Applied Statistics · From Data to ANOVA

The whole subject,
in one sitting.

Nine units, stripped to what matters. Every idea gets a plain definition, the formula, a memory hook, and an example simple enough for a beginner. The aim is understanding, not memorising. The subject builds toward hypothesis testing (the engine) and ANOVA (the final boss) — so it's worth pacing yourself there.

Units

Nine

Contact hours

Major exams

Reference

Walpole

Formula 🧠 Mnemonic Picture-this example Color-coded units

01 / Foundations

Introduction to Statistics

Statistics is the science of collecting, organizing, analyzing, and interpreting data to make decisions under uncertainty. Two halves: descriptive (summarize what you have) and inferential (use a sample to make claims about a population).

The four words you must never mix up

Population — the entire group you care about. Sample — the slice you actually measure.
Parameter — a number describing the population (the truth, usually unknown). Statistic — a number from your sample (your estimate).

How to tell them apart in a word problem — ask two questions in order:

Who does the question ultimately want to know about? → that's the population.
Who did the researcher actually measure? → that's the sample (often smaller than the population).
A number describing the population's truth (uses N, μ, σ) → parameter. It's usually unknown — that's the whole reason to sample.
A number computed from the sample (uses n, x̄, s) → statistic. This is your estimate of the parameter.

🧠

Mnemonic

Population = the whole Pizza 🍕; Sample = one Slice. Parameter ↔ Population, Statistic ↔ Sample. Greek letters (μ, σ) = the truth; Roman letters (x̄, s) = your guess.

Types of variables

Qualitative (categorical): names/labels, no math on them (favorite color). Quantitative (numeric): amounts you can actually calculate with (height).
Discrete: countable jumps, whole numbers only — you can't have 2.5 kids in a class. Continuous: any value on a scale, decimals allowed — weight, time, height.

Levels of measurement — four tiers of "how much information does this number actually carry?" Each level keeps everything the one before it could do, and adds one new power:

Level	What it can do	Example
Nominal	label / name only — no order	eye color: brown, blue, green
Ordinal	+ can rank, but gaps aren't equal	race results: 1st, 2nd, 3rd
Interval	+ equal gaps, but no true zero	temperature: 0°C isn't "no temperature"
Ratio	+ a true zero — ratios make sense	weight: 0 kg really is nothing; 20 kg is twice 10 kg

🧠

Mnemonic

Levels = NOIR (Nominal, Ordinal, Interval, Ratio) — each letter adds one new power. Discrete = Dots you count; Continuous = a Curve you measure.

Summation notation

Sigma just means "add them all up"

Σx = x₁ + x₂ + x₃ + … + xₙ

Σx² means square first, then add — it is not the same as (Σx)², which means add first, then square. Mixing these two up is the single most common Unit I error.

📐 Worked example — Σx vs. Σx²

Data: x = {2, 4, 6}

x² = each value multiplied by itself

x	x²
2	4
4	16
6	36
Σ = 12	Σ = 56

Σx = 12 (add the raw values). Σx² = 56 (square each one first, then add). Notice (Σx)² = 12² = 144 — nowhere close to 56. Two different numbers, two different meanings.

02 / Handling Data

Collection & Presentation of Data

Garbage in, garbage out — how you collect data decides whether any of the later math means anything.

Ways to collect

Direct (interview), Indirect (questionnaire), Registration (records), Observation, Experiment.

Ways to present

Textual = words in a paragraph. Tabular = rows & columns. Graphical = charts (bar, pie, histogram).

Frequency Distribution Table (FDT)

Raw scores are noise — 20, 30, 50 numbers with no visible pattern. An FDT groups them into classes so a shape appears: where values cluster, where they thin out.

Class width

Width = RangeNumber of classes , Range = Highest − Lowest

Always round the width up, never to nearest and never down — round down and your last class won't hold the highest value. Rule of thumb: 5–15 classes.

The process — six steps, in order:

Find the Range = Highest value − Lowest value.
Decide the number of classes, k (given in the problem, or your own reasonable choice, 5–15).
Compute the width = Range ÷ k, then round up to a whole number.
Build the class limits: start at the lowest value; each next class starts width-many numbers later.
Add class boundaries: subtract 0.5 from every lower limit, add 0.5 to every upper limit — this closes the gaps between classes.
Tally each raw value into its class, then count the tally marks for the frequency (f).

🧠

Mnemonic

Build an FDT in order: R-C-W-T — find the Range, decide Classes, get the Width, then Tally.

📐 Worked example — raw data to a complete FDT

Raw scores of 20 students: 12, 22, 27, 30, 33, 36, 39, 42, 45, 47, 48, 50, 52, 55, 58, 60, 66, 70, 74, 87

Steps 1–3: Range = 87 − 12 = 75. Choose k = 6 classes. Width = 75/6 = 12.5 → round up to 13.

Steps 4–6: starting at 12, step forward by 13 each time, apply the ±0.5 boundary rule, then tally:

Boundaries = lower limit − 0.5, upper limit + 0.5 · Midpoint = (lower limit + upper limit) ÷ 2

Class limits	Class boundaries	Midpoint	Tally	f
12–24	11.5–24.5	18	II	2
25–37	24.5–37.5	31	IIII	4
38–50	37.5–50.5	44	IIII I	6
51–63	50.5–63.5	57	IIII	4
64–76	63.5–76.5	70	III	3
77–89	76.5–89.5	83	I	1
Total				20

That's a finished FDT — 20 messy raw scores now visibly cluster in the 38–50 class. Hold onto this exact table: Unit III reuses it to compute the grouped mean, median, and mode.

03 / The Center

III

Central Tendency & Location

One number to represent the "center." Three flavors — they agree when data is symmetric, and disagree when it's skewed (the interesting case).

The big three

Mean: x̄ = Σxn

Median: the middle value once data is sorted

Mode: the value that appears most often

Median position = (n + 1) ÷ 2 → the seat number, not the score sitting in it.

🧠

Mnemonic

All start with M: Mean = add & divide · Median = the Middle (sort first!) · Mode = the Most. And: the Mean is a people-pleaser — one billionaire drags it up. The Median doesn't care.

Weighted mean — when not every value counts equally

The regular mean silently assumes every value matters the same amount. Sometimes that's wrong — a final exam should count more than one quiz. The weighted mean lets you say, explicitly, how much each value counts.

Weighted mean

x̄_w = Σ(w·x)Σw

In plain words: multiply each value x by its own weight w, add those up, then divide by the total of the weights — not by how many values there are. If every weight is equal, this collapses right back into the plain mean.

📐 Worked example — weighted mean

A student's grades and each subject's unit-load (the weight):

w·x = weight × grade

Subject	Grade (x)	Units (w)	w·x
Math	90	3	270
English	85	2	170
PE	95	1	95
Total		Σw = 6	Σ(w·x) = 535

x̄_w = 535 / 6 ≈ 89.17 — notice this is not the same as the plain mean of 90, 85, 95 (which is 90). Math pulled the weighted average down toward it less than PE pulled it up, because Math and PE don't carry equal weight.

Measures of location ("non-central")

The median cuts sorted data exactly in half. Quartiles, deciles, and percentiles are the same idea, just cutting into more pieces — useful when you want to know someone's standing, not just the group's center.

Quartiles (Q1, Q2, Q3) cut sorted data into 4 equal parts — Q2 is the median.
Deciles = 10 parts, Percentiles = 100 parts. Same idea, finer cuts.

📐 Worked example — quartiles

Sorted scores of 8 students: 4, 8, 10, 12, 15, 18, 20, 24

Q2 (the median) sits between the 4th and 5th values: (12+15)/2 = 13.5.
Q1 is the median of the lower half (4, 8, 10, 12): (8+10)/2 = 9.
Q3 is the median of the upper half (15, 18, 20, 24): (18+20)/2 = 19.

A student scoring 19 sits at Q3 — better than roughly 75% of the class. That single number tells you standing, which the mean alone never can.

