Essential Tables and Formulas for College Statistics

Chapter 2: Organizing and Summarizing Data

Frequency and Relative Frequency

Organizing data into frequency tables is a fundamental step in descriptive statistics. Frequency refers to the number of times a value or category appears, while relative frequency expresses this as a proportion of the total.

• Frequency: The count of occurrences for each category or class.

• Relative Frequency: Calculated as frequency divided by the sum of all frequencies.

• Class Midpoint: The average of consecutive lower class limits, used in grouped data.

Example: If a class has lower limits 10 and 20, the midpoint is (10+20)/2 = 15.

Chapter 3: Numerically Summarizing Data

Measures of Central Tendency and Spread

Numerical summaries describe the center and variability of data.

• Population Mean:

• Sample Mean: $\bar{x} = \frac{\sum x_i}{n}$

• Range: Largest Data Value - Smallest Data Value

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

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

• Population Variance: $\sigma^2$

• Sample Variance: $s^2$

Empirical Rule (68-95-99.7 Rule)

• For bell-shaped distributions:

• ~68% of data within 1 standard deviation of mean

• ~95% within 2 standard deviations

• ~99.7% within 3 standard deviations

Chapter 4: Describing the Relation Between Two Variables

Grouped Data and Weighted Means

• Population Mean from Grouped Data: $\mu = \frac{\sum x_i f_i}{\sum f_i}$

• Sample Mean from Grouped Data: $\bar{x} = \frac{\sum x_i f_i}{\sum f_i}$

• Weighted Mean: $\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}$

Standard Deviation from Grouped Data

• Population: $\sigma = \sqrt{\frac{\sum (x_i - \mu)^2 f_i}{\sum f_i}}$

• Sample: $s = \sqrt{\frac{\sum (x_i - \bar{x})^2 f_i}{\sum f_i - 1}}$

z-Scores

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

• Sample z-score: $z = \frac{x - \bar{x}}{s}$

Interquartile Range and Five-Number Summary

• 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, $Q_1$, Median, $Q_3$, Maximum

Correlation and Regression

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

• Least-Squares Regression Line: $\hat{y} = b_1 x + b_0$

• Slope: $b_1 = r \frac{s_y}{s_x}$

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

• Residual: $y - \hat{y}$

• Coefficient of Determination: $R^2 = r^2$

Chapter 5: Probability

Probability Rules

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

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

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

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

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

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

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

• General Multiplication Rule: $P(E \text{ and } F) = P(E) \times P(F|E)$

Chapter 6: Discrete Probability Distributions

Counting Principles

• Factorial: $n! = n \times (n-1) \times \ldots \times 1$

• Permutation: $nP_r = \frac{n!}{(n-r)!}$

• Combination: $nC_r = \frac{n!}{r!(n-r)!}$

• Permutations with Repetition: $\frac{n!}{n_1! n_2! \ldots n_k!}$

Discrete Random Variables

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

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

Binomial and Poisson Distributions

• Binomial Mean: $\mu_X = np$

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

• Binomial Probability: $P(x) = nC_x p^x (1-p)^{n-x}$

• Poisson Probability: $P(x) = \frac{e^{-\lambda} \lambda^x}{x!}$

• Poisson Mean and Standard Deviation: $\mu_X = \lambda$, $\sigma_X = \sqrt{\lambda}$

Chapter 7: The Normal Distribution

Standardizing and Finding Scores

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

• Finding Score: $x = \mu + z \sigma$

Chapter 8: Sampling Distributions

Sampling Distribution of the Mean and Proportion

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

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

• Sample Proportion: $\hat{p} = \frac{x}{n}$

• Mean and Standard Deviation of Sample Proportion: $\mu_{\hat{p}} = p$, $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$

Chapter 9: Estimating the Value of a Parameter

Confidence Intervals

• Confidence Interval for Proportion: $\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• Confidence Interval for Mean: $\bar{x} \pm t_{\alpha/2} \frac{s}{\sqrt{n}}$

• Sample Size for Proportion: $n = \frac{z_{\alpha/2}^2 \hat{p}(1-\hat{p})}{E^2}$

• Sample Size for Mean: $n = \frac{z_{\alpha/2}^2 s^2}{E^2}$

Chapter 10: Hypothesis Tests Regarding a Parameter

Test Statistics

• z-Test for Proportion: $z_0 = \frac{\hat{p} - p_0}{\sqrt{p_0(1-p_0)/n}}$

• t-Test for Mean: $t_0 = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$

Chapter 11: Inferences on Two Samples

Comparing Two Proportions and Means

• z-Test for Two Proportions: $z_0 = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$

