Discrete probability distributions are based on discrete variables, which have a finite or countable number of values. In this post, I show you how to perform goodness-of-fit tests to determine how well your data fit various discrete probability distributions. [Read more…] about Goodness-of-Fit Tests for Discrete Distributions

# interpreting results

## Examples of Hypothesis Tests: Busting Myths about the Battle of the Sexes

In my house, we love the Mythbusters TV show on the Discovery Channel. The Mythbusters conduct scientific investigations in their quest to test myths and urban legends. In the process, the show provides some fun examples of when and how you should use statistical hypothesis tests to analyze data. [Read more…] about Examples of Hypothesis Tests: Busting Myths about the Battle of the Sexes

## Making Predictions with Regression Analysis

If you were able to make predictions about something important to you, you’d probably love that, right? It’s even better if you know that your predictions are sound. In this post, I show how to use regression analysis to make predictions and determine whether they are both unbiased and precise. [Read more…] about Making Predictions with Regression Analysis

## Curve Fitting using Linear and Nonlinear Regression

In regression analysis, curve fitting is the process of specifying the model that provides the best fit to the specific curves in your dataset. Curved relationships between variables are not as straightforward to fit and interpret as linear relationships. [Read more…] about Curve Fitting using Linear and Nonlinear Regression

## P values and Statistical Significance

P values determine whether your hypothesis test results are statistically significant. Statistics use them all over the place. You’ll find P values in t-tests, distribution tests, ANOVA, and regression analysis. P values have become so important that they’ve taken on a life of their own. They can determine which studies are published, which projects receive funding, and which university faculty members become tenured!

Ironically, despite being so influential, P values are misinterpreted very frequently. What *is* the correct interpretation of P values? What do P values *really* mean? That’s the topic of this post! [Read more…] about P values and Statistical Significance

## How To Interpret R-squared in Regression Analysis

R-squared is a goodness-of-fit measure for linear regression models. This statistic indicates the percentage of the variance in the dependent variable that the independent variables explain collectively. R-squared measures the strength of the relationship between your model and the dependent variable on a convenient 0 – 100% scale. [Read more…] about How To Interpret R-squared in Regression Analysis

## How Hypothesis Tests Work: Significance Levels (Alpha) and P values

Hypothesis testing is a vital process in inferential statistics where the goal is to use sample data to draw conclusions about an entire population. In the testing process, you use significance levels and p-values to determine whether the test results are statistically significant.

You hear about results being statistically significant all of the time. But, what do significance levels, P values, and statistical significance actually represent? Why do we even need to use hypothesis tests in statistics? [Read more…] about How Hypothesis Tests Work: Significance Levels (Alpha) and P values

## How to Interpret P-values and Coefficients in Regression Analysis

P-values and coefficients in regression analysis work together to tell you which relationships in your model are statistically significant and the nature of those relationships. The coefficients describe the mathematical relationship between each independent variable and the dependent variable. The p-values for the coefficients indicate whether these relationships are statistically significant. [Read more…] about How to Interpret P-values and Coefficients in Regression Analysis

## How Hypothesis Tests Work: Confidence Intervals and Confidence Levels

A confidence interval is calculated from a sample and provides a range of values that likely contains the unknown value of a population parameter. In this post, I demonstrate how confidence intervals and confidence levels work using graphs and concepts instead of formulas. In the process, you’ll see how confidence intervals are very similar to P values and significance levels. [Read more…] about How Hypothesis Tests Work: Confidence Intervals and Confidence Levels

## How to Interpret Adjusted R-Squared and Predicted R-Squared in Regression Analysis

R-squared tends to reward you for including too many independent variables in a regression model, and it doesn’t provide any incentive to stop adding more. Adjusted R-squared and predicted R-squared use different approaches to help you fight that impulse to add too many. The protection that adjusted R-squared and predicted R-squared provide is critical because too many terms in a model can produce results that you can’t trust. These statistics help you include the correct number of independent variables in your regression model. [Read more…] about How to Interpret Adjusted R-Squared and Predicted R-Squared in Regression Analysis