📐 Picture this

Five kids' scores: 3, 5, 5, 7, 10.

Mean = (3+5+5+7+10) ÷ 5 = 30 ÷ 5 = 6 Median = middle of sorted list = 5 Mode = appears most often = 5

The "average" student is ~5–6. If a sixth kid scored 100, the mean jumps to ~21 (misleading!) but the median barely moves to 6. That's when you report the median.

Mean, median & mode from grouped data (an FDT)

Once data is organized into classes, you can no longer add up individual raw values — they're hidden inside the classes. All three measures adapt the same way: treat every class as if all its values sit exactly at the midpoint, weighted by how many values (f) actually landed there.

Grouped mean

x̄ = Σfx / n

Identical to the ungrouped mean formula — x is just now the class midpoint instead of a raw value.

Grouped median

Median = L + (n/2) − CFf × width

L = lower boundary of the median class (the first class whose cumulative frequency reaches n/2). CF = cumulative frequency before that class. f = frequency of that class.

In plain words — why this formula works: you already know the median class (say the 15 kids before it, plus n/2 more, lands somewhere inside it), but not the exact spot inside that class. So you walk into the class proportionally: (n/2 − CF) tells you how many more values you still need past the class's start, and dividing by f (how many values total live in that class) turns that into a fraction of the way through. Multiply by the class's width and add it to L, the class's starting edge — that's your best estimate of exactly where inside the class the median falls. This technique is called interpolation: estimating a point between known values.

Grouped mode

Mode = L + d₁d₁ + d₂ × width

L = lower boundary of the modal class (highest frequency). d₁ = modal f − f of the class before it. d₂ = modal f − f of the class after it.

In plain words: the mode should lean toward whichever neighboring class is more "crowded." If the class before the modal class is nearly as full (small d₁), the mode sits close to L. If the class after is nearly as full instead (small d₂), the mode sits closer to the far edge. d₁/(d₁+d₂) is just that lean, expressed as a fraction of the class width.

The process — five steps:

From your FDT, list f (each class's frequency), the class midpoint (x), and build a running cumulative frequency (CF) column — CF is just each row's f added to the running total of every f before it.
Mean: multiply f × x for every class → sum → divide by n (the total number of values, i.e. Σf).
Median: find n/2, locate the first class where CF reaches it — that's your median class. Plug in L (that class's lower boundary), CF (the cumulative frequency of the class before it), f (that class's own frequency), and the class width.
Mode: find the class with the highest f — that's your modal class. Plug in L (its lower boundary), d₁ (modal f minus the frequency of the class before it), d₂ (modal f minus the frequency of the class after it), and width.
Sanity check: mode ≤ median ≤ mean (or reversed) should roughly match the skew direction you'd expect from the frequency shape.

📐 Worked example — same FDT as Unit II, now solved

CF = running total of f, top to bottom

Class boundaries	x (midpoint)	f	fx	CF
11.5–24.5	18	2	36	2
24.5–37.5	31	4	124	6
37.5–50.5	44	6	264	12
50.5–63.5	57	4	228	16
63.5–76.5	70	3	210	19
76.5–89.5	83	1	83	20
Total		20	945

Mean: x̄ = 945 / 20 = 47.25

Median: n/2 = 10. Scanning CF (2, 6, 12, …), the first class reaching 10 is 37.5–50.5 (CF = 12) — that's the median class, with CF-before = 6 and f = 6.
Median = 37.5 + [(10 − 6)/6] × 13 = 37.5 + 8.67 = 46.17

Mode: highest f is 6, in the same class 37.5–50.5 — the modal class. d₁ = 6 − 4 = 2 (vs. class before), d₂ = 6 − 4 = 2 (vs. class after).
Mode = 37.5 + [2/(2+2)] × 13 = 37.5 + 6.5 = 44.0

Mode (44.0) < Median (46.17) < Mean (47.25) — a mild right-skew, exactly what you'd expect: the classes trail off more slowly on the high side (4, 3, 1) than they build up on the low side (2, 4).

04 / The Spread

Dispersion & Skewness

The center tells you where; dispersion tells you how spread out. Two datasets can share a mean and tell completely different stories.

Spread, from lazy to powerful

Range = Highest − Lowest

Sample variance: s² = Σx² − (Σx)² / nn − 1

Standard deviation: s = √s²

Coefficient of variation: CV = ( s ÷ x̄ ) × 100%

This is called the computational form — a shortcut. It gives the exact same answer as squaring every (x − x̄) one by one, but you only need two sums, Σx² and Σx, instead of first finding x̄ and then re-visiting every value to subtract it. Same result, far fewer keystrokes. Carry decimals; round only at the end.

🧠

Mnemonic

Variance Vexes (weird squared units), SD Saves it (square-root → real units). Divide by n−1 because Samples are Shy by one. CV lets you Compare Variability across different things (₱ vs kg).

Mean Absolute Deviation — the honest average distance

Before variance, ask the simpler question underneath it: on average, how far does each value sit from the mean? That's the whole idea. Variance gets there by squaring; MAD gets there by just averaging the distances, honestly, with no exaggeration.

Mean Absolute Deviation

MAD = Σ|x − x̄| / n

The bars | | mean "absolute value" — strip the negative sign, keep only the size of the distance. A value 3 below the mean and a value 3 above it are both simply "3 away."

The process — four steps, every time:

Find the mean, x̄.
Subtract the mean from every value: x − x̄. Some will be negative — that's expected, for now.
Drop the sign: take the absolute value, |x − x̄|.
Average those distances: add them all up, divide by n (the number of values).

📐 Worked example — MAD

Data: 4, 6, 8, 10, 12 → x̄ = 40/5 = 8

x − x̄ = value minus the mean · |x − x̄| = that distance, sign removed

x	x − x̄	\|x − x̄\|
4	−4	4
6	−2	2
8	0	0
10	2	2
12	4	4
Σ		12

MAD = 12 / 5 = 2.4 — on average, a value in this dataset sits 2.4 units from the mean. Plain, honest, in the data's original units.

Why variance squares it instead — and why sample divides by n − 1

MAD is honest, but mathematically clumsy — absolute value has a sharp corner at zero, which makes it hard to build more advanced theory on top of. Variance solves the same "negatives cancel out" problem a different way: instead of stripping the sign, it squares every deviation. Squaring also erases the negative — and it behaves far better mathematically, which is why nearly every method later in this course (regression, ANOVA, confidence intervals) is quietly built on squared deviations. The cost: squared units mean nothing on their own (₱² isn't a real thing), so at the end you take a square root — the standard deviation — to land back in real, interpretable units.

Population vs. sample — same idea, two situations

Population variance: σ² = Σ(x − μ)² / N · σ = √σ²

Sample variance: s² = Σ(x − x̄)² / (n − 1) · s = √s²

Data on everyone → population → Greek letters (σ, μ, N) → divide by the full N.
Data on a subset → sample → plain letters (s, x̄, n) → divide by n − 1.

Why n − 1, really — not just a rule to memorize: your sample mean x̄ isn't the true population mean μ; it's an estimate, computed from the very data you're measuring. Because x̄ is built to sit as close as mathematically possible to your own sample, that sample's deviations are, on average, a touch smaller than they'd be from the real μ. Dividing by the slightly smaller n − 1 nudges the result back up to correct for that built-in optimism. This has a name — Bessel's correction — it's a genuine bias fix, not an arbitrary quirk.

The simplest version of that idea: imagine grading your own test using an answer key you wrote after seeing your own answers — you'd naturally score a little better than if a stranger's fixed key were used. Using x̄ (built from your own sample) as the "target" for variance has the same effect: it flatters your own data slightly. Dividing by n − 1 instead of n is how statisticians dock a small penalty to correct for grading your own paper.

🧠

Mnemonic

Population = Greek, whole group, ÷N. Sample = plain letters, subset, ÷(n−1). Can't measure everyone? You're estimating — and an estimate costs you one degree of freedom.

