Statistical Analysis Essentials Cheat Sheet

Mastering statistical concepts from data visualization to regression and probability is key to understanding and interpreting data effectively. This cheat sheet covers core principles for describing data, analyzing relationships, and calculating probabilities.

Core Principles

Data Description: Differentiate between population parameters and sample statistics. Use mean and standard deviation for symmetric distributions, and the 5-number summary (min, Q1, median, Q3, max) with IQR for skewed distributions or when outliers are present.
Correlation vs. Regression: Correlation (r) measures linear association strength and direction, while regression (slope b) models the relationship to predict outcomes and understand the rate of change.
Normal Distribution: The 68-95-99.7% rule states that for a normal distribution, approximately 68% of data falls within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.
Chi-Square Tests: Use the chi-square goodness-of-fit test to compare observed categorical data to expected frequencies, and the chi-square test of independence to assess relationships between two categorical variables.

Action Steps

1. Identify Population vs. Sample: Determine if data represents the entire group of interest (population) or a subset (sample).
2. Classify Variables: Distinguish between categorical (e.g., color, gender) and numerical (e.g., height, age) variables.
3. Choose Appropriate Graphs: Use bar charts or pie charts for categorical data, histograms or stem-leaf plots for numerical data.
4. Calculate Descriptive Statistics: Compute mean and standard deviation for symmetric numerical data; find the 5-number summary and IQR for skewed data or to identify outliers.
5. Analyze Scatterplots: Examine scatterplots for linear patterns, strength, direction, and outliers to inform correlation and regression analysis.
6. Interpret Regression Output: Understand the slope (change in y for a one-unit change in x) and R-squared (proportion of variance explained) in context.
7. Apply Normal Distribution Rule: Use the 68-95-99.7% rule to estimate percentages within standard deviations or find values corresponding to given percentages.
8. Perform Chi-Square Tests: Set up null/alternative hypotheses, calculate the chi-square statistic, and compare to critical values or p-values to draw conclusions about categorical data.

Formulas

Mean: $\\bar{x} = \frac{\\sum x_i}{n}$
Standard Deviation (Sample): $s = \sqrt{\\frac{\\sum (x_i - \\bar{x})^2}{n-1}}$
Interquartile Range (IQR): $IQR = Q3 - Q1$
Outlier Rule: Outliers are values below $Q1 - 1.5*IQR$ or above $Q3 + 1.5*IQR$
Correlation Coefficient (r): Measures linear association strength (-1 to +1).
Regression Slope (b): $b = r \frac{s_y}{s_x}$
R-squared (R^2): The proportion of variance in the response variable explained by the explanatory variable.
Z-score: $z = \frac{x - \\mu}{\\sigma}$
Chi-Square Statistic (Goodness-of-Fit): $\\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$
Chi-Square Statistic (Independence): $\\chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}}$

Key Terms

Parameter: A numerical characteristic of a population (e.g., population mean \\mu).
Statistic: A numerical characteristic of a sample (e.g., sample mean \\bar{x}).
Distribution: Describes the frequency or likelihood of different outcomes for a variable.
Explanatory Variable: The variable used to predict or explain changes in the response variable (often plotted on the x-axis).
Response Variable: The variable being predicted or explained (often plotted on the y-axis).
Correlation Coefficient (r): A measure of the strength and direction of a linear relationship between two numerical variables.
Least Squares Regression: A method to find the line that best fits the data by minimizing the sum of the squared vertical distances between the data points and the line.
Residual: The difference between an observed value and the value predicted by the regression model (residual = observed - predicted).
Extrapolation: Using a regression model to make predictions outside the range of the observed data.
Random Variable: A variable whose value is a numerical outcome of a random phenomenon.
Standard Normal Distribution: A normal distribution with a mean of 0 and a standard deviation of 1 (often denoted as Z).
Chi-Square Statistic (χ²): A statistic used in chi-square tests to measure the difference between observed and expected frequencies.
Degrees of Freedom (d.f.): A parameter that determines the shape of the chi-square distribution, related to the number of categories or variables.

Real World Examples

A study measures the heights of 100 randomly selected students from a university.: The population is all students at the university. The heights of the 100 students are a sample. The average height of the 100 students is a statistic, while the average height of all students at the university would be a parameter.
Analyzing the relationship between hours studied and exam scores.: Hours studied is the explanatory variable, and exam score is the response variable. A scatterplot can show the relationship, and a regression model can predict scores based on study hours. The correlation coefficient 'r' would indicate the strength and direction of the linear association.
Predicting a student's final grade based on their midterm score.: If the regression model's R-squared is 0.75, it means 75% of the variation in final grades can be explained by the midterm score. Using a midterm score outside the range of those in the original study to predict the final grade is extrapolation.
Determining if a new drug is effective compared to a placebo.: A chi-square test of independence could be used if the outcome is categorical (e.g., 'improved' vs. 'not improved') and the groups are 'drug' vs. 'placebo'.