Statistics you can move with your hands.
Six core ideas of MTH 155, each taught the same way: watch a worked example build one step at a time, check that it clicked, then explore the live tool yourself. No prior statistics needed, and every key term explains itself the moment you hover over it.
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These six tools aren't six separate topics, they're one connected argument. The normal and binomial distributions are the shapes; the Central Limit Theorem is the engine that makes sample means predictable; confidence intervals are what that engine powers; regression finds the relationship between two variables; and hypothesis testing is how you weigh the evidence. If this is your first pass, work through them in this order:
Every module follows the same four steps: see a worked example, check your understanding, explore the tool, then take a graded checkpoint. Any colored, underlined word explains itself when you hover or tab to it, so you never have to memorize vocabulary up front. Your checkpoint scores save to your progress report, which you turn in as a PDF, and every formula lives in the quick reference.
The Central Limit Theorem
Real data is usually messy and lopsided, not a tidy curve. Yet something surprising happens when you take small groups and average them: those averages settle into the same smooth, predictable shape every single time. This page builds that idea up slowly, starting from a plain list of numbers. By the end you will know what a population, a sample, and a normal curve are, and why that one fact quietly powers the rest of the course.
Any colored, underlined word on this page is a key term. Hover over it, or move to it with the keyboard and it will explain itself, in plain language, the moment you need it. You do not have to memorize anything up front.
Here is the whole story in one sentence: even when a population is lopsided, the averages of many samples pile up into a tidy normal curve. The walkthrough below shows exactly how, one small step at a time. Start there.
💡 The takeaway
The shape of the population almost stops mattering once you average. Lopsided, flat, or two humps, push the sample size up and the pile of averages always slides toward the same smooth normal curve, centered on the truth.
That is why the rest of the course can lean on a single sample average: we know, in advance, the predictable shape it came from. Smaller samples wobble more; collecting four times the data only halves the standard error, because of that square root.
Hypothesis Testing
A hypothesis test asks one question: is the gap between what you observed and what someone claimed big enough to be surprising, or could plain sampling luck explain it? This page builds the test step by step, like a skeptic weighing evidence, so the t-curve, the threshold, and the p-value all make sense together.
Hover any colored word for its meaning, and start with the walkthrough below, which builds the test up one step at a time.
The logic in one breath: assume the skeptical claim, the null hypothesis, is true. Measure how far your sample falls from it in standard errors, which gives the test statistic. Find the p-value: how often a result this surprising would happen if the claim were true. If it falls below your significance level, you reject the claim.
💡 The takeaway
A hypothesis test never "proves" anything, it weighs evidence. A small p-value means your data would be surprising if the claim were true, which is grounds to reject the claim. A large one just means "not surprising enough," so we fail to reject, we never "accept" the null.
Notice three levers: bigger gaps, smaller spread, and larger samples all shrink the p-value. And the same data can be significant at a 0.05 threshold but not at 0.01, the threshold is a choice you make before looking, not after.
Confidence Intervals
"95% confident" is the most misread phrase in statistics. It is a statement about the method: repeat the whole sampling process over and over and about 95% of the intervals you build will capture the truth. This page builds that idea from a single best guess, and here you can run that "over and over" and count.
Hover any colored word for its meaning, and start with the walkthrough below, which builds the interval up one piece at a time.
In one sentence: take your point estimate (your best single guess from a sample), add a cushion on each side called the margin of error, and you get a confidence interval: a believable range for the true value. The confidence level (like 95%) is how often that recipe captures the truth.
💡 The takeaway
Confidence is a property of the procedure. Once you have built one specific interval, the true average is either in it or it is not, there is no probability left. The "95%" describes how often this recipe succeeds across many uses.
Two tradeoffs show up immediately: cranking the confidence level to 99% captures more often but gives wider, vaguer intervals; growing the sample size tightens the intervals without sacrificing capture rate. You cannot just demand "narrow and very confident" for free, you pay with sample size.
The Normal Distribution
The bell curve is the most common shape in all of statistics, and it lets you answer one kind of question really well: how likely is a value like this one? This page builds the bell up from a plain list of numbers, then shows how a single idea, the z-score, lets you read any bell curve at all.