📐 Worked example — sample variance & standard deviation

Sample data: 7, 9, 10, 12 → x̄ = 38/4 = 9.5

x − x̄ = value minus the mean · (x − x̄)² = that distance, squared

x	x − x̄	(x − x̄)²
7	−2.5	6.25
9	−0.5	0.25
10	0.5	0.25
12	2.5	6.25
Σ		13

s² = 13 / (4 − 1) = 13/3 ≈ 4.33 → s = √4.33 ≈ 2.08

📐 Worked example — one table, MAD → variance → SD together

Six exam scores: 8, 12, 15, 15, 18, 22 → x̄ = 90/6 = 15

x − x̄ = value minus the mean · |x − x̄| = that distance, sign removed (feeds MAD) · (x − x̄)² = that distance, squared (feeds variance)

x	x − x̄	\|x − x̄\|	(x − x̄)²
8	−7	7	49
12	−3	3	9
15	0	0	0
15	0	0	0
18	3	3	9
22	7	7	49
Σ		20	116

MAD = 20 / 6 ≈ 3.33 (n on the bottom — MAD always divides by the full count).
Variance: s² = 116 / (6 − 1) = 116/5 = 23.2 (n − 1 on the bottom — Bessel's correction, since this is sample data).
SD: s = √23.2 ≈ 4.82

One dataset, one table, every spread measure — and notice SD (4.82) > MAD (3.33) here, which is typical: squaring gives extra weight to the two values furthest from the mean (8 and 22), so SD usually comes out a bit larger than MAD on the same data.

Grouped data — the same formulas, weighted by frequency

Once raw data is organized into an FDT, you no longer have individual values in hand — you have classes, each hiding several values you can't see one by one. The fix: treat every value inside a class as if it sat exactly at that class's midpoint (x), and weight everything by how many values are actually in there (f).

Grouped-data formulas

Grouped mean: x̄ = Σfx / n

Grouped MAD: MAD = Σf|x − x̄| / n

Grouped variance: s² = Σf(x − x̄)² / (n − 1) · s = √s²

The process — build the table left to right; each column feeds the next:

f — the frequency of each class, straight from your FDT.
x — the class mark (midpoint) of each class.
fx — multiply f × x, then sum the whole column → gives Σfx.
Compute the grouped mean: x̄ = Σfx / n.
x − x̄ — each class mark's distance from that mean.
|x − x̄| for MAD, or (x − x̄)² for variance.
Multiply that column by f again, sum it, then divide by n (MAD) or n − 1 (variance).

📐 Worked example — one table, mean → MAD → variance → SD

x = class midpoint · fx = f × x · x−x̄ = midpoint minus grouped mean · f|x−x̄| = f × |x−x̄| · f(x−x̄)² = f × (x−x̄)²

Class	f	x	fx	x−x̄	\|x−x̄\|	f\|x−x̄\|	(x−x̄)²	f(x−x̄)²
30–34	3	32	96	−6.79	6.79	20.37	46.10	138.30
35–39	5	37	185	−1.79	1.79	8.95	3.20	16.02
40–44	4	42	168	3.21	3.21	12.84	10.30	41.22
45–49	2	47	94	8.21	8.21	16.42	67.40	134.81
Total	14		543			58.58		330.35

Mean: x̄ = 543 / 14 = 38.79
MAD: 58.58 / 14 = 4.18
Variance: s² = 330.35 / (14 − 1) = 330.35/13 = 25.41
SD: s = √25.41 ≈ 5.04

One table, every spread measure you need — because MAD, variance, and SD are all just different lenses on the same question: how far, on average, from the center?

📐 Worked example — finishing the story from Units II & III

Remember the 20 students' raw scores you organized into an FDT back in Unit II, then found the mean (47.25), median (46.17), and mode (44.0) for in Unit III? Same table — let's find its spread too.

x = class midpoint · fx = f × x · x−x̄ = midpoint minus 47.25 · f|x−x̄| = f × |x−x̄| · f(x−x̄)² = f × (x−x̄)²

Class boundaries	f	x	fx	x−x̄	\|x−x̄\|	f\|x−x̄\|	(x−x̄)²	f(x−x̄)²
11.5–24.5	2	18	36	−29.25	29.25	58.50	855.56	1711.13
24.5–37.5	4	31	124	−16.25	16.25	65.00	264.06	1056.25
37.5–50.5	6	44	264	−3.25	3.25	19.50	10.56	63.38
50.5–63.5	4	57	228	9.75	9.75	39.00	95.06	380.25
63.5–76.5	3	70	210	22.75	22.75	68.25	517.56	1552.69
76.5–89.5	1	83	83	35.75	35.75	35.75	1278.06	1278.06
Total	20		945			286.00		6041.75

MAD = 286.00 / 20 = 14.3 — on average, a student's score sits 14.3 points from the mean of 47.25.
Variance: s² = 6041.75 / (20 − 1) = 6041.75/19 ≈ 318.0
SD: s = √318.0 ≈ 17.83

Compare the two worked examples side by side: the 30–49 class-interval data (tightly packed, SD ≈ 5.04) versus these 20 students (spread across almost the whole 12–89 range, SD ≈ 17.83). Same formulas, wildly different spread — which is exactly what SD is supposed to tell you at a glance.

Skewness — the lopsidedness

Long tail on the right → positive / right-skew (few high outliers).
Long tail on the left → negative / left-skew (few low outliers).
Symmetric → skew ≈ 0; mean ≈ median.

Diagram — the three shapes, side by side

Left-skew
mean < median < mode

Symmetric
mean = median = mode

Right-skew
mode < median < mean

The dashed lines mark where mode, median, and mean fall — watch how the mean always gets pulled furthest toward the long tail.

🧠

Mnemonic

The tail tells the tale. Skew is named after the long tail's direction — and the mean chases the tail (pulled toward the outliers).

📐 Picture this

Two classes both average 80. Class A: everyone 78–82 (tight, tiny SD — average is trustworthy). Class B: 50 to 100 (huge SD — half lost, half bored). Same mean, opposite reality. That's why SD is never optional.

05 / The Keystone

Sampling Distribution & CLT

The keystone unit. Understand why sample averages behave predictably and every test in Units VI–IX stops being magic.

Why & how we sample

Why: cheaper, faster, and sometimes the test destroys the item (you can't crash-test every car).
Random (pure lottery), Systematic (every kth one), Stratified (split into layers, sample each), Cluster (grab whole groups).

🧠

Mnemonic

Really Smart Stats Cluster → Random, Systematic, Stratified, Cluster. Stratified = slice into LAYERS then sample each. Cluster = pick whole GROUPS (entire classrooms).

How big a sample? — Slovin's Formula

A staple of research methods for deciding how many people to survey. When the population size N is known but little is known about its behaviour, Slovin's formula gives the minimum sample size for a chosen margin of error.

Slovin's sample-size formula

n = N1 + N·e²

n = sample size · N = population size · e = margin of error as a decimal (5% → 0.05). Always round the result up — round down and you'd fall short of the margin of error you asked for.

🧠

Mnemonic

Slovin's Sizes the Sample: N on top, (1 + N·e²) on the bottom. Smaller error (e↓) → bigger sample (n↑). It's a quick heuristic — it ignores how spread-out the population is, so it's for fast estimates, not high-stakes precision.

The process — three steps:

Identify N (the whole population) and e (your acceptable margin of error, as a decimal).
Plug into n = N / (1 + N·e²) and compute.
Round up to the next whole number — always up, regardless of the decimal.

📐 Worked example

A study covers all 5,000 students in a college, and you'll accept a 5% margin of error.

N = 5000 , e = 0.05 n = 5000 / (1 + 5000 × 0.05²) = 5000 / (1 + 5000 × 0.0025) = 5000 / (1 + 12.5) = 5000 / 13.5 = 370.4 → round UP → 371 students

You don't need all 5,000 — 371 well-chosen students represent them within 5%. (370.4 rounds up to 371, never down to 370 — the sample-size rounding rule always rounds up.)

