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Topeduprep Field Manual · AP Statistics

The Data Explorer's
Field Journal

Unit 1 — Exploring One-Variable Data
Compiled & annotated by Mr. Minho Kim
Issued by TopEduPrep Expedition Society
Data Explorer's Journal
LOG 01 / 09
00

Expedition Briefing

Welcome, Field Researcher

"Every great expedition begins with a single, messy pile of numbers. Your job isn't to be afraid of the pile — it's to learn how to read it. By the end of this journal, a spreadsheet of numbers will look less like noise and more like a map." — Mr. Minho Kim

You've just joined the TopEduPrep Expedition Society, a team of field researchers who travel into the "Data Frontier" to study whatever numbers they find — quiz scores, streaming habits, heights, you name it. This journal is your field manual for Unit 1: Exploring One-Variable Data.

How this journal works
  • Field Notes — the core idea, explained in plain language first.
  • Specimen examples — small, real datasets you'll see again and again so the numbers feel familiar.
  • Explorer's Solution — a short, clean walkthrough of how to get the answer, no jargon.
  • Field Tips — the mistakes past explorers made, so you don't have to repeat them.
🧭
How to travel: Use the tabs on the left (or the ☰ button on mobile) to jump between chapters. Each chapter builds on the last, so first-time explorers should go in order.
Your route through the frontier
LogChapter
01Categorical vs. Quantitative Data
02Shape, Center, Spread & Outliers
03Charting the Terrain (dotplot, histogram, boxplot, stem-leaf)
04Measures of Center (mean, median)
05Measures of Spread (range, IQR, variance, SD)
06Standardizing & the Normal Curve
07The 1.5×IQR Outlier Rule
08Field Summary & Formula Sheet
01

Log Entry 01

Categorical vs. Quantitative

Before an explorer measures anything, they sort their specimens into two bins. Some things you can only label. Other things you can actually measure. Mixing the two up is the first mistake every rookie makes.
Field Notes

Categorical (qualitative) data places an individual into a group or category. You can count how many are in each group, but the values themselves aren't numbers you can do math with.

Quantitative data takes numerical values that measure or count something, and it makes sense to do arithmetic with them (add them, average them, and so on).

🔎
Quick test: Ask "does averaging these values make sense?" A jersey number or zip code is stored as a number but isn't truly quantitative — averaging two zip codes gives you nonsense. Context decides the type, not just the digits.
Specimen Examples
Specimen A
A field survey asks 50 students: "What is your favorite subject?" Answers: Math, English, Science, History...
Categorical. Each answer sorts a student into a labeled group — there's no numerical order or average "favorite subject."
Specimen B
The same survey also records each student's score on a 20-question quiz.
Quantitative. The scores are counts, and averaging them (finding a class mean) is meaningful.
Specimen C
A researcher records each hiker's t-shirt size (S, M, L, XL).
Categorical — and specifically an ordinal category, since the sizes have a natural order (S < M < L < XL) even though you still can't average them.
Specimen D
A researcher records each hiker's resting heart rate (beats per minute).
Quantitative. It's a measured numerical amount, and differences between values are meaningful (72 bpm is truly 6 more than 66 bpm).
Field Guide Table
SignalCategoricalQuantitative
What it recordsA group or labelA count or measurement
Typical graphsBar chart, pie chartDotplot, histogram, boxplot
Summaries usedCounts, proportionsMean, median, standard deviation
02

Log Entry 02

Shape, Center, Spread & Outliers

Once you've measured your specimens, lay all the values out side by side. The overall silhouette they form — the distribution — tells you almost everything before you calculate a single number.
Field Notes: describing any distribution

Whenever you look at quantitative data, describe it using four things — explorers remember this as S.O.C.S.:

  • Shape — symmetric, skewed left, skewed right, unimodal, bimodal, uniform.
  • Outliers — any unusually small or large values, far from the rest.
  • Center — a typical, middle value (mean or median).
  • Spread — how stretched out the values are (range, IQR, or standard deviation).
The four shapes explorers see most
Symmetric / Unimodal
Left and right sides mirror each other. Mean ≈ median.
Skewed Right
A long tail stretches to the right (high values). Mean > median.
Skewed Left
A long tail stretches to the left (low values). Mean < median.
Bimodal
Two separate peaks — often a sign of two hidden groups.
⚠️
Skew points toward the tail, not the hump. If the long thin tail reaches toward high values, that's "skewed right" — even though the tall peak sits on the left.
Specimen Example
Field Case
A researcher records the weekly streaming hours for 10 students: mostly small numbers clustered together, with one student who streams far more than everyone else.
Description: The distribution is skewed right (a long tail toward high values) with a likely outlier on the high end. Because of the skew and outlier, median is the safer measure of center, and IQR the safer measure of spread. (We'll calculate this exact dataset in Log 07.)
03

Log Entry 03

Charting the Terrain

A pile of numbers is hard to read. A good chart turns that pile into a map. Every explorer carries four kinds of maps — let's chart the same specimen with all four.
📋
Specimen Set (used all chapter): exam scores out of 100 for 16 students:
62, 68, 71, 74, 75, 78, 79, 80, 82, 83, 85, 86, 88, 90, 93, 95
1. Dotplot

Every individual value gets its own dot above a number line. Great for small datasets — you can see every specimen at once.

