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Editorial disclosure: This commentary was generated by ChatGPT in response to a prompt supplied by the site author. It is not an independent review or institutional endorsement.

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Editorial disclosure: This commentary was generated by Grok in response to a prompt supplied by the site author. It is not an independent review or institutional endorsement.

I took a closer look at https://statisticstextbook.com/ and what makes it distinctive is how it blends old-school clarity with surprisingly modern topics.

Some free statistics resources remain basic, while others emphasize heavier mathematics. This one keeps the language simple — no calculus needed — while still covering t-tests, ANOVA, regression, and chi-square in a very example-heavy way. But then it goes further with a whole section on modern statistics: big data, resampling, simulation, machine learning basics, neural networks, and even data ethics. This extends the scope beyond conventional high-school topics.

The author, Dr. Michael Nikoletseas, brings an interdisciplinary background in neuroscience, philosophy of science, and math, which shows in the emphasis on real understanding rather than just formulas. The site is clean, designed for use on phones and tablets, uses nice visuals, and includes optional interactive quizzes (some need a free signup).

It's not flashy, but it's a free statistics site designed to prepare students for traditional classes and today's data-driven world.

Solid find.

— Grok

Editorial disclosure: This commentary was generated by Google Gemini in response to a prompt supplied by the site author. It is not an independent review or institutional endorsement.

In this story we will develop the concept of mixed designs and give a practice example.

Elegant research avoids complex

designs also called split-plot designs 

or mixed designs. However, you may not

 be spared of these monsters in your

student or research life.

Let’s get a whiff of these monsters.

A psychiatrist wanted to see, if two

new drugs improve the condition

of depressive and schizophrenic

patients.

He randomly assigned 4

depressive patients to Drug1 and

Drug2 conditions. That is, each of

the depressive patients will be

serving as a subject in both the

Drug conditions. This is a repeated

measures design.

He did the same with the

schizophrenic patients. He

randomly assign 4 schizophrenic

patients to Drug1 and Drug2

conditions. That is, each of the 

schizophrenic patients will be

serving as a subject in both the

Drug conditions. This is a repeated

measures design.

As you see, here we have two

independent groups (depressive

patients, and schizophrenic

patients) but each patient is given

two treatments, that is he is tested

repeatedly, i.e., in both drug

conditions. We have a hybrid

situation, you would say. Both

independence and non-

independence in the same

experiment.

Here is the layout; X stands for scores.

The analysis of data in complex

designs like the above, is, as

always, an operation involving the

calculation of variance. The

interpretation of the results of such

an analysis is like the interpretations

we considered in this book so far.

ANOVA mixed split plot - formula and practice example

What is ANOVA mixed split plot design

ANOVA mixed split plot designs are complex designs that employ both independent and repeated measures. The best way to explain this is to present the layout of these experimental designs.

TABLE SHOWING THE LAYOUT

 OF MIXED SPLIT-PLOT DESIGNS

Observe that there are two independent groups, depressive, and schizophrenic. Also observe that each subject of the depressive and schizophrenic groups is repeatedly tested, once with Drug 1, and later with Drug 2. This is a repeated measures arrangement So here we have a design in which independent and repeated measures are mixed. The name split plot comes from the fact that this design is extensively used in agricultural research.

ANOVA mixed split plot designs formula

As in all ANOVA, the formula for these designs is: 

We read this as follows: Mean square between over mean square within. What is mean square, you ask? It is the mean of squares. What is squares, you ask. Squares is the statistical term for squared deviations (of squared differences) of each score X from the mean. What are the squared differences, you ask. Remember the formula for variance?

Look at the numerator

These are the squared differences summed. To complete our reasoning, we go back to where we started, the F formula, or F ratio, the formula for ANOVA. Why mean sums of squares? Simple because like all averages, we divide each sum of squares by its degrees of freedom. If you are observant, you will notice that the F formula is a modified t formula.