Standard error & the Central Limit Theorem

The two ideas that run everything

Standard error: SE = σ ÷ √n

CLT: for n ≥ 30, the distribution of x̄ is ≈ Normal — whatever shape the population has

A normal distribution is the symmetric, bell-shaped curve where most values sit near the mean and extremes are rare. SE is the SD of the sample mean — bigger n → smaller SE → a steadier, more reliable average.

🧠

Mnemonic

The average of averages goes bell-shaped. n ≥ 30 is the magic number. SE shrinks as n grows — but only as the square root of n, never in direct proportion.

📐 Worked example — SE shrinks with the square root of n

SE = σ ÷ √n

σ	n	√n	SE = σ/√n
20	16	4	5.0
20	64	8	2.5

Quadrupling n (16 → 64) only halved SE (5.0 → 2.5), not quartered it — because SE depends on √n, not n itself. To cut your error in half, you need four times the data.

📐 Picture this — why the CLT works

Roll one die: any number 1–6, totally flat, no bell. Now roll five dice and write the average, hundreds of times. Those averages pile up around 3.5 in a bell shape — all-1s or all-6s are rare. The original was flat, yet the averages went bell-curve. That's the CLT, and it's why a sample mean is trustworthy.

Diagram — the CLT in action

Population
could be any shape

→

n ≥ 30

Sampling distribution of x̄
always bell-shaped

The population on the left can be lopsided, flat, anything. Once n reaches 30, the distribution of sample means drawn from it looks like the bell curve on the right — every time.

06 / The Engine

Test of Hypothesis

A courtroom for data. Assume "nothing's going on," then check whether the evidence is strong enough to overturn that assumption.

The two hypotheses

H₀ (null) — the boring status quo: "no difference / no effect." Always wears the = sign.
H₁ (alternative) — the new claim you're trying to prove: "there IS a difference."

The 5 steps (every test, same dance)

The process — memorize this order, apply it to literally every test in this unit:

State H₀ (the null — "no difference," always with =) and H₁ (the alternative — the claim you're testing for).
Pick α (the significance level — your accepted false-alarm risk, usually 0.05).
Compute the test statistic — z if σ is known, t if you only have the sample SD s instead.
Find the critical value from the table, using α (and df = n − 1, for t only).
Decide: compare statistic to critical value, then conclude in plain words.

The two workhorse statistics

z (σ known / large n): z = x̄ − μσ / √n

t (σ unknown / small n): t = x̄ − μs / √n , df = n − 1

Quick terms — α (significance level): the false-alarm risk you accept up front, usually 0.05. df (degrees of freedom): the number of values free to vary, here n−1. Critical value: the cut-off from the table. p-value: the chance of data this extreme if H₀ were true (small p = evidence against H₀). By hand you compare your statistic to the critical value — reject H₀ when |statistic| > critical (it landed in the tail).

🧠

Mnemonic

Steps = H-A-T-C-D (Hypotheses, Alpha, Test-stat, Compare, Decide). p Low → null must Go; p High → null gets by. Tails: ≠ → two-tailed, < or > → one-tailed.

📐 Worked example — the 5 steps, fully applied

A machine is supposed to fill bottles with 100 mL on average. Population SD is known: σ = 7 mL. A sample of n = 49 bottles averages x̄ = 101 mL. Test at α = 0.05 (two-tailed) whether the machine is off-target.

H₀: μ = 100 (on target) · H₁: μ ≠ 100 (off-target — two-tailed, since "off" could mean too high or too low).
α = 0.05, two-tailed → split between both tails, 0.025 each.
Test statistic: z = (101 − 100) / (7/√49) = 1 / (7/7) = 1 / 1 = 1.0
Critical value for α = 0.05 two-tailed: z = ±1.96.
Decide: |1.0| is not beyond 1.96 — it doesn't reach the tail. Fail to reject H₀. There isn't enough evidence the machine is off-target; a 1 mL difference is within normal sampling wobble.

Notice this example fails to reject — not every test ends in rejection. A statistic has to genuinely land in the tail, not just differ slightly from H₀.

Two ways to be wrong

Type I (α): reject a true H₀ — a false alarm.
Type II (β): fail to reject a false H₀ — you missed a real effect.

🧠

Mnemonic

Type I = cry wolf when there's none (false alarm). Type II = miss the wolf that's really there. And: statistically significant ≠ important — a tiny effect looks "significant" if n is huge.

📐 Picture this

A candy company swears each bag holds 50 pieces → H₀: μ = 50. You suspect shorting → H₁: μ < 50. You count 30 bags; average is 47. The question: is 47 "far enough" below 50 to prove cheating, or just random bag luck? Deep in the tail → reject H₀ → guilty. Type I = accuse an honest company. Type II = let real cheaters walk.

07 / The Line

VII

Correlation & Regression

Correlation measures how tightly two things move together. Regression draws the best straight line so you can predict.

Correlation coefficient r (lives between −1 and +1)

r = nΣxy − ΣxΣy√[ nΣx² − (Σx)² ] [ nΣy² − (Σy)² ]

+1 = perfect upward line · 0 = random cloud · −1 = perfect downward line.

Regression line ŷ = a + bx

slope: b = nΣxy − ΣxΣynΣx² − (Σx)²

intercept: a = ȳ − b·x̄

coefficient of determination: r² = (r)²

A scientific calculator's regression mode returns a, b and r directly — use the formulas as a sanity check.

🧠

Mnemonic

ŷ = a + bx is just y = mx + b in a lab coat. b is the boost — how much y jumps per +1 of x. Square r to get the share of variation explained (r = 0.9 → r² = 0.81 → 81%).

🧠

Mnemonic — the trap

Correlation ≠ causation. Ice-cream sales and drownings rise together — but the SUN causes both, not each other. Always hunt for the hidden third factor.

The process — build one table, read four numbers off it:

List each pair (x, y), then add columns for xy, x², and y².
Sum every column: Σx, Σy, Σxy, Σx², Σy².
Plug the five sums into r (the correlation coefficient) to measure the relationship's strength, from −1 to +1.
Plug the same sums into b (the slope — how much y changes per +1 of x) and a (the intercept — predicted y when x = 0) to get the regression line.
Square r to get r² — the share of y's variation that x explains.

📐 Worked example — one table, full regression

Hours studied (x) vs. exam score (y), five students:

xy = x × y · x² = x × x · y² = y × y

x	y	xy	x²	y²
1	52	52	1	2,704
2	58	116	4	3,364
3	65	195	9	4,225
4	70	280	16	4,900
5	80	400	25	6,400
Σ = 15	Σ = 325	Σ = 1,043	Σ = 55	Σ = 21,593

r = [5(1043) − (15)(325)] / √{[5(55) − 15²][5(21593) − 325²]} = 340 / √(50 × 2340) = 340/342.05 ≈ 0.99 — an extremely strong positive relationship.

b = 340/50 = 6.8 · a = (325/5) − 6.8(15/5) = 65 − 20.4 = 44.6 → ŷ = 44.6 + 6.8x

r² = 0.99² ≈ 0.988 → about 99% of the variation in exam scores is explained by hours studied. Prediction check: at x = 3, ŷ = 44.6 + 6.8(3) = 65.0 — matching the actual data point almost exactly, which is what a high r promises.

Diagram — the same data from the worked example

Five dots (the raw data), one line (the regression fit) — r ≈ 0.99 means the line barely misses any of them.

📐 Picture this — the causation trap

Kids with bigger feet tend to read better. Do big feet cause good reading? No — age is the hidden variable: older kids have both bigger feet and better reading, unrelated to each other directly. Whenever a correlation surprises you, ask what third factor might be driving both sides at once.

08 / The Setup

VIII

Elements of Experimentation

Before you can analyze an experiment (Unit IX), you have to design it so the results actually mean something.