Hover any colored, underlined word to see what it means. The walkthrough below introduces the terms one at a time, so start there and you will not need to memorize anything in advance.
In one sentence: a normal curve is set by its average and standard deviation, probability is just area under it, and the z-score turns every bell into the same simple ruler.
💡 The takeaway
Every "what is the chance of a value above, below, or between?" question on a normal curve is really an area question. Convert to a z-score and you can read any bell off the same standard scale, which is why one method covers test scores, heights, and commute times alike.
Watch the famous benchmarks fall out on their own, the 68-95-99.7 rule: about 68% of the area sits within one standard deviation of the mean, 95% within two, and 99.7% within three. Slide the standard deviation up and the same z-scores keep the same areas, even as the raw values spread out.
Scatter Plot & Regression
When two things move together, like hours studied and exam scores, we want to measure how tightly they track and turn the pattern into a prediction. This page builds the idea from a single dot: a scatter plot shows the relationship, the regression line summarizes it, and two numbers tell you how much to trust it.
Hover any colored word for its meaning, and start with the walkthrough below, which builds the picture up one dot at a time.
The story in one breath: plot the pairs as a scatter plot, measure the tilt of the cloud with the correlation r (from -1 to +1), draw the regression line to predict y from x, and check r-squared to see how much the line explains. One rule outlasts all the math: a strong relationship is never proof of causation.
💡 The takeaway
The line gives you a prediction; the correlation and r-squared tell you how seriously to take it. A steep slope with a tiny r-squared is a confident-looking line drawn through a cloud: impressive slope, useless predictions.
Two habits to keep: never trust a relationship you have not plotted (the same r can come from wildly different shapes), and never read causation into a correlation. The line describes how the points move together, not why.
The Binomial Distribution
Flip a coin, shoot a free throw, check whether each customer buys. Any time you repeat the same yes/no situation a fixed number of times and count the wins, you get a binomial distribution. This page builds it up from a single trial, and ends by showing how it quietly turns into the normal curve you already know.
Hover any colored word for its meaning, and start with the walkthrough below, which introduces each idea one at a time.
You only need three things for a binomial: a fixed number of trials (n), the same success chance (p) every time, and trials that do not affect each other. From those, three facts do almost all the work: the average count is n × p, the lean of the shape comes from p, and with enough trials the whole thing becomes a normal curve. There is an exact formula for any single probability, but the tool computes it for you, so you can focus on reasoning.
💡 The takeaway
The binomial answers "how many successes?" precisely, one bar per outcome. Its lean is set entirely by p: below 0.5 it tilts right (a long tail of high counts is rare), above 0.5 it tilts left, and at exactly 0.5 it is perfectly symmetric.
Most importantly, it is the Central Limit Theorem in miniature. A count of successes is really a sum of many yes/no trials, and sums of many independent pieces go normal. That is why, once you have enough trials, you can hand the problem to the normal curve and use everything from the Normal module.
Notation, terms & formulas
The symbols and definitions that show up across every module, in one place. The single most common stumble in this course is mixing up population parameters (the fixed truth we rarely know) with sample statistics (what we actually measure and use to estimate them). Start there.
A parameter describes a whole population and is usually unknown, it's the number we're after. A statistic describes a sample and is what we compute from data to estimate the parameter. Greek letters are parameters; Latin letters are statistics.
| Quantity | Parameter (population) | Statistic (sample) |
|---|---|---|
| Mean | μ, "mew" | x̄, "x-bar" |
| Standard deviation | σ, "sigma" | s |
| Proportion | p | p̂, "p-hat" |
| Size | N | n |
Every term used across the site, color-coded the same way everywhere. Inside any module you can hover over a colored, underlined word, or move to it with the keyboard, to see its definition right where you need it. This page collects them all in one place.
Looking for the syllabus, due dates, or your grades? Head to the course page.
Progress report
A running record of the concept checks you have passed and the graded checkpoints you have finished. When you are ready to turn it in, save this page as a PDF and upload it to the course page.