FORMAT OF ANOVA MIXED SPIT PLOT SUMMARY TABLE

HOW TO CALCULATE df OF ANOVA MIXED SPIT PLOT SUMMARY TABLE

ANOVA mixed split plot- practice examples

ANOVA mixed split plot- practice example 1

An experimenter wanted to test drugs (factor A), Drug 1 (A1) and Drug 2 (A2) for their effect on serotonin level in the blood of patients (factor B) suffering from depression (B1) and schizophrenia (B2) . He randomly selected six patients suffering from depression and gave them Drug 1. He waited for one hour and then he measured the level of serotonin in nanograms per liter (ng/lt) of each subject. He recorded the data. One week later he gave these subjects Drug 2. He waited for one hour and measured the level of serotonin of each subject. He also randomly selected six patients suffering from schizophrenia and repeated the same experiment that he performed with the depressive patients. The data are presented in the table below.

ANOVA MIXED SPIT PLOT SUMMARY TABLE

We have already developed the

concept of independence. In those

experiments in which each subject

is used only in one group or

condition, we say that the groups

are independent. So far in this

book we have considered only

independent-groups statistical

designs and experiments.

In designs in which the groups are

not independent, a subject is used

in more than one group or treatment.

That is, each subject experiences

more than one treatment.

For example, John may first be

given behavioral therapy, and

later, several months later, he may

also be given psychoanalytic

therapy. The effects of the two

therapies are then compared.

A variation of this arrangement is

to match each subject with

another subject on the basis of

similarity in some measure. This is

done to eliminate carryover effects

that may, obviously, be present in

giving one subject both treatments.

There are obviously advantages

and disadvantages in choosing

matched groups designs over

independent groups designs.

However, this issue is beyond the

goals of the present book. In

general, independent groups

designs are safer, and should, in

my opinion, be preferred.

The concepts in matched groups

designs are the same as those in

independent groups designs. We

will, therefore, confine ourselves to

giving examples of these designs.

First an example for t-test, and

then an example for ANOVA

repeated measures.

An example of t-test for

matched groups

In comparing two new anti-anxiety

drugs, a pharmaceutical company

selected 5 pairs of patients, each

pair matched on the basis of their

anxiety score.

Here is the layout and data of the

experiment.

  Mean for difference=0.4

The formula for the t-test for

dependent groups is

We read it as follows:

t for paired observations equals

mean of differences divided by the

standard deviation over the square

root of the n. (The standard

deviation divided by the square

root of the n is the standard error

of the mean, SEM, remember?)

You know all of the terms of the 

t-formula. You also recognize that it

is the same old story, our old

friend, the z formula.

t=+0.49 df=4

Entering the t-table with df 4 for a two-tailed 0.05 test, we

find that the required |t|=2.776

Our obtained t 0.49 is smaller

than the required, therefore we do

not have significance. We say that

the difference we observed is not

significant (p>0.05).

Study the table below..

It adds to our effort toward integration

and understanding beyond a mechanistic

use of a plethora of formulas. 

Study the table below..

It adds to our effort toward integration

and understanding beyond a mechanistic

use of a plethora of formulas. 

Example of ANOVA Repeated

Measures

Four patients with damage in the

hippocampus were treated with

two new drugs in order to see if

their memory improved.

Here is the layout as well as the

scores of the experiment. High

scores indicate improvement in

memory.

ANOVA SUMMARY TABLE

Repeated Measures

Entering the F table in Appendix

with df 1 and 3, we find an F of

10.12. This is the required F in

order to have significance. Our

obtained F (see ANOVA summary

table above) is 7.71. It is less than

the required F, therefore, we do

not have significance. We say:

There was no significant difference

between the means of the two

conditions (p>0.05). 

P greater than point o five.

I see there are questions.

What is Between Columns? You ask.

It is the usual Between variance

that you know. The variance that

our treatments produce. The

variance of the means.

What is Between Rows? you ask.

If you look at the layout above,

you see that the rows are subjects,

one subject per row. The mean of

each subject is the mean of each

row. The variance of these means

are the variance between the

rows.

Why you did not calculate an F for

the Rows? you ask.

There is no reason that I can think

of, that would justify my wanting to

know whether there is a statistical

significant difference between

subjects. That would be an

absurd statement. 

Once again you see that our

conceptual approach allowed us to

attack this design too, without the

need for new formulas. What is of

course more important is the fact

that we understand the logic of this

design too. We feel in command,

comfortable to handle any issue.