## How to Interpret the Constant (Y Intercept) in Regression Analysis

The constant term in regression analysis is the value at which the regression line crosses the y-axis. The constant is also known as the y-intercept. That sounds simple enough, right? Mathematically, the regression constant really is that simple. However, the difficulties begin when you try to interpret the *meaning* of the y-intercept in your regression output. [Read more…] about How to Interpret the Constant (Y Intercept) in Regression Analysis

## How to Interpret the F-test of Overall Significance in Regression Analysis

The F-test of overall significance indicates whether your linear regression model provides a better fit to the data than a model that contains no independent variables. In this post, I look at how the F-test of overall significance fits in with other regression statistics, such as R-squared. R-squared tells you how well your model fits the data, and the F-test is related to it. [Read more…] about How to Interpret the F-test of Overall Significance in Regression Analysis

## Multicollinearity in Regression Analysis: Problems, Detection, and Solutions

Multicollinearity occurs when independent variables in a regression model are correlated. This correlation is a problem because independent variables should be *independent*. If the degree of correlation between variables is high enough, it can cause problems when you fit the model and interpret the results. [Read more…] about Multicollinearity in Regression Analysis: Problems, Detection, and Solutions

## Benefits of Welch’s ANOVA Compared to the Classic One-Way ANOVA

Welch’s ANOVA is an alternative to the traditional analysis of variance (ANOVA) and it offers some serious benefits. One-way analysis of variance determines whether differences between the means of at least three groups are statistically significant. For decades, introductory statistics classes have taught the classic Fishers one-way ANOVA that uses the F-test. It’s a standard statistical analysis, and you might think it’s pretty much set in stone by now. Surprise, there’s a significant change occurring in the world of one-way analysis of variance! [Read more…] about Benefits of Welch’s ANOVA Compared to the Classic One-Way ANOVA

## Standard Error of the Regression vs. R-squared

The standard error of the regression (S) and R-squared are two key goodness-of-fit measures for regression analysis. While R-squared is the most well-known amongst the goodness-of-fit statistics, I think it is a bit over-hyped. [Read more…] about Standard Error of the Regression vs. R-squared

## Chi-Square Test of Independence and an Example

The Chi-square test of independence determines whether there is a statistically significant relationship between categorical variables. It is a hypothesis test that answers the question—do the values of one categorical variable depend on the value of other categorical variables? [Read more…] about Chi-Square Test of Independence and an Example

## Multivariate ANOVA (MANOVA) Benefits and When to Use It

Multivariate ANOVA (MANOVA) extends the capabilities of analysis of variance (ANOVA) by assessing multiple dependent variables simultaneously. ANOVA statistically tests the differences between three or more group means. For example, if you have three different teaching methods and you want to evaluate the average scores for these groups, you can use ANOVA. However, ANOVA does have a drawback. It can assess only one dependent variable at a time. This limitation can be an enormous problem in certain circumstances because it can prevent you from detecting effects that actually exist. [Read more…] about Multivariate ANOVA (MANOVA) Benefits and When to Use It

## Repeated Measures Designs: Benefits and an ANOVA Example

Repeated measures designs, also known as a within-subjects designs, can seem like oddball experiments. When you think of a typical experiment, you probably picture an experimental design that uses mutually exclusive, independent groups. These experiments have a control group and treatment groups that have clear divisions between them. Each subject is in only one of these groups. [Read more…] about Repeated Measures Designs: Benefits and an ANOVA Example

## Hypothesis Testing and the Mythbusters: Are Yawns Contagious?

When it comes to hypothesis testing, statistics help you avoid opinions about when an effect is large and how many samples you need to collect. Opinions about these things can be *way* off—even among those who regularly perform experiments and collect data! This can lead you to draw the incorrect conclusions. Always perform the correct hypothesis tests so you understand the strength of your evidence.

[Read more…] about Hypothesis Testing and the Mythbusters: Are Yawns Contagious?

## Statistical Analysis of the Republican Establishment Split

Back in 2014, House Speaker John Boehner resigned, and then Kevin McCarthy refused the position of Speaker of the House before the vote. The Republican’s search for a new speaker ultimately led to Paul Ryan. Simultaneously, the Republican Freedom Caucus was making the news with a potential shutdown of the government that was controversial even amongst some Republicans. [Read more…] about Statistical Analysis of the Republican Establishment Split