process. It is used in reliability theory and waiting times.

where:

• is the rate of occurrence of the events.

Uniform Distribution

Models a variable that has the same probability of taking any value within a defined interval.

where:

• andare the limits of the interval.

Gamma Distribution

Generalizes the exponential distribution. Models the time untilevents occur in a Poisson process.

where:

• is a shape parameter
• is a rate parameter

Beta Distribution

Primarily used in Bayesian statistics to model probability distributions in proportions or probabilities.

where:

• andare shape parameters
• is the beta function

Cauchy Distribution

Models phenomena where the mean and variance are undefined or infinite.

where:

• is the location
• is the scale parameter

Student’s t-distribution

Used to estimate the mean of a normally distributed population when the sample size is small and the variance is unknown.

where:

• are the degrees of freedom, which determine the shape of the distribution. Asincreases, the t-distribution converges to a standard normal distribution.

Pearson Correlation Coefficient ()

Regression Line Equation

where:

• = slope
• = intercept

Slope

where:

• andare the standard deviations ofand, respectively.

Point Estimation (sample mean)

Confidence Interval for the Mean

whereis the critical value from the normal distribution.

Hypothesis Test (p-value)

• Ifis the null hypothesis andis the alternative hypothesis:

Test Statistic for the Mean

Pearson Correlation Coefficient ()

Regression Line Equation

where:

• = slope
• = intercept

Slope

where:

• andare the standard deviations ofand, respectively.