• Confidence Interval for Difference of Proportions: $(\hat{p}_1 - \hat{p}_2) \pm z_{\alpha/2} \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$

• t-Test for Matched Pairs: $t_0 = \frac{\bar{d} - \mu_d}{s_d/\sqrt{n}}$

• Confidence Interval for Matched Pairs: $\bar{d} \pm t_{\alpha/2} \frac{s_d}{\sqrt{n}}$

• t-Test for Two Means (Independent): $t_0 = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

• Confidence Interval for Difference of Means: $(\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2} \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$

Chapter 12: Inference on Categorical Data

Chi-Square Tests

• Expected Counts: $E_i = n p_i$

• Chi-Square Test Statistic: $\chi^2_0 = \sum \frac{(O_i - E_i)^2}{E_i}$

• Expected Frequency for Independence: $\frac{\text{row total} \times \text{column total}}{\text{table total}}$

• Test Statistic for Dependent Proportions: $\chi^2_0 = \frac{(f_{12} - f_{21})^2}{f_{12} + f_{21}}$

Chapter 13: Comparing Three or More Means (ANOVA)

One-Way ANOVA and Tukey's Test

• ANOVA F-Test: $F = \frac{\text{Mean Square Treatment}}{\text{Mean Square Error}}$

• Mean Square Treatment: $MST = \frac{\sum n_i (\bar{x}_i - \bar{x})^2}{k-1}$

• Mean Square Error: $MSE = \frac{\sum (n_i - 1) s_i^2}{n - k}$

• Tukey's Test: $q = \frac{\bar{x}_2 - \bar{x}_1}{s \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$

Chapter 14: Inference on Regression Models

Least-Squares Regression and Multiple Regression

• Standard Error of Estimate: $s_e = \sqrt{\frac{\sum (y_i - \hat{y}_i)^2}{n-2}}$

• Standard Error of Slope: $s_{b_1} = \frac{s_e}{\sqrt{\sum (x_i - \bar{x})^2}}$

• t-Test for Slope: $t_0 = \frac{b_1 - 0}{s_{b_1}}$

• Confidence Interval for Slope: $b_1 \pm t_{\alpha/2} s_{b_1}$

• Confidence Interval for Mean Response: $\hat{y} \pm t_{\alpha/2} s_e \sqrt{\frac{1}{n} + \frac{(x^* - \bar{x})^2}{\sum (x_i - \bar{x})^2}}$

• Prediction Interval for Individual Response: $\hat{y} \pm t_{\alpha/2} s_e \sqrt{1 + \frac{1}{n} + \frac{(x^* - \bar{x})^2}{\sum (x_i - \bar{x})^2}}$

Statistical Tables

Random Numbers Table

Used for random sampling and simulation. Each cell contains a five-digit random number.

Critical Values for Correlation Coefficient

n	CV
3	0.997
10	0.632
17	0.482
24	0.404
30	0.361

Additional info: CV is the minimum value for r to be considered statistically significant at a given sample size.

Critical Values for Normal Probability Plots

Sample Size, n	Critical Value
5	0.880
13	0.932
21	0.952
30	0.960

Standard Normal Distribution Table (z-table)

Provides cumulative probabilities for z-scores. Used to find probabilities and percentiles for normal distributions.

Confidence Interval Critical Values

Level of Confidence	Critical Value, $z_{\alpha/2}$
90%	1.645
95%	1.96
98%	2.33
99%	2.575

Hypothesis Testing Critical Values

Level of Significance	Left-Tailed	Right-Tailed	Two-Tailed
0.10	-1.28	1.28	±1.645
0.05	-1.645	1.645	±1.96
0.01	-2.33	2.33	±2.575

t-Distribution Table

Used for small sample sizes or unknown population standard deviation. Values depend on degrees of freedom (df) and area in the right tail.

Chi-Square Distribution Table

Used for categorical data analysis, goodness-of-fit, and independence tests. Values depend on degrees of freedom and area to the right of the critical value.

df	0.05	0.01
1	3.841	6.635
2	5.991	9.210
10	18.307	23.209
20	31.410	37.566

Additional info: These tables and formulas are essential for statistical analysis, hypothesis testing, and interpreting results in college-level statistics courses.