Statistics & Probability Reference Sheet

1. Descriptive Statistics

Mean (Arithmetic Average)

• Formula: $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$xˉ=n∑i=1n​xi​​
• Variables:
  • $x_i$xi​ = each value in the dataset
  • $n$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: $s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}$s2=n−1∑i=1n​(xi​−xˉ)2​
• Variables:
  • $s^2$s2 = variance
  • $x_i$xi​ = each value in the dataset
  • $\bar{x}$xˉ = mean of the dataset
  • $n$n = number of values
• Explanation: Measures the dispersion of the dataset. Larger values indicate more spread.

Standard Deviation

• Formula: $s = \sqrt{s^2}$s=s2​
• Explanation: The square root of variance, representing data spread in the same units as the data.

2. Probability Basics

Probability Rules

• Addition Rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$P(A∪B)=P(A)+P(B)−P(A∩B)
• Multiplication Rule: $P(A \cap B) = P(A) \times P(B|A)$P(A∩B)=P(A)×P(B∣A)

Conditional Probability

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

Bayes' Theorem

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

3. Common Distributions

Normal Distribution

• Formula: $f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$f(x)=σ2π​1​e−2σ2(x−μ)2​
• Variables:
  • $\mu$μ = mean
  • $\sigma$σ = standard deviation
• Explanation: Bell-shaped curve; applicable when data is symmetrically distributed.

Binomial Distribution

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

Poisson Distribution

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

Exponential Distribution

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

4. Hypothesis Testing

Z-test

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

t-test

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

p-value

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

Confidence Intervals

• Formula: $\bar{x} \pm Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}$xˉ±Zα/2​×n​σ​
• 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: $y = \beta_0 + \beta_1 x$y=β0​+β1​x
• Variables:
  • $\beta_0$β0​ = y-intercept
  • $\beta_1$β1​ = slope
• Explanation: Models the relationship between two variables by fitting a linear equation.

Correlation Coefficient (Pearson's r)

• Formula: $r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$r=∑(xi​−xˉ)2∑(yi​−yˉ​)2​∑(xi​−xˉ)(yi​−yˉ​)​
• 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.