The Ultimate Statistics Formula Cheat Sheet

A quick reference guide for the most essential formulas in statistics. Find the equation you need, understand when to use it, and translate the variables.

Adjusted R-Squared ($R^2_{adj}$): Formula:
$$R^2_{adj} = 1 – \left[ \frac{(1 – R^2)(n – 1)}{n – k – 1} \right]$$

When to use it:
To evaluate a multiple regression model. Regular $R^2$ artificially goes up every time you add a new variable, even if it’s useless. Adjusted $R^2$ penalizes you for adding bad variables, giving you a much more honest accuracy score.

Variables:
$R^2$ = The standard Coefficient of Determination
$n$ = The sample size
$k$ = The number of independent variables (predictors) in the model

ANOVA (Analysis of Variance) F-Test: Formula:
$$F = \frac{MST}{MSE}$$

When to use it:
To compare the means of three or more distinct groups at the same time to see if at least one of them is significantly different from the rest (doing this prevents the compounded error of running multiple t-tests).

Variables:
$F$ = The calculated F-statistic
$MST$ = Mean Square Treatment (the variance between the different groups)
$MSE$ = Mean Square Error (the variance within the individual groups)

Autocorrelation Function (ACF): Formula:
$$r_k = \frac{\sum (y_t – \bar{y})(y_{t-k} – \bar{y})}{\sum (y_t – \bar{y})^2}$$

When to use it:
To measure how strongly a data point in a time series is correlated with its own past values (e.g., seeing if today’s stock price is heavily influenced by the price exactly 7 days ago).

Variables:
$r_k$ = The autocorrelation at lag $k$
$y_t$ = The data value at time $t$
$y_{t-k}$ = The data value at $k$ time periods in the past (the “lag”)
$\bar{y}$ = The overall mean of the time series

Autoregressive Model (AR): Formula:
$$y_t = c + \phi_1 y_{t-1} + \phi_2 y_{t-2} + \dots + \phi_p y_{t-p} + \epsilon_t$$

When to use it:
To forecast future values based strictly on a linear combination of its own past values. It assumes that past patterns will continue into the future.

Variables:
$y_t$ = The forecasted value at time $t$
$c$ = A constant (intercept)
$\phi_1, \phi_2, \dots$ = The coefficients for each lagged time period
$y_{t-1}, y_{t-2}, \dots$ = The actual values from past time periods
$\epsilon_t$ = White noise (random error)

Bayes’ Theorem: Formula:
$$P(A\vert{}B) = \frac{P(B\vert{}A) \times P(A)}{P(B)}$$

When to use it:
To update the probability of a hypothesis (Event A) based on new, incoming evidence (Event B). It is famously used in medical testing to determine the likelihood you actually have a disease given a positive test result.

Variables:
$P(A\vert{}B)$ = The posterior probability (probability of A given B)
$P(B\vert{}A)$ = The likelihood (probability of B given A)
$P(A)$ = The prior probability (initial probability of A)
$P(B)$ = The total or marginal probability of B

Binomial Distribution: Formula:
$$P(X = x) = \binom{n}{x} p^x (1 – p)^{n – x}$$

When to use it:
To calculate the probability of getting a specific number of “successes” in a fixed number of trials, where there are only two possible outcomes (like flipping a coin 10 times and wanting exactly 7 heads).

Variables:
$P(X = x)$ = The probability of getting exactly $x$ successes
$\binom{n}{x}$ = The number of combinations (how many ways you can choose $x$ successes from $n$ trials)
$n$ = The total number of trials
$x$ = The number of successful trials you are looking for
$p$ = The probability of success on a single trial
$(1 – p)$ = The probability of failure on a single trial

Chi-Square Test ($\chi^2$): Formula:
$$\chi^2 = \sum \frac{(O_i – E_i)^2}{E_i}$$

When to use it:
To analyse categorical data (like eye colour or voting preference). It tells you if there is a significant difference between the frequencies you expected to see and what you actually observed.

