Essential Formulas, Tables, and Procedures for College Statistics

Descriptive Statistics

Measures of Central Tendency and Dispersion

Descriptive statistics summarize and describe the main features of a dataset. The most common measures include the mean, weighted mean, and standard deviation.

• Mean (Arithmetic Average): The sum of all values divided by the number of values. $x = \frac{\Sigma x}{n}$

• Weighted Mean: Used when different values have different weights. $\text{Weighted Mean} = \frac{\Sigma w x}{\Sigma w}$

• Mean from Frequency Table: $x = \frac{\Sigma (f \cdot x)}{\Sigma f}$

• Standard Deviation: Measures the spread of data around the mean. $s = \sqrt{\frac{\Sigma (x - \bar{x})^2}{n - 1}}$

• Variance: The square of the standard deviation. $\text{Variance} = s^2$

Example: If a dataset has values 2, 4, 6, the mean is $\frac{2+4+6}{3} = 4$.

Percentiles and Ranking

Percentiles indicate the relative standing of a value within a dataset.

• Percentile Rank: $L = \frac{k}{100} \cdot n$ $\text{Percentile} = \frac{\text{# of values less than } x}{\text{total number of values}} \times 100$

Probability

Basic Probability Rules

Probability quantifies the likelihood of events occurring. Key rules include:

• Mutually Exclusive Events: $P(A \text{ or } B) = P(A) + P(B)$

• Not Mutually Exclusive: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

• Independent Events: $P(A \text{ and } B) = P(A) \cdot P(B)$

• Dependent Events: $P(A \text{ and } B) = P(A) \cdot P(B|A)$

• Conditional Probability: $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$

• Rule of Complements: $P(A') = 1 - P(A)$

Counting Principles: Permutations and Combinations

• Permutations (no elements alike): $nPr = \frac{n!}{(n - r)!}$

• Permutations (some elements alike): $\frac{n!}{n_1! n_2! \cdots n_k!}$

• Combinations: $nCr = \frac{n!}{(n - r)! r!}$

Binomial Probability

• Binomial Probability Formula: $P(x) = nCr \cdot p^x \cdot q^{n-x}$

• Mean: $m = n \cdot p$

• Variance: $s^2 = n \cdot p \cdot q$

• Standard Deviation: $s = \sqrt{n \cdot p \cdot q}$

Normal Distribution

Standard Scores (z-scores)

The normal distribution is a symmetric, bell-shaped curve describing many natural phenomena. Standard scores (z-scores) measure how many standard deviations a value is from the mean.

• z-score: $z = \frac{x - \mu}{\sigma}$

• Reverse formula: $x = \mu + z \sigma$

Central Limit Theorem

• Mean of sample means: $\mu_x = \mu$

• Standard error: $\sigma_x = \frac{\sigma}{\sqrt{n}}$

Confidence Intervals and Sample Size

Confidence Intervals for Proportions and Means

Confidence intervals estimate population parameters with a specified probability.

• Proportion: $\hat{p} - E \leq p \leq \hat{p} + E$ $E = z_{\alpha/2} \sqrt{\frac{\hat{p} \hat{q}}{n}}$

• Mean (s unknown): $E = t_{\alpha/2} \frac{s}{\sqrt{n}}$

• Mean (s known): $E = z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$

Sample Size Determination

• For Proportion: $n = \frac{(z_{\alpha/2})^2 0.25}{E^2}$

• For Mean: $n = \left(\frac{z_{\alpha/2} \sigma}{E}\right)^2$

Hypothesis Testing

Procedure for Hypothesis Tests

Hypothesis testing is a formal procedure for comparing observed data with a claim about a population.

1. Identify the Claim: State the claim to be tested in symbolic form.

2. Give Symbolic Form: Express the claim and its negation symbolically.

3. Identify Null and Alternative Hypothesis: Null hypothesis ($H_0$) contains equality; alternative hypothesis ($H_1$) does not.

4. Select Significance Level ($\alpha$): Common values are 0.05 and 0.01.

5. Identify the Test Statistic: Choose the appropriate statistic (z, t, chi-square).

6. Find Values: Calculate the test statistic and P-value or critical value.

7. Make a Decision: Reject $H_0$ if P-value $\leq \alpha$; otherwise, fail to reject $H_0$.

8. Restate Decision in Nontechnical Terms: Address the original claim in simple language.

Wording of Final Conclusions

Final conclusions should be stated in clear, nontechnical terms, addressing the original claim.

Condition	Conclusion
Original claim does not include equality, and you reject $H_0$	There is sufficient evidence to support the claim that ... (original claim)
Original claim does not include equality, and you fail to reject $H_0$	There is not sufficient evidence to support the claim that ... (original claim)
Original claim includes equality, and you reject $H_0$	There is sufficient evidence to warrant rejection of the claim that ... (original claim)
Original claim includes equality, and you fail to reject $H_0$	There is not sufficient evidence to warrant rejection of the claim that ... (original claim)

Probability Distributions

Binomial and Poisson Distributions

• Binomial Probability: $P(x) = nCr \cdot p^x \cdot q^{n-x}$

• Poisson Distribution: $P(x) = \frac{m^x e^{-m}}{x!}$ where $e = 2.71828$

Correlation and Regression

Pearson and Spearman Correlation

• Pearson Correlation Coefficient: $r = \frac{n \Sigma xy - (\Sigma x)(\Sigma y)}{\sqrt{[n \Sigma x^2 - (\Sigma x)^2][n \Sigma y^2 - (\Sigma y)^2]}}$

• Spearman Rank Correlation: $r_s = 1 - \frac{6 \Sigma d^2}{n(n^2 - 1)}$

Regression Line

• Slope: $b_1 = \frac{n \Sigma xy - (\Sigma x)(\Sigma y)}{n \Sigma x^2 - (\Sigma x)^2}$

• Intercept: $b_0 = \bar{y} - b_1 \bar{x}$

• Regression Equation: $y_n = b_0 + b_1 x$

Chi-Square and Analysis of Variance

Chi-Square Tests

• Goodness-of-Fit: $\chi^2 = \sum \frac{(O - E)^2}{E}$

• Contingency Table: $\chi^2 = \sum \frac{(O - E)^2}{E}$, $E = \frac{\text{row total} \times \text{column total}}{\text{grand total}}$

One-Way Analysis of Variance (ANOVA)

• Procedure:

  1. Use software or calculator to obtain results.

  2. Identify the P-value.

  3. Form conclusion: If P-value $\leq \alpha$, reject the null hypothesis of equal means; otherwise, fail to reject.

Statistical Tables

Standard Normal (z) Table

The z-table provides cumulative probabilities for the standard normal distribution. Common critical values:

• 90% confidence: $z = 1.645$

• 95% confidence: $z = 1.96$

• 99% confidence: $z = 2.575$

t Distribution Table

The t-table is used for small samples or when the population standard deviation is unknown. Values depend on degrees of freedom and significance level.

Chi-Square Table

The chi-square table is used for tests of variance, goodness-of-fit, and contingency tables. Values depend on degrees of freedom and area to the right of the critical value.

Correlation Coefficient Tables

Tables for Pearson and Spearman correlation coefficients provide critical values for hypothesis testing.

Summary

This guide covers essential formulas, procedures, and tables for college-level statistics, including descriptive statistics, probability, distributions, confidence intervals, hypothesis testing, correlation, regression, and chi-square/ANOVA methods. Use these notes for exam preparation and reference.