The vocabulary

Four terms, and every experiment you'll ever design maps onto them:

Term	Plain meaning	In the fertilizer study below
Treatment	the thing you're testing — the "what-if"	fertilizers A, B, C
Experimental unit	what receives the treatment — the "who-gets-it"	each pot
Response variable	what you measure afterward — the "what-happens"	plant height
Factor	a variable you deliberately change; its settings are levels	fertilizer type (3 levels)

Fisher's three principles (the heart of design)

The process — apply all three before you collect a single data point:

Randomization — assign treatments by chance, so no group gets an unfair edge.
Replication — repeat each treatment many times, so one fluke can't fool you.
Local control / Blocking — group similar units together before comparing, to cancel out nuisance differences.

🧠

Mnemonic

Randomize to be fair, Replicate to be sure, Block to be smart. Roles: Treatment = the WHAT-IF · Unit = WHO-GETS-IT · Response = WHAT-HAPPENS.

📐 Picture this

Which fertilizer grows the tallest plants? Treatments = fertilizers A, B, C. Units = the pots. Response = height. Replication: many pots per fertilizer (one could be lucky). Randomization: don't put all of A on the sunny sill — assign spots by chance. Blocking: group pots by sunlight first, then compare fertilizers within each sun-group. Now it's fair.

09 / The Final Boss

Analysis of Variance (ANOVA)

The largest topic in most applied-statistics courses, and a frequent centrepiece of final exams. The good news: every design (one-way, two-way, RCBD, factorial, split-plot) is the same template with the Sum of Squares sliced differently.

What ANOVA does

Comparing 3+ group means with many t-tests is messy and error-prone. ANOVA compares them all at once by asking: is the difference BETWEEN the groups bigger than the random wobble WITHIN each group?

The ANOVA pipeline — memorize this skeleton, not five procedures

SS_total = SS_between (treatment) + SS_within (error)

MS = SS ÷ df

F = MS_between ÷ MS_within

df: treatment = k − 1 · error = N − k · total = N − 1 (k = #groups, N = total obs). The F-table needs two df — numerator AND denominator — plus α.

🧠

Mnemonic

The pipeline is SS → df → MS → F. Logic: Between bigger than Within → big F → groups really differ (reject H₀). One factor = one-way; two factors (and do they team up?) = two-way.

The process — six steps to a finished ANOVA table:

Find each group's mean, and the grand mean (mean of every value, ignoring groups).
SS_between: for each group, (group mean − grand mean)², times that group's size — sum across groups.
SS_within: for each value, (value − its own group's mean)² — sum every one of these, across all groups.
df: between = k − 1, within = N − k, total = N − 1 (k = number of groups, N = total observations).
MS = SS ÷ its own df, for both rows.
F = MS_between ÷ MS_within. Compare to the critical F; larger F favors rejecting H₀.

📐 Worked example — full one-way ANOVA

Three fertilizers, four plants each — plant heights in cm:

A: 5, 7, 6, 8 (mean = 6.5) B: 9, 10, 8, 9 (mean = 9.0) C: 4, 5, 3, 4 (mean = 4.0) Grand mean = (26 + 36 + 16) / 12 = 78/12 = 6.5

SS = sum of squares · MS = SS ÷ df · F = MS_between ÷ MS_within

Source	SS	df	MS	F
Between (treatment)	50	2	25	25
Within (error)	9	9	1
Total	59	11

SS_between = 4(6.5−6.5)² + 4(9.0−6.5)² + 4(4.0−6.5)² = 0 + 25 + 25 = 50, on df = 3−1 = 2 → MS = 50/2 = 25.
SS_within = sum of every (value − its group mean)² inside A, B, and C = 5 + 2 + 2 = 9, on df = 12−3 = 9 → MS = 9/9 = 1.
F = 25/1 = 25 — with critical F around 4.26 (df 2, 9 at α = 0.05), 25 is far beyond it. Reject H₀: the fertilizers genuinely produce different average heights; this isn't just plant-to-plant noise.

F-tables have two degrees of freedom (numerator = df_between across the top, denominator = df_within down the side), so a full table needs one grid per α level — too large to reproduce here. Your exam will supply the F-table; just make sure you're reading numerator df along the correct axis, matched against denominator df.

Interaction (the two-way twist)

🧠

Mnemonic

Interaction = "it depends." Coffee helps you focus — but coffee + no sleep = jitters. The combo matters, not just each factor alone.

📐 Picture this

Three brands of plant food, several pots each. A averages 30 cm, B 32, C 31. Are they really different, or normal plant-to-plant variation? ANOVA stacks the between-brand gap against the within-brand wobble:

Brands differ a lot, plants inside each barely vary → between ≫ within → BIG F → brands genuinely differ ✓ Brands differ a little, plants inside each vary wildly → between ≈ within → small F → it's just noise ✗

One F-value, one verdict, no messy pile of t-tests.

10 / Look It Up

Z-Table — Area Under the Curve

This table answers one question, over and over: "what fraction of a normal distribution lies below this z-score?" Every value inside it is a cumulative area — P(Z < z) — the proportion of the curve to the left of z. It's the reference table behind every critical value and every probability you've computed by hand in Units V and VI.

How to read it

Split your z-score into two parts. The first decimal (e.g. the 1.9 in 1.96) picks the row. The second decimal (the .06 in 1.96) picks the column. Where they meet is P(Z < z). Example: z = 1.96 → row 1.9, column .06 → 0.9750.

The four question types — and how to get each area from the table

The table only ever gives you area to the left. Every other kind of area is a short conversion away:

① Left of z

P(Z < z) = table value

Read it straight off the table — no conversion needed.

② Right of z

P(Z > z) = 1 − table value

The whole curve is area 1, so "everything else" is 1 minus what's to the left.

③ Between two z's

P(z₁ < Z < z₂) = larger − smaller

Look up both, subtract. This is the classic "confidence interval" computation.

④ Two tails (outside ±z)

2 × [1 − table value for +z]

Find the single right-tail area, then double it — both tails are identical by symmetry.

📐 Worked example — the critical values you already use

Every critical value in Unit VI comes straight out of this table, read backwards (area known, z unknown):

You want	Look for area	Closest table value	z ≈
95% one-tailed (α = 0.05)	0.9500	0.9495 / 0.9505	1.645
95% two-tailed (α = 0.05, one side)	0.9750	0.9750	1.96
99% one-tailed (α = 0.01)	0.9900	0.9901	2.33
99% two-tailed (α = 0.01, one side)	0.9950	0.9951	2.58

This is exactly where 1.96, 1.645, 2.33, and 2.58 — the critical values from Unit VI — actually come from. They aren't arbitrary; they're just the z-scores whose cumulative area matches your confidence level.

📐 Worked example — applying all four recipes to z = 1.50

Table lookup: z = 1.50 → row 1.5, column .00 → 0.9332 ① Left of 1.50: P(Z < 1.50) = 0.9332 ② Right of 1.50: P(Z > 1.50) = 1 − 0.9332 = 0.0668 ③ Between 0 and 1.50: P(0 < Z < 1.50) = 0.9332 − 0.5000 = 0.4332 ④ Outside ±1.50: 2 × 0.0668 = 0.1336

One lookup, four different answers — the table never changes, only what you do with the number after you find it.

📐 Quick drill — right of z = 1.32

Row 1.3, column .02 → table gives 0.9066 (that's ① left). The question wants ②, the right tail: 1 − 0.9066 = 0.0934. About 9.3% of the curve lies beyond z = 1.32.

Diagram — area to the left of z = 1.32

The shaded region is everything the table gives you directly (0.9066). The right tail is whatever's left over (1 − 0.9066 = 0.0934) — same picture as the worked example above.