Variables:
$\chi^2$ (chi-square) = The test statistic
$O_i$ = The observed frequency (actual count) for a specific category
$E_i$ = The expected frequency for that category
$\sum$ = The sum of these calculations across all categories

Coefficient of Determination ($R^2$): Formula:
$$R^2 = 1 – \frac{\sum (y_i – \hat{y}_i)^2}{\sum (y_i – \bar{y})^2}$$

When to use it:
To evaluate how well your regression line actually fits the data. It tells you the exact percentage of the variation in your Y variable that can be explained by your X variable. (In simple linear regression, this is simply the square of the Pearson correlation coefficient, $r$).

Variables:
$R^2$ = The coefficient of determination
$y_i$ = The actual observed values of Y
$\hat{y}_i$ = The predicted values of Y (from your regression line)
$\bar{y}$ = The average of all actual Y values
$\sum (y_i – \hat{y}_i)^2$ = The Sum of Squares of Residuals (unexplained variance)
$\sum (y_i – \bar{y})^2$ = The Total Sum of Squares (total variance)

Coefficient of Variation (CV): Formula:
$$CV = \frac{s}{\bar{x}} \times 100$$

When to use it:
To compare the relative spread or volatility of two different datasets, especially when they are measured in completely different units (e.g., comparing the price volatility of a ₹10 stock versus a ₹500 stock).

Variables:
$CV$ = The Coefficient of Variation (usually expressed as a percentage)
$s$ = The sample standard deviation
$\bar{x}$ = The sample mean

Cohen’s d (Effect Size): Formula:
$$d = \frac{\bar{x}_1 – \bar{x}_2}{s_{pooled}}$$

When to use it:
To measure the effect size (the actual practical significance) between two group means. While a T-test just tells you if a difference exists, Cohen’s d tells you how massive or trivial that difference actually is, standardizing it so it isn’t skewed by large sample sizes.

Variables:
$d$ = The calculated effect size (generally, 0.2 is small, 0.5 is medium, 0.8 is large)
$\bar{x}_1, \bar{x}_2$ = The sample means of group 1 and group 2
$s_{pooled}$ = The pooled standard deviation (the combined spread of both groups)

Combinations (Order Doesn’t Matter): Formula:
$$C(n, r) = \frac{n!}{r!(n – r)!}$$

When to use it:
To calculate the total number of possible selections when picking a group from a larger pool, where the order is completely irrelevant (like dealing a 5-card poker hand).

Variables:
$n$ = The total number of items in the pool
$r$ = The number of items being chosen
$!$ = Factorial

Conditional Probability: Formula:
$$P(A\vert{}B) = \frac{P(A \cap B)}{P(B)}$$

When to use it:
To find the probability of Event A happening, given that Event B has already happened and reduced the pool of possible outcomes.

Variables:
$P(A\vert{}B)$ = The probability of A given B
$P(A \cap B)$ = The probability of both A and B occurring
$P(B)$ = The total probability of Event B occurring

Confidence Interval for a Mean (Using T): Formula:
$$CI = \bar{x} \pm t^* \left( \frac{s}{\sqrt{n}} \right)$$

When to use it:
To find the confidence interval for a population mean when you have a small sample size ($n < 30$) and do not know the true population standard deviation.

Variables:
$CI$ = The Confidence Interval
$\bar{x}$ = The sample mean
$t^*$ = The critical T-value (based on your confidence level and degrees of freedom, $n-1$)
$s$ = The sample standard deviation
$n$ = The sample size

Confidence Interval for a Mean (Using Z): Formula:
$$CI = \bar{x} \pm z^* \left( \frac{\sigma}{\sqrt{n}} \right)$$

When to use it:
To find a range of values that you are confident contains the true population mean. Use this version when your sample size is large ($n \ge 30$) or you know the exact population standard deviation.

Variables:
$CI$ = The Confidence Interval (lower and upper bound)
$\bar{x}$ = The sample mean
$z^*$ = The critical Z-value (e.g., 1.96 for 95% confidence)
$\sigma$ = The population standard deviation
$n$ = The sample size

Confidence Interval for a Proportion: Formula:
$$CI = \hat{p} \pm z^* \sqrt{\frac{\hat{p}(1 – \hat{p})}{n}}$$

When to use it:
To find a range of values that likely contains the true population percentage based on a sample percentage (like estimating the true percentage of voters who support a candidate).

