BTech Statistical Analysis Formulas and Cheat Sheet

Creating a comprehensive cheat sheet for statistical analysis in a Bachelor of Technology (BTech) program involves covering a wide range of topics and formulas. Below is a condensed version of this essential reference:

Descriptive Statistics

Measures of Central Tendency

• Mean: \(\bar{x} = \frac{\sum{x}}{n}\)
• Median: Depends on data arrangement.
• Mode: Most frequent value.

Measures of Dispersion

• Variance: \(s^2 = \frac{\sum(x - \bar{x})^2}{n-1}\)
• Standard Deviation: \(s = \sqrt{s^2}\)
• Range: \(Range = Max(x) - Min(x)\)
• Interquartile Range (IQR): \(Q3 - Q1\)

Measures of Shape

• Skewness
• Kurtosis

Probability

Basic Probability

• \(P(A)\) = \(\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\)
• \(P(A \cup B)\) = \(P(A) + P(B) - P(A \cap B)\)

Conditional Probability

• \(P(A|B)\) = \(\frac{P(A \cap B)}{P(B)}\)

Bayes' Theorem

• \(P(A|B) = \frac{P(B|A) \times P(A)}{P(B)}\)

Random Variables and Distributions

Discrete Random Variables

• Probability Mass Function (PMF)
• Expected Value: \(E(X) = \sum(x \times P(X=x))\)
• Variance: \(Var(X) = E((X-E(X))^2)\)

Continuous Random Variables

• Probability Density Function (PDF)
• Expected Value: \(E(X) = \int_{-\infty}^{\infty} x \times f(x) \, dx\)
• Variance: \(Var(X) = E((X-E(X))^2)\)

Sampling Distributions

Central Limit Theorem

• \(\bar{X} \sim N(\mu, \frac{\sigma^2}{n})\) for large \(n\).

Estimation

Point Estimation

• Estimators: \(\hat{\theta}\)
• Sample Mean: \(\bar{X}\)
• Sample Variance: \(s^2\)

Interval Estimation

• Confidence Interval for \(\mu\): \(\bar{X} \pm z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}\)

Hypothesis Testing

Null and Alternative Hypotheses

• \(H_0: \mu = \mu_0\)
• \(H_1: \mu \neq \mu_0\) (or \(\mu > \mu_0\) or \(\mu < \mu_0\))

Test Statistics

• \(t = \frac{\bar{X} - \mu_0}{s/\sqrt{n}}\)

Critical Region and P-values

• P-value: Probability of observing the test statistic or more extreme values, assuming the null hypothesis is true.

Regression and Correlation

Simple Linear Regression

• \(y = \beta_0 + \beta_1 x + \epsilon\)

Multiple Linear Regression

• \(y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_n x_n + \epsilon\)

Correlation Coefficient

• \(r = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sqrt{\sum{(x_i - \bar{x})^2} \times \sum{(y_i - \bar{y})^2}}}\)

ANOVA (Analysis of Variance)

One-Way ANOVA

• \(F = \frac{MS_{\text{between}}}{MS_{\text{within}}}\)

Non-parametric Tests

Chi-Square Test

• \(\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}\)