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Statistics Cheatsheet

Definitions, Symbols, Formulas, and Notes — All in One Place.

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Measures of Central Tendency

Term		Definition		Formula
Mean		The arithmetic average of a set of values.		$\frac{\sum_{i = 1}^{n} x_{i}}{n}$
Median		The middle value of a dataset.		For $n$ even: $\tilde{x} = \frac{Value at position (\frac{n}{2}) + Value at position (\frac{n}{2} + 1)}{2}$ For $n$ odd: $\tilde{x} = Value at position (\frac{n + 1}{2})$
Mode		The most frequently occurring value in a dataset.
Central Limit Theorem		A theorem stating that the sampling distribution of the sample mean of a random variable approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution, provided that the sample size is sufficiently large.
Weighted Mean		The average of a set of values, each multiplied by a corresponding value called its weight.		$\frac{\sum_{i = 1}^{n} (x_{i} \times w_{i})}{\sum_{i = 1}^{n} w_{i}}$
Trimmed Mean		A trimmed mean is calculated by removing a certain percentage of the lowest and highest values from a dataset and then calculating the mean of the remaining values.		$\frac{\sum_{i = p + 1}^{n - p} x_{i}}{n - 2 p}$
Mean Absolute Deviation		A measure of the average absolute deviation of a set of values from their mean.		$\frac{1}{n} \sum_{i = 1}^{n} \| x_{i} - \bar{x} \|$ Where $\bar{x}$ is the mean of the dataset.
Median Absolute Deviation		A measure of the spread of data points around the median.		$median (\| X_{i} - median (X) \|)$
Moving Average		A statistical technique used to smooth out fluctuations in data by averaging data points over a specified period. It helps identify trends by reducing the impact of random variations.		${SMA}_{k} = \frac{x_{t - k + 1} + x_{t - k + 2} + \dots + x_{t}}{k}$ Where ${SMA}_{k}$ is the simple moving average for period $k$ , $x_{t}$ represents the value at time $t$ , and $k$ is the number of periods over which the averaging is performed.

Inferential Statistics

Term		Definitions		Formula
Hypothesis Testing		A statistical method for making inferences about population parameters based on sample data.
p-value		A measure of the strength of evidence against the null hypothesis in hypothesis testing.
Type I Error		Occurs when the null hypothesis is incorrectly rejected when it is actually true.
Type II Error		Occurs when the null hypothesis is incorrectly accepted when it is actually false.
Effect Size		A measure of the magnitude of the difference or relationship between two groups or variables in a statistical analysis.
Z-test		A statistical test used to compare a sample mean to a known population mean when the population standard deviation is known.		$Z = \frac{\bar{x} - μ}{\frac{σ}{\sqrt{n}}}$
F-test		A statistical test used to compare the variances or standard deviations of two or more groups.
False Positive Rate
True Positive Rate
T-test		A statistical test used to compare the means of two groups and determine whether they are significantly different from each other.		$t = \frac{{\bar{x}}_{1} - {\bar{x}}_{2}}{\sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}}$
Linear Regression		A statistical method for modeling the relationship between a dependent variable and one or more independent variables by fitting a linear equation to observed data.
Recall
Accuracy
Precision

Experimental Design

Measures of Variability

Term		Definitions		Formula
Minimum		The smallest value in a dataset.
Maximum		The largest value in a dataset.
Range		The difference between the maximum and minimum values in a dataset.
Interquartile Range		The range within which the middle 50% of the data values in a dataset lie.		$IQR = Q_{3} - Q_{1}$ where $Q_{3}$ is the third quartile and $Q_{1}$ is the first quartile.
Outliers		Data points that significantly differ from the rest of the dataset.
Sum of Squares		The sum of the squared differences between each data point and the mean of the dataset.		$\sum_{i = 1}^{n} (x_{i} - \bar{x})^{2}$ where $\bar{x}$ is the sample mean.
Standard Deviation		A measure of the amount of variation or dispersion in a set of values.		$\sqrt{\frac{1}{n} \sum_{i = 1}^{n} (x_{i} - \bar{x})^{2}}$
Coefficient of Variation		A measure of relative variability, calculated as the ratio of the standard deviation to the mean, expressed as a percentage.		$\frac{Standard Deviation}{Mean} \times 100 %$
Upper Fence		The threshold beyond which data points are considered potential outliers.		$Q_{3} + 1.5 \times IQR$
Lower Fence		The threshold below which data points are considered potential outliers.		$Q_{1} - 1.5 \times IQR$
Standard Error of the Mean		An estimate of how much the sample mean is expected to vary from the true population mean.		$\frac{Standard Deviation}{\sqrt{n}}$
Confidence Intervals		A range of values that is likely to contain the true population parameter, with a certain level of confidence.		$CI = \bar{x} \pm z \times \frac{s}{\sqrt{n}}$ where $\bar{x}$ is the sample mean, $s$ is the sample standard deviation, $n$ is the sample size, and $z$ is the critical value from the standard normal distribution for the desired confidence level.