Variables:
$CI$ = The Confidence Interval
$\hat{p}$ = The sample proportion (e.g., 0.55 for 55%)
$z^*$ = The critical Z-value
$n$ = The sample size

Confidence Interval for the Difference Between Two Means: Formula:
$$CI = (\bar{x}_1 – \bar{x}_2) \pm t^* \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$$

When to use it:
To estimate a range of values for the true difference between the averages of two independent groups (e.g., comparing the average test scores of Class A versus Class B).

Variables:
$CI$ = The Confidence Interval
$\bar{x}_1, \bar{x}_2$ = The sample means of group 1 and group 2
$t^*$ = The critical T-value
$s_1^2, s_2^2$ = The sample variances of group 1 and group 2
$n_1, n_2$ = The sample sizes of group 1 and group 2

Confidence Interval for the Difference Between Two Proportions: Formula:
$$CI = (\hat{p}_1 – \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1 – \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 – \hat{p}_2)}{n_2}}$$

When to use it:
To estimate a range for the true difference in percentages between two independent groups (e.g., estimating the difference in conversion rates between website design A and design B).

Variables:
$CI$ = The Confidence Interval
$\hat{p}_1, \hat{p}_2$ = The sample proportions of group 1 and group 2
$z^*$ = The critical Z-value
$n_1, n_2$ = The sample sizes of group 1 and group 2

Expected Value: Formula:
$$E(X) = \sum [x_i \times P(x_i)]$$

When to use it:
To calculate the long-term average outcome of a random scenario if you were to repeat it infinitely (like calculating the average payout of a lottery ticket).

Variables:
$E(X)$ = The Expected Value
$x_i$ = The numerical value of a specific outcome
$P(x_i)$ = The probability of that specific outcome occurring

Exponential Distribution: Formula:
$$f(x) = \lambda e^{-\lambda x}$$

When to use it:
To calculate the probability of the amount of time passing between two events occurring. It is the continuous cousin of the Poisson distribution (e.g., predicting how many minutes until the next customer walks through the door).

Variables:
$f(x)$ = The probability density
$\lambda$ = The rate parameter (average number of events per time period)
$x$ = The specific amount of time passed
$e$ = Euler’s number ($\approx 2.71828$)

Exponential Smoothing (Simple): Formula:
$$F_{t+1} = \alpha y_t + (1 – \alpha) F_t$$

When to use it:
To forecast data without a clear trend or seasonality. Unlike a simple moving average where all past data carries equal weight, exponential smoothing gives more weight to recent observations, making it react faster to sudden changes.

Variables:
$F_{t+1}$ = The forecast for the next time period
$\alpha$ = The smoothing constant (a chosen number between 0 and 1; higher means more weight on recent data)
$y_t$ = The actual observed value in the current period
$F_t$ = The previous forecast for the current period

Geometric Distribution: Formula:
$$P(X = x) = (1 – p)^{x – 1} p$$

When to use it:
To calculate the probability that your very first success occurs exactly on the $x$-th trial (e.g., flipping a coin and wanting to know the exact probability that your first “Heads” happens on the 4th flip).

Variables:
$P(X = x)$ = The probability that the first success is on trial $x$
$p$ = The probability of success on any single trial
$(1 – p)$ = The probability of failure on any single trial
$x$ = The specific trial number where the first success happens

Independent Two-Sample T-Test: Formula:
$$t = \frac{\bar{x}_1 – \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$$

When to use it:
To compare the means of two completely distinct, unrelated groups (like a control group and a treatment group) to see if they are significantly different from each other.

Variables:
$t$ = The calculated t-statistic
$\bar{x}_1, \bar{x}_2$ = The means of sample 1 and sample 2
$s_1^2, s_2^2$ = The variances of sample 1 and sample 2
$n_1, n_2$ = The sample sizes of group 1 and group 2

Interquartile Range (IQR): Formula:
$$IQR = Q_3 – Q_1$$

When to use it:
To measure the spread of the middle 50% of your data. Unlike standard deviation, IQR completely ignores extreme outliers, making it perfect for skewed data like income or housing prices.

