Statistics Formulas Overview

Basic Statistics Formulas

• Mean (Average): Calculates the average of a set of values.

  • Formula: (\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i)
  • Variables:
    • (n): Total number of values.
    • (x_i): Each individual value.
    • (\sum): Summation symbol (add up all the values).
  • Use case: Finding the average of a sample or population.

• Sample Variance (s²): Measures the spread of values around the mean for a sample.

  • Formula: (s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2)
  • Variables:
    • (x_i): Each value in the sample.
    • (\bar{x}): Sample mean.
    • (n): Sample size.
    • (n-1): Degrees of freedom.
  • Use case: Estimating the variance when only sample data is available.

Binomial Distribution Formulas

• Mean of Binomial Distribution: Expected number of successes in a given number of trials.

  • Formula: (E(X) = np)
  • Variables:
    • (n): Number of trials.
    • (p): Probability of success in a single trial.
  • Use case: Binomial experiments (e.g., coin flips).

• Variance of Binomial Distribution: Measures the spread of the binomial distribution.

  • Formula: (\text{Var}(X) = np(1 - p))
  • Variables:
    • (n): Number of trials.
    • (p): Probability of success.
    • (1-p): Probability of failure.
  • Use case: Assessing variability in binomial experiments.

Uniform Distribution Formulas

• Mean of Uniform Distribution: Average of a uniform distribution.

  • Formula: (\mu = \frac{a + b}{2})
  • Variables:
    • (a): Lower bound.
    • (b): Upper bound.
  • Use case: Finding the average when all values within a range are equally likely (e.g., dice rolls).

• Standard Deviation of Uniform Distribution: Measures spread in a uniform distribution.

  • Formula: (\sigma = \sqrt{\frac{b - a}{12}})
  • Variables:
    • (a): Lower bound.
    • (b): Upper bound.
  • Use case: Calculating the variability of uniform data.

• Probability Density Function (PDF) of Uniform Distribution: Height of the uniform distribution curve.

  • Formula: (f(x) = \frac{1}{b - a})
  • Variables:
    • (a): Lower bound.
    • (b): Upper bound.
  • Use case: Calculating probabilities over ranges in a uniform distribution.

Standardization and Z-Scores

• Z-Score Formula: Converts a value into standard deviations from the mean.

  • Formula: (z = \frac{x - \mu}{\sigma})
  • Variables:
    • (x): Observed value.
    • (\mu): Population mean.
    • (\sigma): Population standard deviation.
  • Use case: Comparing data points to a normal distribution.

• Standard Error of the Mean: Standard deviation of the sample mean.

  • Formula: (\text{SE} = \frac{\sigma}{\sqrt{n}})
  • Variables:
    • (\sigma): Population standard deviation.
    • (n): Sample size.
  • Use case: Measuring the uncertainty in the sample mean.

• Standard Error for Proportion: Measures the spread of sample proportions.

  • Formula: (\text{SE}_p = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}})
  • Variables:
    • (\hat{p}): Sample proportion
    • (n): Sample size.
  • Use case: Confidence intervals or hypothesis tests for proportions.

Confidence Intervals

• Z Confidence Interval for Mean: Builds a range where the population mean likely lies. (when population standard deviation is known)

  • Formula: (\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}})
  • Variables:
    • (\bar{x}): Sample mean.
    • (z_{\alpha/2}): Z critical value.
    • (\sigma): Population standard deviation.
    • (n): Sample size.
  • Use case: Creating a range for the population mean.

• Z Confidence Interval for Proportion: Confidence interval for a proportion.

  • Formula: (\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}})
  • Variables:
    • (\hat{p}): Sample proportion.
    • (z_{\alpha/2}): Z critical value.
    • (n): Sample size.
  • Use case: Confidence range for a sample proportion.

• T Confidence Interval for Mean: Confidence interval for mean. (when population standard deviation is unknown)

  • Formula: (\bar{x} \pm t_{\alpha/2, n-1} \frac{s}{\sqrt{n}})
  • Variables:
    • (\bar{x}): Sample mean.
    • (t_{\alpha/2, n-1}): T critical value.
    • (s): Sample standard deviation.
    • (n): Sample size.
  • Use case: Creating an interval for a population mean when the standard deviation is unknown.

Two-Sample Comparisons

• Pooled Standard Deviation: Combines variances of two samples (assuming variances are equal).

  • Formula: (s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2})
  • Variables:
    • (n_1, n_2): Sample sizes.
    • (s_1^2, s_2^2): Sample variances.
  • Use case: Two-sample t-tests assuming equal variances.

• Standard Error for Two Proportions: Measures uncertainty in the difference between two proportions.

  • Formula: (\text{SE} = \sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}})
  • Variables:
    • (\hat{p}_1, \hat{p}_2): Sample proportions.
    • (n_1, n_2): Sample sizes.
  • Use case: Assessing uncertainty in the differences between sample proportions.