Negative z? This table only lists z ≥ 0 — that's standard, and it's enough. The normal curve is perfectly symmetric, so: P(Z < −z) = 1 − P(Z < z). Look up the positive version, then subtract from 1.

z	.00	.01	.02	.03	.04	.05	.06	.07	.08	.09
0.0	0.5000	0.5040	0.5080	0.5120	0.5160	0.5199	0.5239	0.5279	0.5319	0.5359
0.1	0.5398	0.5438	0.5478	0.5517	0.5557	0.5596	0.5636	0.5675	0.5714	0.5753
0.2	0.5793	0.5832	0.5871	0.5910	0.5948	0.5987	0.6026	0.6064	0.6103	0.6141
0.3	0.6179	0.6217	0.6255	0.6293	0.6331	0.6368	0.6406	0.6443	0.6480	0.6517
0.4	0.6554	0.6591	0.6628	0.6664	0.6700	0.6736	0.6772	0.6808	0.6844	0.6879
0.5	0.6915	0.6950	0.6985	0.7019	0.7054	0.7088	0.7123	0.7157	0.7190	0.7224
0.6	0.7257	0.7291	0.7324	0.7357	0.7389	0.7422	0.7454	0.7486	0.7517	0.7549
0.7	0.7580	0.7611	0.7642	0.7673	0.7704	0.7734	0.7764	0.7794	0.7823	0.7852
0.8	0.7881	0.7910	0.7939	0.7967	0.7995	0.8023	0.8051	0.8078	0.8106	0.8133
0.9	0.8159	0.8186	0.8212	0.8238	0.8264	0.8289	0.8315	0.8340	0.8365	0.8389
1.0	0.8413	0.8438	0.8461	0.8485	0.8508	0.8531	0.8554	0.8577	0.8599	0.8621
1.1	0.8643	0.8665	0.8686	0.8708	0.8729	0.8749	0.8770	0.8790	0.8810	0.8830
1.2	0.8849	0.8869	0.8888	0.8907	0.8925	0.8944	0.8962	0.8980	0.8997	0.9015
1.3	0.9032	0.9049	0.9066	0.9082	0.9099	0.9115	0.9131	0.9147	0.9162	0.9177
1.4	0.9192	0.9207	0.9222	0.9236	0.9251	0.9265	0.9279	0.9292	0.9306	0.9319
1.5	0.9332	0.9345	0.9357	0.9370	0.9382	0.9394	0.9406	0.9418	0.9429	0.9441
1.6	0.9452	0.9463	0.9474	0.9484	0.9495	0.9505	0.9515	0.9525	0.9535	0.9545
1.7	0.9554	0.9564	0.9573	0.9582	0.9591	0.9599	0.9608	0.9616	0.9625	0.9633
1.8	0.9641	0.9649	0.9656	0.9664	0.9671	0.9678	0.9686	0.9693	0.9699	0.9706
1.9	0.9713	0.9719	0.9726	0.9732	0.9738	0.9744	0.9750	0.9756	0.9761	0.9767
2.0	0.9772	0.9778	0.9783	0.9788	0.9793	0.9798	0.9803	0.9808	0.9812	0.9817
2.1	0.9821	0.9826	0.9830	0.9834	0.9838	0.9842	0.9846	0.9850	0.9854	0.9857
2.2	0.9861	0.9864	0.9868	0.9871	0.9875	0.9878	0.9881	0.9884	0.9887	0.9890
2.3	0.9893	0.9896	0.9898	0.9901	0.9904	0.9906	0.9909	0.9911	0.9913	0.9916
2.4	0.9918	0.9920	0.9922	0.9925	0.9927	0.9929	0.9931	0.9932	0.9934	0.9936
2.5	0.9938	0.9940	0.9941	0.9943	0.9945	0.9946	0.9948	0.9949	0.9951	0.9952
2.6	0.9953	0.9955	0.9956	0.9957	0.9959	0.9960	0.9961	0.9962	0.9963	0.9964
2.7	0.9965	0.9966	0.9967	0.9968	0.9969	0.9970	0.9971	0.9972	0.9973	0.9974
2.8	0.9974	0.9975	0.9976	0.9977	0.9977	0.9978	0.9979	0.9979	0.9980	0.9981
2.9	0.9981	0.9982	0.9982	0.9983	0.9984	0.9984	0.9985	0.9985	0.9986	0.9986
3.0	0.9987	0.9987	0.9987	0.9988	0.9988	0.9989	0.9989	0.9989	0.9990	0.9990
3.1	0.9990	0.9991	0.9991	0.9991	0.9992	0.9992	0.9992	0.9992	0.9993	0.9993
3.2	0.9993	0.9993	0.9994	0.9994	0.9994	0.9994	0.9994	0.9995	0.9995	0.9995
3.3	0.9995	0.9995	0.9995	0.9996	0.9996	0.9996	0.9996	0.9996	0.9996	0.9997
3.4	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9998

Table gives P(Z < z), area to the left of z, for the standard normal distribution (μ = 0, σ = 1). Values computed to 4 decimal places, the standard textbook precision.

10 / Train

Practice Problems

Three problems per unit, built to climb: ① Warm-up for fluency, ② Exam-level for the full procedure, ③ Challenge for whether you truly get it. Solutions are hidden on purpose — attempt each one cold, then click to reveal.

How to use this — the method that works for math

Try before you peek. Attempting and struggling first (then checking) builds memory far better than re-reading a solution. Reveal the answer only after a real attempt.
Estimate first. Before computing, predict the ballpark or the direction. Catching "that can't be right" is half the skill.
The three levels are a ladder. ① locks the mechanics, ② is what an exam asks, ③ removes the scaffolding and tests understanding — most ③s target the exact mistake students make.
Read the solution even when you're right. Check your reasoning, not just the final number.
Re-do a missed problem a day later. Spacing it out is what makes it stick.

Unit I — Introduction

① Warm-up

For the data x = {4, 7, 7, 10, 12}, compute Σx and Σx².

Show solution

Σx = 4 + 7 + 7 + 10 + 12 = 40 Σx² = 16 + 49 + 49 + 100 + 144 = 358

Remember: square first, then add. Σx² ≠ (Σx)².

② Exam-level

A researcher records the blood type of 500 students to estimate the proportion of type O in the whole city. Identify the (a) population, (b) sample, (c) variable and its type, (d) whether "proportion of type O in the city" is a parameter or a statistic.

Show solution

(a) Population = all people in the city. (b) Sample = the 500 students. (c) Variable = blood type — qualitative, nominal. (d) The city proportion is a parameter (unknown truth); the proportion from the 500 would be the statistic estimating it.

③ Challenge

A set of 10 values has Σx = 50. You add a new value of 17. Find the new mean. Then judge a classmate's claim: "adding a value always raises the mean."

Predict first: will the mean go up or down — and what does that depend on?

Show solution

Old mean = 50/10 = 5. New: Σx = 67, n = 11, mean = 67/11 ≈ 6.09 — it rose, because the new value (17) is above the old mean. The claim is false: adding a value below the current mean lowers it. The mean always shifts toward the value you add, relative to where it already sits.

Unit II — Collection & Presentation

① Warm-up

Data ranges from a low of 12 to a high of 87. Using 6 classes, find the class width.

Show solution

Range = 87 − 12 = 75 Width = 75 / 6 = 12.5 → round UP to 13

Always round the width up, or the top values won't fit in any class.

② Exam-level

Using width 13 and starting at the lowest value 12, write the class limits, class boundaries, and midpoint of the first two classes.

Show solution

Class 1: limits 12–24 boundaries 11.5–24.5 midpoint 18 Class 2: limits 25–37 boundaries 24.5–37.5 midpoint 31

Lower limits step by the width (12 → 25). Boundaries = limits ± 0.5 (they close the gaps). Midpoint = (lower + upper limit)/2.

③ Challenge

A dataset of 50 values (range 200) could be grouped into 4 classes (width 50) or 20 classes (width 10). Give one drawback of each, and state the trade-off in choosing the number of classes.

Show solution

4 classes → too coarse: it over-smooths and hides the real shape of the data. 20 classes → too fine: many near-empty classes, a jagged, noisy picture with no clear pattern. The trade-off is detail vs. clarity — roughly 5–15 classes usually balances them.