Variables:
$IQR$ = The Interquartile Range
$Q_3$ = The Third Quartile (75th percentile)
$Q_1$ = The First Quartile (25th percentile)

Kruskal-Wallis H Test: Formula:
$$H = \frac{12}{N(N+1)} \sum \frac{R_i^2}{n_i} – 3(N+1)$$

When to use it:
To compare the medians of three or more independent groups to see if they come from different populations. You use this instead of a One-Way ANOVA when your data is ordinal (ranked) or violates the assumption of normal distribution.

Variables:
$H$ = The Kruskal-Wallis test statistic (which approximates a Chi-Square distribution)
$N$ = The total number of observations across all groups combined
$R_i$ = The sum of the ranks for a specific group (group $i$)
$n_i$ = The sample size of that specific group
$\sum$ = The sum of $\frac{R_i^2}{n_i}$ calculated for every single group

Logistic Regression (Log-Odds / Logit Equation): Formula:
$$\ln\left(\frac{p}{1-p}\right) = b_0 + b_1x$$

When to use it:
To model the relationship between an independent variable and a binary outcome (like Pass/Fail or Spam/Not Spam). Unlike linear regression which predicts a continuous number, this predicts the log-odds of an event happening.

Variables:
$p$ = The probability of the target event occurring (a “success”)
$\frac{p}{1-p}$ = The odds of the event occurring
$\ln$ = The natural logarithm
$b_0$ = The y-intercept
$b_1$ = The coefficient (slope) of the independent variable
$x$ = The known value of the independent variable

Logistic Regression (Probability Form / Sigmoid Function): Formula:
$$p = \frac{1}{1 + e^{-(b_0 + b_1x)}}$$

When to use it:
To convert the raw output of your logistic regression model into a clean, readable probability score strictly between 0 and 1 (e.g., finding out there is an 82% chance a customer clicks “Buy”).

Variables:
$p$ = The predicted probability (between 0 and 1)
$e$ = Euler’s number ($\approx 2.71828$)
$b_0$ = The y-intercept
$b_1$ = The coefficient of the independent variable
$x$ = The known value of the independent variable

Margin of Error (MOE): Formula:
$$E = z^* \left( \frac{s}{\sqrt{n}} \right)$$

When to use it:
To calculate the maximum expected difference between the true population parameter and your sample estimate (like saying a poll is accurate “plus or minus 3%”).

Variables:
$E$ = The Margin of Error
$z^*$ = The critical value (usually from a Z-table, like 1.96 for a 95% confidence level)
$s$ = The sample standard deviation
$n$ = The sample size

Mean Absolute Error (MAE): Formula:
$$MAE = \frac{1}{n} \sum \vert{}y_i – \hat{y}_i\vert{}$$

When to use it:
To measure the average size of the errors in a set of predictions, without considering their direction (positive or negative). It is highly resilient to massive outliers.

Variables:
$n$ = The sample size
$y_i$ = The actual observed value
$\hat{y}_i$ = The predicted value
$\vert{}…\vert{}$ = Absolute value (makes all negative errors positive)

Mean Squared Error (MSE): Formula:
$$MSE = \frac{1}{n} \sum (y_i – \hat{y}_i)^2$$

When to use it:
To measure the average squared difference between estimated values and actual values. Because the errors are squared, MSE heavily penalizes models for making large mistakes (outliers).

Variables:
$n$ = The sample size
$y_i$ = The actual observed value
$\hat{y}_i$ = The predicted value

Minimum Sample Size for Estimating a Mean: Formula:
$$n = \left( \frac{z^* \sigma}{E} \right)^2$$

When to use it:
To figure out exactly how many people or items you need to survey before you start a study, ensuring your final estimate will fall within a specific Margin of Error ($E$).

