You cannot see the population, so you measure a sample — and the whole discipline is saying how close the sample gets. Mean, median and standard deviation by hand; the normal curve, its seven landmarks, the 68–95–99.7 rule and z-scores; surveys, observational studies and experiments and what each is allowed to conclude; random sampling against bias; margin of error and why 52% ± 3 is not a lead; a simulation that puts a claim to the test. Study cards, hints, a practice quiz at three levels, and four workshop activities that check themselves.
One normal curve. The mean sits at the peak; everything is measured out from there.
68 inside one standard deviation, 95 inside two, 99.7 inside three. What is left over is split between two tails, so beyond μ + 2σ is 2.5%, not 5% — the 5% is both ends together.
What people get wrong
⚠️People often think…
Above μ + 2σ lies 5% of the data.
95% sits inside two deviations, so 5% is OUTSIDE — and it is split between the low tail and the high tail. One tail is half of that: 2.5%.
Two tails. Halve what is left over.
⚠️People often think…
If the squared deviations average to 16, the standard deviation is 16.
16 is the VARIANCE — it is still in squared units. Take the root: √16 = 4. The standard deviation has to be in the same units as the data, or it cannot be marked on the axis.
Variance is squared. Root it.
⚠️People often think…
52% with a margin of ±3 points means that side is ahead.
The interval runs 49% to 55%. It contains 49, which is behind. When the margin reaches across the halfway line the honest report is: too close to call.
Read the whole interval, not the middle.
⚠️People often think…
A sample of 5,000 is big enough that bias stops mattering.
Size shrinks random error, not bias. 5,000 people who chose to answer are still 5,000 volunteers — a bigger sample of the wrong group gives you a more precise wrong answer.
Size fixes noise. Only randomness fixes bias.
Landmarks first, then read the bands
1Watch one
Scores are normal with μ = 70 and σ = 5. What percent scored above 80?
Mark the landmarks: 60, 65, 70, 75, 80.
80 is two standard deviations above the mean.
95% lies inside two, so 5% lies outside — in both tails.
One tail is half of that: 2.5% scored above 80.
2Do one with me
Same curve: μ = 70, σ = 5, variance 25.
Percent between 65 and 75 is
Percent above 80 is
If the variance is 16, σ is
💬One sentence, then you move on
Why is one tail 2.5% and not 5%?
3Try one
A 1300 on one test, a 28 on another. What one number lets you compare them?
I want a hint first
Count the distance from the mean in standard deviations instead of points. That number has a one-letter name.
💬Last one — then you're done here
Why does a bigger sample not fix a biased one?
Where this goes
Where this lives
Every poll, every medical study, every “9 out of 10 people” claim. The number is easy; the question is who got asked and how far the answer could be off.
What this feeds
Next unit is financial math, where the numbers are your own.
Name one claim you have heard that came from a survey.
One card at a time — tap “Show me” to check yourself, then Next. Start at Foundation; when those feel easy, climb.
Helpful Hints
🧭 Any normal-curve question — the routine
Write the seven landmarks first: μ − 3σ, μ − 2σ, μ − σ, μ, μ + σ, μ + 2σ, μ + 3σ (for μ = 170, σ = 8: 146, 154, 162, 170, 178, 186, 194). Mark the bands 68 · 95 · 99.7 between them and the tails 16 · 2.5 · 0.15 outside. Then find where the question’s number sits and add up the pieces. For a value between landmarks, compute z = (x − μ)/σ and use a table.
Seven landmarks, then read the bands.
💡 The whole unit in one idea
You cannot see the population, so you measure a sample — and a random sample’s statistic lands NEAR the parameter, not on it. Statistics is the discipline of saying how near: the margin of error says how far a poll can miss by chance, and a simulation says how rare your result would be if a claim were true. Two randoms do two jobs: random sampling lets you generalize, random assignment lets you say “caused.”
⚠️ Traps the test loves
Giving 5% for one tail. Beyond μ + 2σ is 2.5%; 5% is both tails together.
Stopping at the variance. 80 ÷ 5 = 16 is the variance; the standard deviation is √16 = 4.
Calling 52% ± 3 a lead. 49% is in the interval — the race is within the margin.
Believing a huge sample fixes bias. 5,000 volunteers are still volunteers.
Reading “associated with” as “causes.” Only random assignment earns the word cause.
Comparing raw scores from different tests. 1300 vs 28 says nothing; z = 1.25 vs z = 1.4 does.
📐 The numbers to know — the chart
Question
Formula or rule
Worked (μ = 170, σ = 8; poll n = 1,000)
Within 1σ / 2σ / 3σ
68% / 95% / 99.7%
162–178 / 154–186 / 146–194
One tail beyond 1σ / 2σ
16% / 2.5%
above 178: 16%; above 186: 2.5%
z-score
z = (x − μ)/σ
x = 182 → z = 12/8 = 1.5
Percentile of z = −1, 0, 1, 2
16th, 50th, 84th, 97.5th
z = 1 → 84% are at or below
Standard deviation
√(sum of squared deviations ÷ n)
10, 12, 14, 18, 21 → √(80/5) = 4
Margin of error (95%)
≈ 1/√n, or 2√(p̂(1 − p̂)/n)
n = 1,000 → 3.2%; n = 400 → 5%
Halve the margin
× 4 the sample
1,000 → 4,000 gives 1.6%
🎯 How the test will ask
A mean and σ — “what percent is above / below / between?” Landmarks, then bands.
Two scores from two tests — “who did better?” Convert both to z.
A small data set — “find the mean and standard deviation.” Mean, deviations, square, average, root.
A described study — “survey, observational or experiment? What can it conclude?” Did anyone ASSIGN a treatment?
A poll with a margin — “does A lead?” Build the interval; check whether 50% is inside.
A sampling method — “what is wrong with it?” Name the bias: convenience, voluntary, undercoverage, wording.
A simulation result — “is 15 heads evidence?” Compare the real result with how often chance produced it.
✅ Before the test, can you…
Tell population from sample and parameter from statistic in a described study?
Compute a mean, a median and a standard deviation by hand for five numbers?
Draw the normal curve with its seven landmarks and read any 68–95–99.7 question off it?
Find a z-score and say what percentile it is?
Classify a study as survey, observational or experiment, and say what it can and cannot conclude?
Read 52% ± 3 correctly and explain why it is not a lead?
Describe a simulation that tests a claim, and say what makes a result strong evidence?
Pick your level
Look back at anything you missed — the hint that appeared is exactly what to reread tonight.
How sure did you feel?
Workshop
Work like a statistician: label the normal curve with the numbers it always carries, sort twelve studies by how their data were collected, put the standard-deviation computation in order, then read a poll’s margin of error and argue what it does and does not show. Every activity checks itself, and hints are free.
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Your practice record — saved on this device
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