Catalog / Statistics Cheat Sheet
Statistics Cheat Sheet
A quick reference guide covering fundamental statistical concepts, formulas, and distributions. This cheat sheet provides a concise overview for students, researchers, and data analysts.
Descriptive Statistics
Measures of Central Tendency
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Mean |
Average of all values: \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} |
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Median |
Middle value when data is ordered. If n is even, average of the two middle values. |
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Mode |
Most frequent value. A dataset can have multiple modes or no mode. |
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Weighted Mean |
Average where each data point contributes unequally: \frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i} |
Measures of Dispersion
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Range |
Difference between the maximum and minimum values: Range = max(x_i) - min(x_i) |
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Variance |
Average squared difference from the mean: Sample Variance: s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1} Population Variance: \sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N} |
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Standard Deviation |
Square root of the variance: Sample Standard Deviation: s = \sqrt{s^2} Population Standard Deviation: \sigma = \sqrt{\sigma^2} |
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Coefficient of Variation |
Relative measure of dispersion: CV = \frac{\sigma}{\mu} (for population), CV = \frac{s}{\bar{x}} (for sample) |
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Interquartile Range (IQR) |
The difference between the 75th percentile (Q3) and the 25th percentile (Q1): IQR = Q3 - Q1 |
Measures of Shape
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Skewness |
Measure of asymmetry of the distribution. Positive skew (right-skewed) indicates a longer tail on the right side. Negative skew (left-skewed) indicates a longer tail on the left side. \text{Skewness} = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^3}{n s^3} |
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Kurtosis |
Measure of the ‘tailedness’ of the distribution. High kurtosis indicates heavy tails (more outliers). Low kurtosis indicates light tails. \text{Kurtosis} = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^4}{n s^4} - 3 |
Probability
Basic Probability Concepts
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Probability of an Event |
P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} |
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Complement Rule |
P(A') = 1 - P(A) |
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Addition Rule |
P(A \cup B) = P(A) + P(B) - P(A \cap B) |
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Conditional Probability |
P(A|B) = \frac{P(A \cap B)}{P(B)} |
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Multiplication Rule |
P(A \cap B) = P(A|B)P(B) = P(B|A)P(A) |
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Independent Events |
If A and B are independent: P(A \cap B) = P(A)P(B), and P(A|B) = P(A) |
Discrete Probability Distributions
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Bernoulli Distribution |
Probability of success (p) or failure (1-p) in a single trial. P(X=x) = p^x (1-p)^{(1-x)}, where x = 0 or 1 |
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Binomial Distribution |
Number of successes in n independent trials. P(X=k) = \binom{n}{k} p^k (1-p)^{(n-k)} |
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Poisson Distribution |
Number of events in a fixed interval of time or space. P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!} |
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Geometric Distribution |
Number of trials until the first success. P(X=k) = (1-p)^{k-1} p |
Continuous Probability Distributions
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Uniform Distribution |
Probability is constant over a given interval [a, b]. f(x) = \frac{1}{b-a} for a \le x \le b |
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Normal Distribution |
Bell-shaped curve, defined by mean (\mu) and standard deviation (\sigma). f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} |
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Exponential Distribution |
Time until an event occurs. f(x) = \lambda e^{-\lambda x} for x \ge 0 |
Inferential Statistics
Confidence Intervals
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General Form |
Estimate \pm (Critical Value * Standard Error) |
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CI for Population Mean (\mu) with known \sigma |
\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}} |
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CI for Population Mean (\mu) with unknown \sigma |
\bar{x} \pm t_{\alpha/2, n-1} \frac{s}{\sqrt{n}} |
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CI for Population Proportion (p) |
\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} |
Hypothesis Testing
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Null Hypothesis (H₀) |
Statement being tested. |
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Alternative Hypothesis (H₁) |
Statement to be supported if H₀ is rejected. |
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Test Statistic |
Value calculated from sample data to test the hypothesis. |
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P-value |
Probability of observing a test statistic as extreme as, or more extreme than, the one computed, assuming H₀ is true. |
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Significance Level (\alpha) |
Probability of rejecting H₀ when it is true (Type I error). |
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Decision Rule |
If p-value \le \alpha, reject H₀. Otherwise, fail to reject H₀. |
Common Hypothesis Tests
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Z-test |
Testing population mean with known \sigma or large sample size. z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}} |
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t-test |
Testing population mean with unknown \sigma and small sample size. t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} |
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Chi-Square Test |
Testing association between categorical variables. \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} |
Regression Analysis
Simple Linear Regression
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Regression Equation |
y = \beta_0 + \beta_1 x + \epsilon |
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Estimating Coefficients |
\hat{\beta_1} = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n} (x_i - \bar{x})^2} \hat{\beta_0} = \bar{y} - \hat{\beta_1} \bar{x} |
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Coefficient of Determination (R²) |
Proportion of variance in dependent variable explained by the independent variable. R^2 = \frac{SSR}{SST} = 1 - \frac{SSE}{SST} |
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SSR, SSE, SST |
Sum of Squares Regression (SSR), Sum of Squares Error (SSE), Total Sum of Squares (SST) SST = \sum (y_i - \bar{y})^2 |
Multiple Linear Regression
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Regression Equation |
y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_p x_p + \epsilon |
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Adjusted R² |
Adjusts R² for the number of predictors in the model. R_{adj}^2 = 1 - \frac{(1-R^2)(n-1)}{n-p-1} |
Assumptions of Linear Regression
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