Variables:
$n$ = The required minimum sample size
$z^*$ = The critical Z-value for your desired confidence level
$\sigma$ = The estimated population standard deviation
$E$ = The desired Margin of Error (the maximum acceptable distance from the true mean)

Minimum Sample Size for Estimating a Proportion: Formula:
$$n = \hat{p}(1 – \hat{p}) \left( \frac{z^*}{E} \right)^2$$

When to use it:
To determine how many subjects you need to poll to estimate a population percentage within a specific Margin of Error. (Note: If you have no historical data to estimate $\hat{p}$, researchers safely use 0.50, which maximizes the required sample size).

Variables:
$n$ = The required minimum sample size
$\hat{p}$ = An initial estimate of the population proportion
$z^*$ = The critical Z-value for your desired confidence level
$E$ = The desired Margin of Error (e.g., 0.03 for a ±3% error)

Minimum Sample Size for Estimating a Proportion: Formula:
$$n = \hat{p}(1 – \hat{p}) \left( \frac{z^*}{E} \right)^2$$

When to use it:
To determine how many subjects you need to poll to estimate a population percentage within a specific Margin of Error. (Note: If you have no historical data to estimate $\hat{p}$, researchers safely use 0.50, which maximises the required sample size).

Variables:
$n$ = The required minimum sample size
$\hat{p}$ = An initial estimate of the population proportion
$z^*$ = The critical Z-value for your desired confidence level
$E$ = The desired Margin of Error (e.g., 0.03 for a ±3% error)

Normal Distribution (Probability Density Function): Formula:
$$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2} \left( \frac{x – \mu}{\sigma} \right)^2}$$

When to use it:
To describe how continuous data is distributed when it perfectly follows a symmetrical “bell curve.” (Note: In practice, most people use the Z-score formula to look up probabilities for the normal curve rather than calculating this raw equation by hand, but it is the foundational mathematical engine behind the curve).

Variables:
$f(x)$ = The probability density at a given value $x$
$x$ = The specific point on the horizontal axis
$\mu$ = The mean (center) of the distribution
$\sigma$ = The standard deviation (spread) of the distribution
$\pi$ = Pi (approximately 3.14159)
$e$ = Euler’s number (approximately 2.71828)

One-Sample T-Test: Formula:
$$t = \frac{\bar{x} – \mu}{s / \sqrt{n}}$$

When to use it:
To determine if your sample mean is significantly different from a known or hypothesised population mean.

Variables:
$t$ = The calculated t-statistic
$\bar{x}$ = The sample mean
$\mu$ = The hypothesised population mean
$s$ = The sample standard deviation
$n$ = The sample size

One-Sample Z-Test for a Mean: Formula:
$$z = \frac{\bar{x} – \mu}{\sigma / \sqrt{n}}$$

When to use it:
Exactly like a one-sample t-test, but used when you actually know the true population standard deviation ($\sigma$) and you have a large sample size (typically $n > 30$).

Variables:
$z$ = The calculated z-statistic
$\bar{x}$ = The sample mean
$\mu$ = The hypothesised population mean
$\sigma$ = The known population standard deviation
$n$ = The sample size

One-Sample Z-Test for a Proportion: Formula:
$$z = \frac{\hat{p} – p_0}{\sqrt{\frac{p_0(1 – p_0)}{n}}}$$

When to use it:
To test if a sample percentage (like a 55% approval rating in a poll) is significantly different from an expected or historical population percentage.

Variables:
$z$ = The calculated z-statistic
$\hat{p}$ (p-hat) = The sample proportion (e.g., 0.55)
$p_0$ = The hypothesised population proportion
$n$ = The sample size

Paired T-Test: Formula:
$$t = \frac{\bar{d}}{s_d / \sqrt{n}}$$

When to use it:
To compare the means of the exact same group of subjects at two different times (such as taking a baseline measurement, applying a treatment, and measuring the same subjects again).

Variables:
$t$ = The calculated t-statistic
$\bar{d}$ = The mean of the differences between the paired observations
$s_d$ = The standard deviation of those differences
$n$ = The total number of pairs

Pearson Correlation Coefficient ($r$): Formula:
$$r = \frac{\sum (x_i – \bar{x})(y_i – \bar{y})}{\sqrt{\sum (x_i – \bar{x})^2 \sum (y_i – \bar{y})^2}}$$

When to use it:
To measure the strength and direction of a linear relationship between two continuous variables. The result always falls between -1 (perfect negative correlation) and 1 (perfect positive correlation).

