Statistics & Probability Reference Sheet

1. Descriptive Statistics

Mean (Arithmetic Average)

• Formula: (\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n})
• Variables:
  • (x_i) = each value in the dataset
  • (n) = number of values
• Explanation: The mean provides the central value of a dataset. Use it for symmetric distributions without outliers.

Median

• Explanation: The middle value of a dataset when ordered. Use it for skewed distributions or when outliers are present.

Mode

• Explanation: The most frequently occurring value in a dataset. Useful for categorical data.

Variance

• Formula: (s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1})
• Variables:
  • (s^2) = variance
  • (x_i) = each value in the dataset
  • (\bar{x}) = mean of the dataset
  • (n) = number of values
• Explanation: Measures the dispersion of the dataset. Larger values indicate more spread.

Standard Deviation

• Formula: (s = \sqrt{s^2})
• Explanation: The square root of variance, representing data spread in the same units as the data.

2. Probability Basics

Probability Rules

• Addition Rule: (P(A \cup B) = P(A) + P(B) - P(A \cap B))
• Multiplication Rule: (P(A \cap B) = P(A) \times P(B|A))

Conditional Probability

• Formula: (P(A|B) = \frac{P(A \cap B)}{P(B)})
• Explanation: Probability of (A) occurring given that (B) has occurred.

Bayes' Theorem

• Formula: (P(A|B) = \frac{P(B|A) \times P(A)}{P(B)})
• Explanation: Used to update the probability of (A) given new evidence (B).

3. Common Distributions

Normal Distribution

• Formula: (f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}})
• Variables:
  • (\mu) = mean
  • (\sigma) = standard deviation
• Explanation: Bell-shaped curve; applicable when data is symmetrically distributed.

Binomial Distribution

• Formula: (P(X = k) = \binom{n}{k} p^k (1-p)^{n-k})
• Variables:
  • (n) = number of trials
  • (k) = number of successes
  • (p) = probability of success
• Explanation: Used for discrete data with two possible outcomes (success/failure).

Poisson Distribution

• Formula: (P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!})
• Variables:
  • (\lambda) = average number of occurrences
• Explanation: Models the number of events in a fixed interval of time or space.

Exponential Distribution

• Formula: (f(x|\lambda) = \lambda e^{-\lambda x})
• Variables:
  • (\lambda) = rate parameter
• Explanation: Time between events in a Poisson process.

4. Hypothesis Testing

Z-test

• Formula: (Z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}})
• Explanation: Used when the population variance is known and the sample size is large ((n > 30)).

t-test

• Formula: (t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}})
• Explanation: Used when the population variance is unknown and the sample size is small ((n \leq 30)).

p-value

• Explanation: Probability of observing test results as extreme as the observed results, under the null hypothesis.

Confidence Intervals

• Formula: (\bar{x} \pm Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}})
• Explanation: Range of values within which the population parameter is expected to lie with a certain level of confidence.

5. Regression & Correlation

Linear Regression

• Formula: (y = \beta_0 + \beta_1 x)
• Variables:
  • (\beta_0) = y-intercept
  • (\beta_1) = slope
• Explanation: Models the relationship between two variables by fitting a linear equation.

Correlation Coefficient (Pearson's 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}})
• Explanation: Measures the strength and direction of a linear relationship between two variables.

6. Sampling

Types of Sampling

• Simple Random Sampling: Every member has an equal chance of being selected.
• Stratified Sampling: Population divided into strata, and random samples taken from each stratum.
• Cluster Sampling: Population divided into clusters, and entire clusters are randomly selected.

Central Limit Theorem

• Explanation: With a sufficiently large sample size, the sampling distribution of the mean will be normally distributed, regardless of the shape of the population distribution.

This reference sheet provides a concise overview of essential statistics and probability concepts, allowing quick access to formulas, definitions, and explanations for practical use in both academic and professional settings.