Unit III — Central Tendency

① Warm-up

Find the mean, median, and mode of {3, 8, 8, 9, 12}.

Show solution

Mean = (3+8+8+9+12)/5 = 40/5 = 8 Median = middle of sorted list = 8 Mode = most frequent value = 8

Here all three agree — a sign the data is fairly symmetric.

② Exam-level

Monthly incomes (₱ thousands): 18, 20, 22, 25, 200. Compute the mean and median. Which better represents a "typical" income here, and why?

Estimate first: which will be larger, the mean or the median?

Show solution

Mean = 285 / 5 = 57 Median = 22 (the middle value)

The median (22) is the better summary. The single value 200 is an outlier that drags the mean up to 57 — far above what four of the five people actually earn.

③ Challenge

Seven numbers have a median of 10. You append the value 50. Explain why the median barely moves, even though 50 is large — and contrast this with what 50 would do to the mean.

Show solution

The median is position-based: appending one value makes n = 8, so the median becomes the average of the 4th and 5th sorted values — both still near 10. A single large number only nudges the middle. The mean, by contrast, sums every value, so 50 pulls it up substantially. This is exactly why the median is called resistant to outliers and the mean is not.

Unit IV — Dispersion & Skewness

① Warm-up

For the sample {2, 4, 6}, compute the sample variance s² and standard deviation s (use the computational form).

Show solution

Σx = 12 , Σx² = 4+16+36 = 56 , n = 3 s² = [Σx² − (Σx)²/n] / (n−1) = [56 − 144/3] / 2 = [56 − 48] / 2 = 8/2 = 4 s = √4 = 2

Divide by n−1 (not n) — it's a sample.

② Exam-level

Two machines fill bottles. Machine A: mean 500 mL, SD 2 mL. Machine B: mean 500 mL, SD 8 mL. Which is more consistent? Compute each coefficient of variation and interpret.

Show solution

CV_A = (2/500)×100% = 0.4% CV_B = (8/500)×100% = 1.6%

Machine A is more consistent — its spread relative to the mean is four times smaller. Same mean, but A's values cluster far tighter around 500 mL.

③ Challenge

A dataset is right-skewed. Without any calculation, order the mean, median, and mode from smallest to largest, and justify it with the "tail" idea.

Show solution

mode < median < mean. The long tail points right, and the mean is dragged toward the tail (the few large values pull it up). The mode sits at the tall peak on the left; the median lands between the two. (Left-skew is the mirror image: mean < median < mode.)

Unit V — Sampling & CLT

① Warm-up

A population has N = 2,400. Using Slovin's formula with a 5% margin of error, find the required sample size.

Show solution

n = N / (1 + N·e²) = 2400 / (1 + 2400 × 0.05²) = 2400 / (1 + 2400 × 0.0025) = 2400 / (1 + 6) = 2400 / 7 ≈ 342.86 → 343

② Exam-level

A population has σ = 15. (a) Find the standard error for n = 25. (b) For n = 100. (c) By what factor did SE shrink, and why didn't it shrink to a quarter when n grew four-fold?

Show solution

(a) SE = 15/√25 = 15/5 = 3 (b) SE = 15/√100 = 15/10 = 1.5

(c) SE halved (3 → 1.5). Because SE = σ/√n depends on √n, quadrupling n divides SE by √4 = 2, not by 4. The "square-root law": to halve your error you need four times the data.

③ Challenge

A population is heavily skewed. A student insists: "the sample mean can't be normal because the population isn't." For a sample of n = 50, is the student right? What changes for n = 5?

Show solution

For n = 50 the student is wrong: the Central Limit Theorem says the distribution of the sample mean is approximately normal for large n (≥ 30) regardless of the population's shape. For n = 5, the CLT hasn't kicked in — with so few values the sample mean's distribution still mirrors the population's skew. The magic is in the sample size, not the population.

Unit VI — Hypothesis Testing

① Warm-up

Write H₀ and H₁ for: "A company claims its batteries last 40 hours on average; a consumer group suspects they last less." Is it one- or two-tailed?

Show solution

H₀: μ = 40 (the status quo, always with =). H₁: μ < 40 (the suspicion). Because H₁ uses "<", it's a one-tailed (left-tailed) test.

② Exam-level

A sample of n = 36 gives x̄ = 38, with population σ = 6. Test at α = 0.05 whether the true mean is below 40 (H₀: μ = 40, H₁: μ < 40). The critical value is z = −1.645. Compute the test statistic and decide.

Show solution

z = (x̄ − μ) / (σ/√n) = (38 − 40) / (6/√36) = −2 / (6/6) = −2 / 1 = −2.0

Since −2.0 < −1.645, the statistic lands in the rejection tail → reject H₀. There's evidence the mean is below 40 hours.

③ Challenge

You rejected H₀ above. Suppose in reality the true mean is 40 (the company was honest). (a) What error did you make? (b) If α had been 0.01 (critical z = −2.33), would your decision change? (c) What does this reveal about α?

Show solution

(a) Rejecting a true H₀ is a Type I error (a false alarm). (b) At α = 0.01 the cutoff is −2.33; since −2.0 is not beyond −2.33, you'd now fail to reject. (c) A smaller α makes rejection harder, lowering the Type I error rate — but it raises the chance of missing a real effect (Type II). You can't shrink both at once; it's a trade-off.

Unit VII — Correlation & Regression

① Warm-up

Given n = 5, Σx = 15, Σy = 25, Σxy = 88, Σx² = 55, compute the regression slope b.

Show solution

b = [nΣxy − ΣxΣy] / [nΣx² − (Σx)²] = [5·88 − 15·25] / [5·55 − 15²] = [440 − 375] / [275 − 225] = 65 / 50 = 1.3

(Bonus: intercept a = ȳ − b·x̄ = 5 − 1.3·3 = 1.1, so ŷ = 1.1 + 1.3x.)

② Exam-level

A study finds r = 0.85 between hours studied and exam score. (a) Interpret the strength and direction. (b) Compute r² and interpret it. (c) If the line is ŷ = 50 + 8x, predict the score for 5 hours.

Show solution

(a) A strong positive linear relationship — more study, higher score. (b) r² = 0.85² = 0.7225 → about 72% of the variation in scores is explained by hours studied. (c) ŷ = 50 + 8(5) = 90.

③ Challenge

A headline reads: "Cities with more libraries have higher crime (r = 0.7)." Does cutting libraries reduce crime? Name the flaw and a plausible hidden variable.

Show solution

No. This is the correlation ≠ causation trap. A hidden variable — population (city size) — drives both: larger cities have more libraries and more total crime. The headline mistakes a shared cause for a direct one. Cutting libraries would do nothing to crime.

Unit VIII — Experimentation

① Warm-up

A farmer tests 3 fertilizers on tomato plants and measures yield (kg). Identify the treatments, the experimental units, and the response variable.

Show solution

Treatments = the 3 fertilizers (the "what-if"). Experimental units = the tomato plants/plots (what receives the treatment). Response = yield in kg (what's measured).

② Exam-level

A researcher compares a new teaching method vs. the old one on test scores, but lets students choose their group. Name the flaw, the Fisher principle it violates, and the fix.

Show solution

The flaw is self-selection bias — more motivated students may flock to the new method, so any score difference could be due to who chose it, not the method. It violates randomization. Fix: randomly assign students to the two groups.

③ Challenge

An experiment tests 3 fertilizers but uses only one plant per fertilizer. Even if the yields differ, why can't we trust the conclusion? Which principle is missing, and what does it protect against?

Show solution

With one plant each, a yield difference could come from that individual plant (natural plant-to-plant variation) rather than the fertilizer — you can't tell signal from noise. The missing principle is replication. It protects against mistaking random variation for a real treatment effect, and it's what lets you estimate the experimental error in the first place.