Variables:
$r$ = The correlation coefficient
$x_i, y_i$ = Individual data points for variables X and Y
$\bar{x}, \bar{y}$ = The sample means of variables X and Y

Permutations (Order Matters): Formula:
$$P(n, r) = \frac{n!}{(n – r)!}$$

When to use it:
To calculate the total number of possible arrangements when selecting a smaller group from a larger pool, where the sequence or order of selection is strictly important (like a PIN code or finishing a race in 1st, 2nd, and 3rd place).

Variables:
$n$ = The total number of items in the pool
$r$ = The number of items being chosen
$!$ = Factorial (e.g., $4! = 4 \times 3 \times 2 \times 1$)

Poisson Distribution: Formula:
$$P(X = x) = \frac{\lambda^x e^{-\lambda}}{x!}$$

When to use it:
To calculate the probability of a given number of events happening in a fixed interval of time or space (like predicting the number of customer emails you will receive in the next hour, assuming they arrive at a known constant rate).

Variables:
$P(X = x)$ = The probability of exactly $x$ events occurring
$\lambda$ (lambda) = The average rate of value or expected number of events in that interval
$x$ = The specific number of events you are checking the probability for
$e$ = Euler’s number (a mathematical constant approximately equal to 2.71828)
$!$ = Factorial

Population Mean: Formula:
$$\mu = \frac{\sum x_i}{N}$$

When to use it:
To find the exact average when you have collected data from every single member of the entire population.

Variables:
$\mu$ (mu) = The population mean
$\sum x_i$ = The sum of all individual values
$N$ = The total number of items in the population

Population Standard Deviation: Formula:
$$\sigma = \sqrt{\frac{\sum (x_i – \mu)^2}{N}}$$

When to use it:
To measure the spread of a dataset when you have collected data from every single member of the entire population (notice it divides by $N$, not $n-1$).

Variables:
$\sigma$ (sigma) = The population standard deviation
$x_i$ = Each individual value
$\mu$ = The population mean
$N$ = The total number of items in the population

Probability of a Single Event: Formula:
$$P(A) = \frac{\text{Number of Favourable Outcomes}}{\text{Total Number of Possible Outcomes}}$$

When to use it:
To find the basic, foundational chance of a single, specific event occurring (like rolling a 4 on a six-sided die).

Variables:
$P(A)$ = The probability of Event A happening

Regression Line Slope ($b_1$): Formula:
$$b_1 = r \left( \frac{s_y}{s_x} \right)$$

When to use it:
To calculate how steep the line of best fit is. It tells you exactly how much Y is expected to change for every one-unit increase in X.

Variables:
$b_1$ = The slope
$r$ = The Pearson correlation coefficient
$s_y$ = The sample standard deviation of the Y variable
$s_x$ = The sample standard deviation of the X variable

Regression Y-Intercept ($b_0$): Formula:
$$b_0 = \bar{y} – b_1\bar{x}$$

When to use it:
To find the exact starting point of your line of best fit. You must calculate the slope ($b_1$) before you can calculate the intercept.

Variables:
$b_0$ = The y-intercept
$\bar{y}$ = The sample mean of the Y variable
$b_1$ = The calculated slope of the regression line
$\bar{x}$ = The sample mean of the X variable

Root Mean Squared Error (RMSE): Formula:
$$RMSE = \sqrt{\frac{1}{n} \sum (y_i – \hat{y}_i)^2}$$

When to use it:
Exactly like MSE, but you take the square root at the end. This converts the error metric back into the original units of your data (e.g., dollars, inches), making it much easier to explain to non-technical stakeholders.

Variables:
$n$ = The sample size
$y_i$ = The actual observed value
$\hat{y}_i$ = The predicted value

Sample Mean (Average): Formula:
$$\bar{x} = \frac{\sum x_i}{n}$$

When to use it:
To find the mathematical centre (average) of a specific sample of data you have collected.