Basic Probability

Term		Definition		Symbol
Sample Space		The set of all possible outcomes of a random experiment.
Event		A subset of the sample space, representing a collection of outcomes of interest.
Independence		Two events are independent if the occurrence of one event does not affect the occurrence of the other.
Conditional Probability		The probability of an event $A$ occurring given that another event $B$ has already occurred.		$P (A \| B)$
Random Variables		A variable whose possible values are numerical outcomes of a random phenomenon.
Bayes' Theorem		A formula that calculates the probability of an event A given that event B has occurred.		$P (A \| B) = \frac{P (B \| A) \times P (A)}{P (B)}$
Addition Rule		The probability of the union of two independent events A and B is the sum of their individual probabilities		$P (A \cup B) = P (A) + P (B)$
Multiplication Rule		The probability of the intersection of two independent events A and B is the product of their individual probabilities.		$P (A \cap B) = P (A) \times P (B \| A)$

Discrete Probability Distributions

Term		Definition		Probability Mass Function
Binomial Distribution		A discrete probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials.		$P (X = k) = (\binom{n}{k}) p^{k} (1 - p)^{n - k}$
Poisson Distribution		A discrete probability distribution that describes the number of events occurring in a fixed interval of time or space.		$P (X = k) = \frac{e^{- λ} λ^{k}}{k!}$
Bernoulli Distribution		A probability distribution representing a single trial with two possible outcomes: success (usually denoted as 1) and failure (usually denoted as 0).		$P (X = x) = p^{x} (1 - p)^{1 - x}$
Geometric Distribution		A probability distribution representing the number of trials needed to achieve the first success in a sequence of independent and identically distributed Bernoulli trials, each with probability $p$ of success.		$P (X = k) = (1 - p)^{k - 1} p$
Hypergeometric Distribution		A discrete probability distribution that describes the probability of a specified number of successes in a fixed number of draws without replacement from a finite population.		$P (X = k) = \frac{(\binom{K}{k}) (\binom{N - K}{n - k})}{(\binom{N}{n})}$
Negative Binomial Distribution		A probability distribution that models the number of failures that occur before a specified number of successes $r$ in a sequence of independent Bernoulli trials, each with probability $p$ of success.		$P (X = k) = (\binom{k + r - 1}{k}) p^{r} (1 - p)^{k}$
Multinomial Distribution		A probability distribution that generalizes the binomial distribution to more than two categories.		$P (X_{1} = x_{1}, X_{2} = x_{2}, . . ., X_{k} = x_{k}) = \frac{n!}{x_{1}! x_{2}! . . . x_{k}!} p_{1}^{x_{1}} p_{2}^{x_{2}} . . . p_{k}^{x_{k}}$
Probability Mass Function		A function that gives the probability of a discrete random variable taking on a particular value.		$P (X = x)$

Continuous Probability Distributions

Term		Definitions		Probability Density Function
Normal Distribution		A probability distribution that is symmetric, bell-shaped, and characterized by its mean and standard deviation.		$f (x) = \frac{1}{\sqrt{2 π} σ} e^{- \frac{(x - μ)^{2}}{2 σ^{2}}}$
Log-Normal Distribution		A probability distribution of a random variable whose logarithm is normally distributed.		$f (x) = \frac{1}{x σ \sqrt{2 π}} e^{- \frac{(\ln (x) - μ)^{2}}{2 σ^{2}}}$ Where $μ$ and $σ$ are the mean and standard deviation of the natural logarithm of the variable.
Exponential Distribution		A probability distribution that describes the time between events in a Poisson process, where events occur continuously and independently at a constant average rate. It is commonly used to model the waiting times between successive events, such as the time between arrivals of customers at a service center or the lifespan of electronic components.		$f (x) = λ e^{- λ x}$ Where $λ$ is the rate parameter.
Chi-Squared Distribution		A probability distribution that arises in the context of hypothesis testing and is characterized by a single parameter, often denoted as kk, representing the degrees of freedom. The chi-squared distribution is the distribution of the sum of the squares of kk independent standard normal random variables.		$f (x) = \frac{1}{2^{k / 2} Γ (k / 2)} x^{(k / 2) - 1} e^{- x / 2}$ Where $k$ is the degrees of freedom and $Γ$ is the gamma function.
Beta Distribution		A probability distribution defined on the interval [0, 1], often used to model random variables representing proportions, probabilities, or percentages. The beta distribution is characterized by two shape parameters, denoted as αα and ββ, which determine the shape of the distribution.		$f (x) = \frac{1}{Beta (α, β)} x^{α - 1} (1 - x)^{β - 1}$ Where $α$ and $β$ are the shape parameters and $Beta$ is the beta function.
Mixed Probability Distribution		A probability distribution that is a mixture of two or more component distributions. Each component distribution is weighted by a probability, and the mixture distribution is obtained by summing or integrating the component distributions according to their weights
Joint Distribution		A probability distribution that describes the simultaneous behavior of two or more random variables.
Probability Mechanism		A function that describes the likelihood of a continuous random variable falling within a particular range of values.

Data Anomalies

Entropy

Measures of Shape

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