60 70 80 90 100
No two students scored identically, so each dot sits alone in a single row — but you can already see scores clustering in the 70s–80s with a gap near 60.
2. Histogram

Values are grouped into equal-width bins, and bars show how many specimens fall in each bin. Best for larger datasets where individual dots would be too crowded.

6070 8090100 2563
Bins of width 10: 60–70: 2, 70–80: 5, 80–90: 6, 90–100: 3 students. The tallest bar (80s) shows where most scores landed.
3. Boxplot (Box-and-Whisker)

Built from five numbers: minimum, Q1, median, Q3, maximum. The box holds the middle 50% of the data; whiskers reach to the min/max.

60708090100
min = 62, Q1 = 74.5, median = 81, Q3 = 87, max = 95. Box spans the IQR (Q1 to Q3); the line inside is the median. No whisker extends past the 1.5×IQR fences here, so there are no outliers (more on that rule in Log 07).
4. Stem-and-Leaf Plot

The "stem" is every digit except the last; the "leaf" is the last digit. It keeps every original value visible while showing shape, like a sideways histogram.

6 | 2 8
7 | 1 4 5 8 9
8 | 0 2 3 5 6 8
9 | 0 3 5
Key: 6 | 2 means a score of 62.
🗺️
Choosing a map: small dataset → dotplot or stem-leaf (keeps every value). Large dataset → histogram. Comparing two groups or spotting outliers fast → boxplot.
04

Log Entry 04

Measures of Center

If you could only report one number to describe your specimens, what would it be? Explorers reach for one of two "basecamp" numbers: the mean or the median.
📋
Specimen Set (used through Log 05): quiz scores out of 10 for 12 students:
4, 5, 6, 6, 7, 7, 7, 8, 8, 9, 9, 10 (already sorted)
Mean

The mean (x) is the arithmetic average: add every value, divide by how many there are.

x = ( Σx ) / n
Explorer's Solution
Sum: 4+5+6+6+7+7+7+8+8+9+9+10 = 86. Count: n = 12.
x = 86 / 12 = 7.17 (rounded to 2 decimal places)
Median

The median is the middle value once the data is sorted. With an even count, it's the average of the two middle values.

Explorer's Solution
n = 12 (even), so the median sits between the 6th and 7th sorted values.
Sorted: 4, 5, 6, 6, 7, 7, 7, 8, 8, 9, 9, 10 → the 6th and 7th values are both 7.
Median = (7 + 7) / 2 = 7
⚖️
Resistance matters. The median is resistant — a single extreme value barely moves it. The mean is not resistant — it gets pulled toward outliers and long tails. For skewed data, report the median; for roughly symmetric data, the mean and median are usually close, so either works well.
Specimen Example
Field Case
Recall the skewed streaming-hours data from Log 02 (one student streams far more than the rest).
In a right-skewed dataset like that, the mean gets pulled toward the tail (higher) while the median stays near the typical student. We'll compute both exactly in Log 07 — keep this in mind as a preview.
05

Log Entry 05

Measures of Spread

Two campsites can have the same average temperature and still feel completely different — one calm and steady, one wild with swings. Center alone never tells the whole story; you need to measure the spread too.
📋
Same Specimen Set as Log 04: 4, 5, 6, 6, 7, 7, 7, 8, 8, 9, 9, 10 (n = 12, x = 7.17)
Range

The simplest spread measure: maximum minus minimum.

Explorer's Solution
Range = max − min = 10 − 4 = 6
Easy to compute, but it only uses two values — one wild outlier can wreck it.
Quartiles & IQR

Q1 is the median of the lower half of the data; Q3 is the median of the upper half. The interquartile range (IQR = Q3 − Q1) is the width of the middle 50% of the data.

Explorer's Solution
Lower half: 4, 5, 6, 6, 7, 7 → Q1 = (6+6)/2 = 6
Upper half: 7, 8, 8, 9, 9, 10 → Q3 = (8+9)/2 = 8.5
IQR = Q3 − Q1 = 8.5 − 6 = 2.5
Variance & Standard Deviation

These measure the typical distance of each value from the mean. Standard deviation (s) is in the original units, which makes it the go-to number for "typical distance from average."

s² = Σ(x − x)² / (n − 1)     s = √(s²)
Explorer's Solution
Subtract the mean (7.17) from each value, square each result, and add them all up: Σ(x − x)² ≈ 33.67.
s² = 33.67 / (12 − 1) = 33.67 / 11 ≈ 3.06
s = √3.06 ≈ 1.75
🎒
Why n − 1? Dividing by n − 1 instead of n is a small correction that makes s a better estimate of the true population spread when you're working from a sample — which is almost always the case in AP Statistics.
Field Guide: which spread to report
Center usedPair it with
MedianIQR (resistant to outliers)
MeanStandard deviation (sensitive to outliers)
06