Variables:
$\bar{x}$ (x-bar) = The sample mean
$\sum x_i$ = The sum of all individual values
$n$ = The total number of items in your sample

Sample Standard Deviation: Formula:
$$s = \sqrt{\frac{\sum (x_i – \bar{x})^2}{n – 1}}$$

When to use it:
To measure how spread out your data is when working with a subset (sample) of a larger population. The n-1 applies Bessel’s correction.

Variables:
$s$ = The sample standard deviation
$x_i$ = Each individual value
$\bar{x}$ = The sample mean
$n$ = The sample size

Sample Variance: Formula:
$$s^2 = \frac{\sum (x_i – \bar{x})^2}{n – 1}$$

When to use it:
To find the average squared deviation from the mean in a sample. It is the mathematical precursor to standard deviation, heavily used in ANOVA and regression.

Variables:
$s^2$ = The sample variance
$x_i$ = Each individual value
$\bar{x}$ = The sample mean
$n$ = The sample size

Simple Linear Regression Equation: Formula:
$$\hat{y} = b_0 + b_1x$$

When to use it:
To predict the expected value of a dependent variable (Y) based on a given value of an independent variable (X) using the “line of best fit.”

Variables:
$\hat{y}$ (y-hat) = The predicted value of the dependent variable
$b_0$ = The y-intercept (the value of y when x is exactly 0)
$b_1$ = The slope of the regression line
$x$ = The known value of the independent variable

Simple Moving Average (SMA): Formula:
$$SMA = \frac{y_t + y_{t-1} + y_{t-2} + \dots + y_{t-(n-1)}}{n}$$

When to use it:
To smooth out short-term fluctuations and highlight longer-term trends or cycles. It calculates the average of a specific number ($n$) of the most recent data points.

Variables:
$SMA$ = The Simple Moving Average
$y_t, y_{t-1}, \dots$ = The data points for the current and past periods
$n$ = The total number of time periods in your “window” (e.g., $n = 50$ for a 50-day moving average)

Standard Error of the Mean (SEM): Formula:
$$SE = \frac{s}{\sqrt{n}}$$

When to use it:
To estimate how far your sample mean is likely to be from the true population mean. It is the foundational building block for calculating margin of error and confidence intervals.

Variables:
$SE$ = The Standard Error of the mean
$s$ = The sample standard deviation
$n$ = The sample size

The Addition Rule (The “OR” Rule): Formula:
$$P(A \cup B) = P(A) + P(B) – P(A \cap B)$$

When to use it:
To calculate the probability of Event A or Event B occurring. Subtracting the intersection $P(A \cap B)$ prevents you from double-counting outcomes where both events happen at the same time.

Variables:
$P(A \cup B)$ = The probability of A or B occurring
$P(A)$ = The probability of Event A
$P(B)$ = The probability of Event B
$P(A \cap B)$ = The probability of both A and B occurring together

The Complement Rule (Probability of “Not” A): Formula:
$$P(A’) = 1 – P(A)$$

When to use it:
To find the probability of an event not happening. It is often mathematically easier to calculate the chance of something happening and subtract it from 100% (or 1) than to calculate all the ways it could fail.

Variables:
$P(A’)$ = The probability of Event A not happening (also written as $P(A^c)$)
$P(A)$ = The probability of Event A happening

Two-Sample Z-Test for Proportions: Formula:
$$z = \frac{\hat{p}_1 – \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$$

When to use it:
To compare the percentages of two distinct, independent groups to see if there is a significant difference between them (frequently used in A/B testing, like comparing the conversion rates of two different website layouts).

Variables:
$z$ = The calculated z-statistic
$\hat{p}_1, \hat{p}_2$ = The proportions from sample 1 and sample 2
$n_1, n_2$ = The sizes of sample 1 and sample 2
$\hat{p}$ = The pooled sample proportion (the total number of successes in both groups divided by the total sample size of both groups combined)

Z-Score (Standard Score): Formula:
$$z = \frac{x – \mu}{\sigma}$$

When to use it:
To find out exactly how many standard deviations a specific data point is above or below the average.

Variables:
$z$ = The standard score
$x$ = The specific raw data value
$\mu$ = The population mean
$\sigma$ = The population standard deviation

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