Introductory Statistics Formula Sheet and Reference Tables

Chapter 2: Organizing and Summarizing Data

Relative Frequency

The relative frequency of a category is the proportion of observations within that category, calculated as:

$\text{Relative Frequency} = \frac{\text{Frequency}}{\text{Sum of all Frequencies}}$

Chapter 3: Numerically Summarizing Data

Measures of Central Tendency

• Population Mean (μ): $\mu = \frac{\sum X}{N}$

• Sample Mean (\(\bar{x}\)): $\bar{x} = \frac{\sum X}{n}$

Measures of Dispersion

• Sample Standard Deviation (s): $s = \sqrt{\frac{\sum (X - \bar{x})^2}{n-1}}$

• Population Standard Deviation (σ): $\sigma = \sqrt{\frac{\sum (X - \mu)^2}{N}}$

• Sample Variance (s²): $s^2 = \frac{\sum (X - \bar{x})^2}{n-1}$

• Population Variance (σ²): $\sigma^2 = \frac{\sum (X - \mu)^2}{N}$

Empirical Rule (for Bell-Shaped Distributions)

• Approximately 68% of the data lie within 1 standard deviation of the mean.

• Approximately 95% of the data lie within 2 standard deviations of the mean.

• Approximately 99.7% of the data lie within 3 standard deviations of the mean.

Measures of Position

• Interquartile Range (IQR): $IQR = Q_3 - Q_1$

• Lower Fence: $Q_1 - 1.5 \times IQR$

• Upper Fence: $Q_3 + 1.5 \times IQR$

Five-Number Summary

• Minimum, Q1, Median, Q3, Maximum

Chapter 4: Describing the Relation Between Two Variables

Correlation Coefficient (r)

The correlation coefficient measures the strength and direction of a linear relationship between two variables:

$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{(n-1)s_x s_y}$

Chapter 5: Probability

Probability Rules

• Classical Probability: $P(E) = \frac{\text{Number of ways E can occur}}{\text{Number of possible outcomes}}$

• Empirical Probability: $P(E) = \frac{\text{Frequency of E}}{\text{Number of trials}}$

• Complement Rule: $P(E^c) = 1 - P(E)$

• Addition Rule for Disjoint Events: $P(E \text{ or } F) = P(E) + P(F)$

• General Addition Rule: $P(E \text{ or } F) = P(E) + P(F) - P(E \text{ and } F)$

• Multiplication Rule for Independent Events: $P(E \text{ and } F) = P(E) \cdot P(F)$

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

Chapter 6: Discrete Probability Distributions

Mean and Standard Deviation of a Discrete Random Variable

• Mean (Expected Value): $\mu_X = E(X) = \sum x \cdot P(x)$

• Standard Deviation: $\sigma_X = \sqrt{\sum (x - \mu_X)^2 P(x)}$

Binomial Probability Distribution

• Probability of x successes: $P(X = x) = {n \choose x} p^x (1-p)^{n-x}$

• Mean: $\mu = np$

• Standard Deviation: $\sigma = \sqrt{np(1-p)}$

Chapter 7: The Normal Distribution

Standardizing a Normal Random Variable

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

Sampling Distributions

• Mean of the Sampling Distribution of the Sample Mean: $\mu_{\bar{x}} = \mu$

• Standard Deviation (Standard Error): $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

• Mean of the Sampling Distribution of the Sample Proportion: $\mu_{\hat{p}} = p$

• Standard Deviation: $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$

Chapter 8 & 9: Confidence Intervals and Sample Size

Confidence Interval for a Population Proportion

$\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

Confidence Interval for a Population Mean (σ unknown)

$\bar{x} \pm t_{\alpha/2} \frac{s}{\sqrt{n}}$

Sample Size for Proportion

$n = \left( \frac{z_{\alpha/2}}{E} \right)^2 \hat{p}(1-\hat{p})$

Sample Size for Mean

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

Chapter 10: Hypothesis Tests Regarding a Parameter

Test Statistic for a Population Mean (σ unknown)

$t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$

Test Statistic for a Population Proportion

$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

Chapter 11: Inferences on Two Samples

Test Statistic for Matched-Pairs Data

$t = \frac{\bar{d}}{s_d/\sqrt{n}}$

Test Statistic Comparing Two Population Proportions (Independent Samples)

$z = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$

Test Statistic Comparing Two Means (Independent Samples)

$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

Chapter 12: Inference on Categorical Data

Chi-Square Test Statistic

$\chi^2 = \sum \frac{(O - E)^2}{E}$

Where O = observed count, E = expected count.

Expected Counts (Goodness-of-Fit)

$E_i = n p_i$

Chapter 13: Comparing Three or More Means (ANOVA)

Test Statistic for One-Way ANOVA

$F = \frac{\text{Mean Square Between Groups (MST)}}{\text{Mean Square Error (MSE)}}$

Reference Tables

Random Number Table

Used for selecting random samples in statistical studies.

Critical Values for Correlation Coefficient and Normal Probability Plots

Sample Size (n)	Critical Value
5	0.880
10	0.918
20	0.951
30	0.960

Standard Normal (Z) Table

Provides areas (probabilities) to the left of a given z-score under the standard normal curve.

t-Distribution Table

Provides critical values for the t-distribution for various degrees of freedom and significance levels.

Chi-Square Distribution Table

Provides critical values for the chi-square distribution for various degrees of freedom and significance levels.

Chapter 2: Example of Frequency and Relative Frequency Tables

Body Part	Frequency
Back	12
Wrist	2
Elbow	1
Hip	2
Shoulder	4
Knee	5
Hand	2
Groin	1
Neck	1

Example Interpretation: The bar graph and pie chart above display the frequency and relative frequency of different types of rehabilitation required by patients. The most common body part requiring rehabilitation is the back, followed by the knee and shoulder.

Additional info: This formula sheet and table reference is a comprehensive summary of the key formulas, definitions, and tables used throughout an introductory statistics course, covering all major topics from data organization to hypothesis testing and ANOVA.