Unit IX — ANOVA

① Warm-up

An ANOVA has SS_between = 60 (df = 3) and SS_within = 90 (df = 18). Compute MS_between, MS_within, and F.

Show solution

MS_between = 60 / 3 = 20 MS_within = 90 / 18 = 5 F = MS_between / MS_within = 20 / 5 = 4.0

Follow the pipeline: SS → df → MS → F.

② Exam-level

Three diets are compared on weight loss; the ANOVA gives F = 4.0 with critical F = 3.35 (α = 0.05). (a) State H₀ and H₁. (b) Decide. (c) What does rejecting H₀ tell you — and what does it not?

Show solution

(a) H₀: μ₁ = μ₂ = μ₃ (all three diet means equal); H₁: at least one mean differs. (b) F = 4.0 > 3.35 → reject H₀. (c) It tells you at least one diet's mean is different — it does not tell you which one(s). Pinning that down needs a post-hoc test.

③ Challenge

In a two-way ANOVA (factors: Drug × Diet), the interaction term is significant. A student concludes "Drug A is always best." Why might that be wrong? Explain what a significant interaction means here.

Show solution

A significant interaction means the effect of one factor depends on the level of the other. Drug A might be best on Diet 1 but worst on Diet 2 — so no blanket "always best" claim holds. You have to read the Drug effect separately within each Diet. Interaction is the formal version of "it depends."

Goal	Data type	Test	Table
Compare 1 mean to a target number	numeric	z (σ known / large n) or t (σ unknown / small n)	`z / t`
Compare 2 group means	numeric, 2 groups	2-sample t	`t`
Compare 3+ group means	numeric, 3+ groups	ANOVA (F)	`F`
Test one proportion	yes / no	z for proportion	`z`
Compare two proportions	yes / no, 2 groups	z for two proportions	`z`
Are two categories related?	categorical	Chi-square (independence)	`χ²`
Relationship between two numerics	numeric pairs	correlation / regression	`t`

Quantity	Formula	Note
Mean	`x̄ = Σx / n`	—
Weighted mean	`x̄w = Σ(w·x) / Σw`	÷ total weight, not n
Median position	`(n + 1) / 2`	seat, not score
Grouped mean	`x̄ = Σfx / n`	x = class mark
Grouped median	`L + [(n/2 − CF)/f] × width`	L = median class's lower boundary
Grouped mode	`L + [d₁/(d₁+d₂)] × width`	L = modal class's lower boundary
Mean Absolute Deviation	`MAD = Σ\|x−x̄\| / n`	grouped: Σf\|x−x̄\| / n
Sample variance	`s² = [Σx² − (Σx)²/n] / (n−1)`	computational form
Grouped variance	`s² = Σf(x−x̄)² / (n−1)`	x = class mark
Standard deviation	`s = √s²`	real units
Coeff. of variation	`CV = (s / x̄) × 100%`	compare across things
Slovin's sample size	`n = N / (1 + N·e²)`	e = margin of error, round UP
Standard error	`SE = σ / √n`	SD of the mean
z-score	`z = (x − μ) / σ`	standardize a value
One-sample z	`z = (x̄ − μ) / (σ/√n)`	σ known / large n
One-sample t	`t = (x̄ − μ) / (s/√n)`	df = n − 1
Correlation r	`[nΣxy − ΣxΣy] / √([nΣx²−(Σx)²][nΣy²−(Σy)²])`	−1 to +1
Regression slope	`b = [nΣxy − ΣxΣy] / [nΣx²−(Σx)²]`	a = ȳ − bx̄
Chi-square	`χ² = Σ[(O − E)² / E]`	O = observed, E = expected
ANOVA F	`F = MS_between / MS_within`	MS = SS / df

df	0.100	0.050	0.025	0.010	0.005
two-tail →	0.20	0.10	0.05	0.02	0.01
1	3.078	6.314	12.706	31.821	63.657
2	1.886	2.920	4.303	6.965	9.925
3	1.638	2.353	3.182	4.541	5.841
4	1.533	2.132	2.776	3.747	4.604
5	1.476	2.015	2.571	3.365	4.032
6	1.440	1.943	2.447	3.143	3.707
7	1.415	1.895	2.365	2.998	3.499
8	1.397	1.860	2.306	2.896	3.355
9	1.383	1.833	2.262	2.821	3.250
10	1.372	1.812	2.228	2.764	3.169
11	1.363	1.796	2.201	2.718	3.106
12	1.356	1.782	2.179	2.681	3.055
13	1.350	1.771	2.160	2.650	3.012
14	1.345	1.761	2.145	2.624	2.977
15	1.341	1.753	2.131	2.602	2.947
16	1.337	1.746	2.120	2.583	2.921
17	1.333	1.740	2.110	2.567	2.898
18	1.330	1.734	2.101	2.552	2.878
19	1.328	1.729	2.093	2.539	2.861
20	1.325	1.725	2.086	2.528	2.845
21	1.323	1.721	2.080	2.518	2.831
22	1.321	1.717	2.074	2.508	2.819
23	1.319	1.714	2.069	2.500	2.807
24	1.318	1.711	2.064	2.492	2.797
25	1.316	1.708	2.060	2.485	2.787
26	1.315	1.706	2.056	2.479	2.779
27	1.314	1.703	2.052	2.473	2.771
28	1.313	1.701	2.048	2.467	2.763
29	1.311	1.699	2.045	2.462	2.756
30	1.310	1.697	2.042	2.457	2.750
40	1.303	1.684	2.021	2.423	2.704
60	1.296	1.671	2.000	2.390	2.660
120	1.289	1.658	1.980	2.358	2.617
∞ (z)	1.282	1.645	1.960	2.326	2.576

Unit	Best free source	Why
I–II · Foundations & Data	OpenStax Ch. 1; Khan "Analyzing data"	cleanest intro to terms, sampling, frequency tables
III–IV · Center & Spread	OpenStax Ch. 2; NIST §1 (EDA)	worked mean/SD + the why behind n−1 and skew
V · Sampling & CLT	OpenIntro Ch. 5; Khan "Sampling distributions"	the CLT shown by simulation — it finally clicks
VI · Hypothesis testing	Penn State STAT 500; OpenIntro Ch. 5–7	full z/t/χ² procedures with university-grade rigor
VII · Regression & correlation	Penn State STAT 501; NIST §4	least-squares derivation + r² interpretation
VIII–IX · DOE & ANOVA	Penn State STAT 502; NIST §3; Walpole	the one-way / two-way ANOVA tables, every cell

Introduction to Statistics

The four words you must never mix up

Types of variables

Summation notation

Collection & Presentation of Data

Ways to collect

Ways to present

Frequency Distribution Table (FDT)

Central Tendency & Location

Weighted mean — when not every value counts equally

Measures of location ("non-central")

Mean, median & mode from grouped data (an FDT)

Dispersion & Skewness

Mean Absolute Deviation — the honest average distance

Why variance squares it instead — and why sample divides by n − 1

Grouped data — the same formulas, weighted by frequency

Skewness — the lopsidedness

Sampling Distribution & CLT

Why & how we sample

How big a sample? — Slovin's Formula

Standard error & the Central Limit Theorem

Test of Hypothesis

The two hypotheses

The 5 steps (every test, same dance)

Two ways to be wrong

Correlation & Regression

Elements of Experimentation

The vocabulary

Fisher's three principles (the heart of design)

Analysis of Variance (ANOVA)

What ANOVA does

Interaction (the two-way twist)

Z-Table — Area Under the Curve

The four question types — and how to get each area from the table

Practice Problems

Exam Cheat Sheet

1 · Which test do I use?

2 · Master formula list

3 · The z-table lives in its own section

4 · The t-Table (critical values)

5 · Five mistakes that cost points (no-PC edition)

Glossary of Key Terms

Sources & Further Reading

1 · Free open references — zero-cost, exam-grade

2 · Standard course textbooks

3 · Where to go deeper, by unit