Log Entry 06

Standardizing & the Normal Curve

How do you compare a height measured in one camp to a test score measured in another? Explorers convert every measurement into a universal unit: "how many standard deviations from the mean."
Standardizing: the z-score

A z-score tells you how many standard deviations a value sits above (positive) or below (negative) the mean.

z = (x − μ) / σ
📋
Specimen Set: Heights of students at basecamp are approximately Normal with mean μ = 165 cm and standard deviation σ = 6 cm.
Explorer's Solution — Example 1
A student measures 177 cm tall. Find their z-score.
z = (177 − 165) / 6 = 12 / 6 = z = 2
This student is 2 standard deviations above the mean height.
Explorer's Solution — Example 2
Another student has z = −0.5. Find their actual height.
Rearranged: x = μ + z·σ = 165 + (−0.5)(6) = 165 − 3 = 162 cm
The Normal Distribution & the 68-95-99.7 Rule

Many natural measurements (like height) follow a symmetric, bell-shaped Normal distribution. For any Normal distribution, the same three percentages always apply:

μ = 165 159 171 153 177 147 183 68%
68% of heights fall within 1σ of the mean (159–171 cm)
95% fall within 2σ (153–177 cm)
99.7% fall within 3σ (147–183 cm)
Explorer's Solution — Example 3
About what percent of students are taller than 177 cm?
177 cm is exactly 2σ above the mean (z = 2). Since 95% falls within 2σ, 5% falls outside — split evenly between both tails, so 2.5% are taller than 177 cm.
🧭
The rule only works for roughly Normal (symmetric, bell-shaped, unimodal) data. Never apply 68-95-99.7 to a skewed distribution.
07

Log Entry 07

The 1.5×IQR Outlier Rule

Every expedition eventually finds that one specimen that doesn't belong — a boulder in a field of pebbles. Explorers don't decide "that looks weird" by eye; they use a precise rule.
📋
Specimen Set: weekly streaming hours for 10 students (sorted):
2, 3, 3, 4, 4, 5, 5, 6, 7, 22
Field Notes

Build a "fence" on each side of the box using the IQR. Anything outside the fences is flagged as an outlier.

Lower fence = Q1 − 1.5 × IQR     Upper fence = Q3 + 1.5 × IQR
Explorer's Solution
Lower half: 2, 3, 3, 4, 4 → Q1 = 3. Upper half: 5, 5, 6, 7, 22 → Q3 = 6.
IQR = 6 − 3 = 3
Lower fence = 3 − 1.5(3) = 3 − 4.5 = −1.5
Upper fence = 6 + 1.5(3) = 6 + 4.5 = 10.5

Every value sits between −1.5 and 10.5 except 22, which is far above the upper fence.
22 is an outlier.
Watch the center measures move
MeasureWith the outlier (22)Without it
Mean6.104.33
Median4.504.00
Std. dev (s)5.781.58
The single outlier drags the mean up by nearly 2 hours and more than triples the standard deviation — while the median barely moves. This is exactly why median + IQR is the safer pair to report whenever an outlier or skew is present.
🪨
An outlier isn't automatically an error. Sometimes it's a data-entry mistake worth fixing — but sometimes it's a genuinely unusual (and interesting!) specimen worth investigating, not deleting.
08

Log Entry 08 — Final Entry

Explorer's Field Summary

The expedition is complete. Before you head home, copy every key formula and rule into the back page of your journal — this is the page you'll flip to the night before the exam.
Data Types

Categorical = labels/groups. Quantitative = numbers you can meaningfully average.

Describing a Distribution — S.O.C.S.

Shape (symmetric / skewed left / skewed right / uni- or bimodal), Outliers, Center, Spread.

Graphs

Dotplot & stem-leaf (show every value, best for small n) · Histogram (bins, best for large n) · Boxplot (5-number summary, best for spotting outliers / comparing groups).

Center
x = Σx / n   |   median = middle value (sorted data)
Spread
Range = max − min
IQR = Q3 − Q1
s² = Σ(x − x)² / (n − 1)    s = √(s²)
Standardizing
z = (x − μ) / σ

68% within 1σ · 95% within 2σ · 99.7% within 3σ of the mean (Normal distributions only).

Outlier Rule
Lower fence = Q1 − 1.5×IQR    Upper fence = Q3 + 1.5×IQR
Any value beyond either fence is an outlier.
Center + Spread Pairing
If the shape is...Report
Roughly symmetric, no outliersMean & standard deviation
Skewed, or outliers presentMedian & IQR
🏕️
Expedition complete. You've charted every core skill in Unit 1 — data types, shape, graphs, center, spread, standardizing, and outliers. Next expedition: Unit 2, comparing distributions between groups.