Click on a test for an interactive explication and musicrelated application
Warning: I'm not an authority on stats! Treat this page as a basic introduction to ttests, ANOVAs, and twoway ANOVAs.
There are two kinds of ttests: independentsamples ttests and pairedsamples ttests. This page covers independent samples ttests. The independentsamples ttest is also referred to as an independentmeasures ttest, betweensubjects ttest, unpaired ttest, and Student's ttest.
The independentsamples ttest is an inferential statistical test that is used to determine if a difference exists between the means of two independent groups on a single continuous dependent variable.
Suppose you have a friend who teaches an orchestra class and a wind ensemble class at a local high school. Your friend wants to see which group, the orchestra or the wind ensemble, is better at sightreading. So, the teacher designed and administered a sightreading test to all the students in both ensembles. Your friend knows to use a ttest to analyze the data, but is having trouble figuring out what the dependent and independent variables are. Help your friend figure out the variables by clicking and dragging the words above to the appropriate box below.
Levene's Test for Equality of Variances  ttest for Equality of Means  

F  Sig  t  df  Sig. (2tailed)  Mean Difference  Std. Error Difference  95% Confidence Interval of the Difference  
Lower  Upper  
engagement  Equal variances assumed  1.922  .174  2.365  38  .023  .25900  .10954  .03726  .48074 
Equal variances not assumed  2.365  35.055  .024  .25900  .10954  .03664  .48136 
"t" signifies that you're reporting a tstatistic (the results of a ttest).
This number is the degrees of freedom.
This number undicates the obtained value of the tstatistic (obtained tvalue).
Remember: with a ttest the larger the number, the more likely the populations represented by the indepented variables are different. The smaller the number, the more likely the two populations are similar/the same.
This indicates the probability of obtaining the observed tvalue if the null hypothesis is correct.
AKA: betweensubjects ANOVA or onefactor ANOVA.
If you want to determine whether there are any statistically significant differences between the means of two or more independent groups, you can use a oneway analysis of variance (ANOVA).
Imagine that you have a friend that teaches three ensembles at a high school: an orchestra, a wind ensemble, and a jazz band. In order to figure out which group sightreads better, your friend administers a sightreading test and wants to find out if there is a statistically significant difference between the three ensembles. Help your friend figure what the independent and dependent variables are by dragging them to the appropriate box below.
Sum of Squares  df  Mean Square  F  Sig.  

Between Groups  49.033  3  16.344  8.136  .000 
Within Groups  53.066  37  1.965  
Total  102.099  30 
F
Indicates that you are comparing to an Fdistribution (Ftest).
(3, 27)
3: Indicates the Between Groups degrees of freedom ("df1")
27: Indicates the Within Groups [Error] degrees of freedom ("df2")
8.316
Indicates the obtained value of the Fstatistic (obtained Fvalue)
p < .0005
Indicates the probability of obtaining the observed Fvalue if the null hypothesis is true.
The twoway ANOVA is used to determine whether there is an interaction effect between two independent variables on a continuous dependent variable (i.e., if a twoway interaction effect exists).
This time, suppose your friend teaches a total of six ensembles split up between two schools (three at each school): an orchestra at both schools, a wind ensemble at both schools, and a jazz band at both schools. The first school is called "School A" and the second school is called "School B." The teacher wants to see if there's an interaction effect between the ensemble type and the school on sightreading scores. So, at the end of the year the teacher administers a sightreading test to all the students to determine which group as a whole did better, the orchestra, the wind ensemble, or the jazz band. Help your friend correctly categorize all the variables by draging them to the correct box below.
The primary power of the twoway ANOVA is its ability to determine if there is an interaction between two independent variables on a dependent variable. While substantial calculations must be conducted to determine if an interaction is significant, the process of determining the presence an interaction is not. An interaction effect can be seen by simply graphing a “profile plot” (https://statistics.laerd.com/premium/spss/twa/twowayanovainspss12.php). Profile plots can be calculated by hand by calculating the means for each combination of independent variable group. For example, if the twoway ANOVA has one independent variable with three categories (band, orchestra, and jazz band) and another independent variable with two categories (School A and School B), and the dependent variable of sightreading scores, then the means for the following categories need to be calculated to create a profile pot: 1) all band students in School A, 2) all orchestra students in School A, 3) all jazz band students in School B, 4) all band students in School B, 5) all orchestra students in School A, and 6) all jazz band students in School B. Graph the means along two plots, School A and School B. If the resulting lines are parallel then there is no interaction, if the lines are not parallel then there is interaction.
Raw Data Yes, each ensemble has only four people! 
Independent Variable #1 (ensemble type)  

Orchestra  Wind Ensemble  Jazz Band  
Independent Variable #2 (school) 
School A 
School A's orchestra sightreading scores

School A's wind ensemble sightreading scores

School A's jazz band sightreading scores

School B 
School B's orchestra sightreading scores

School B's wind ensemble sightreading scores

School A's jazz band sightreading scores

means  Independent Variable #2  

Orchestra Members  Wind Ensemlbe Members  Jazz Band  rows  
Independent Variable #1  School A  Mean for all of school A: row 1 mean 

School B  Mean for all of school B: row 2 mean 

Means for the columns  Mean for all orchestra members: column 1 mean 
Mean for all wind ensemble members: column 2 mean 
Mean for all jazz ensemble members: column 3 mean 
Mean for everyone 
Source  Degrees of Freedom  Sum of Square  Mean Square  F 

A  a1  SSa  MSa  MSa/MSwithin 
B  b1  SSb  MSb  MSb/MSwithin 
AxB  (a1)(b1)  SS aXb  MSaxb  MSaxb/MSwithin 
Within  ab(r1)  SS within  MSwithin  
Total  abr1  SStotal 
Below is an adapted twoway ANOVA output from https://statistics.laerd.com/premium/spss/twa/twowayanovainspss12.php
Source  Type III Sum of Squares  df  Mean Square  F  Sig.  Partial Eta 

Corrected Model  5645.998  5  1129.200  78.538  .000  .883 
Intercept  132091.906  1  132091.906  9187.227  .000  .994 
gender  8.420  1  8.420  .586  .448  .011 
education_level  5446.697  2  2723.348  189.414  .000  .879 
gender * education_level  210.338  2  105.169  7.315  .002  .220 
Error  52  52  14.378  
Total  58  58  
Corrected Total  57  57 
Indicates that we are comparing to an Fdistribution (Ftest).
2: Indicates the degrees of freedom for the interaction term.
h2: Indicates the degrees of freedom for the error term.
Indicates the obtained value of the Fstatistic (obtained Fvalue).
Indicates the probability of obtaining the observed Fvalue given the null hypothesis is true.
Indicates the probability of obtaining the observed Fvalue if the null hypothesis is true.
A measure of effect size.