Essential Statistics Formulas Cheat Sheet
This cheat sheet provides concise definitions and formulas for key statistical measures used to analyze data, understand distributions, and identify relationships between variables.
Core Principles
- Measures of Central Tendency: Mean (average) and Interquartile Range (IQR) summarize data centers.
- Measures of Dispersion: Standard Deviation (sample) and IQR quantify data spread.
- Relationship Analysis: Correlation Coefficient (r) measures linear association strength, while Regression Slope (b) predicts changes.
- Outlier Detection: Values outside Q1 - 1.5*IQR or Q3 + 1.5*IQR are considered outliers.
- Model Fit: R-squared (R²) indicates the proportion of variance explained by a model.
- Standardization: Z-scores transform data to a standard normal distribution.
- Categorical Data Tests: Chi-Square statistics assess goodness-of-fit and independence.
Formulas
- Mean: $ \bar{x} = \frac{\sum x_i}{n} $
- Standard Deviation (Sample): $ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} $
- Interquartile Range (IQR): $ IQR = Q_3 - Q_1 $
- Outlier Rule: Values below $ Q_1 - 1.5 * IQR $ or above $ Q_3 + 1.5 * IQR $
- Correlation Coefficient (r): Measures linear association strength ( -1 to +1).
- Regression Slope (b): $ b = r \frac{s_y}{s_x} $
- R-squared (R²): Proportion of variance in the response variable explained by the explanatory variable.
- Z-score: $ z = \frac{x - \mu}{\sigma} $
- Chi-Square Statistic (Goodness-of-Fit): $ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} $
- Chi-Square Statistic (Independence): $ \chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}} $
Key Terms
- Mean: The average of a dataset, calculated by summing all values and dividing by the number of values.
- Standard Deviation (Sample): A measure of the amount of variation or dispersion in a set of values, calculated from a sample.
- Interquartile Range (IQR): The range between the first quartile (Q1) and the third quartile (Q3) of a dataset, representing the middle 50% of the data.
- Outlier: A data point that differs significantly from other observations, often identified using the IQR rule.
- Correlation Coefficient (r): A statistical measure that describes the extent of a linear relationship between two variables, ranging from -1 (perfect negative correlation) to +1 (perfect positive correlation).
- Regression Slope (b): The coefficient that represents the change in the response variable for a one-unit change in the explanatory variable in a linear regression model.
- R-squared (R²): The proportion of the variance in the dependent variable that is predictable from the independent variable(s) in a regression model.
- Z-score: A measure of how many standard deviations a particular data point is away from the mean of its distribution.
- Chi-Square Statistic: A statistic used in hypothesis testing to assess the difference between observed and expected frequencies in categorical data.
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