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Statistics I · Week 3 — Numerical data

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Statistics I · Week 3 — Numerical data

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5 min read

2026-08-16

Reading compass

Now · Week map

Week 3 — numerical data

Quiz 2 scope: Weeks 1–8 per IITM May 2026 foundation courses. Source baseline: IITM BS admissions important-dates calendar · May 2026 cycle. Times on assessments are operational conventions — verify hall ticket.

Part of the Quiz 2 prep system · Weeks 1–8 index · Pattern atlas · Formula chains.

Week map

Sort → center (mean/median) → spread → five-number → boxplot read

Classify → Represent → Execute → Trap-check

Recognize: Ask: When prefer median over mean?
Procedure: Sort data for median. Mean uses all values. Even n: median average of two middle values.
Variations / traps: Watch for: Median without sorting.

Formula chain (compressed)

mean vs median → spread (range, IQR) → five-number summary → outlier caution.

Mean — x̄ = Σx / n — symmetric, no extreme outliers
Median — middle of sorted data — skewed / outliers
IQR — Q3 − Q1 — middle 50% spread
Five-number — min, Q1, med, Q3, max — boxplot skeleton
Variance — avg squared deviation — spread measure

Open interactive formula desk · Week 3 tab.

Deep study

Statistics I · Week 3 — Numerical summaries

Numerical data asks for center, spread, and five-number summary — then reading boxplots.

Week map

Sort data → mean vs median → quartiles Q1, Q3 → IQR → five-number summary → boxplot → outlier sensitivity.

Center notation

Mean $\bar{x}$ → sum divided by count → $\bar{x} = \frac{\sum x_i}{n}$ .
Median → middle of sorted list → resists extremes.
$n$ odd → median is middle value.
$n$ even → median average of two middle values.

Mini-data: [3, 7, 8, 12],

n=4

, sorted same. Median =

(7+8)/2 = 7.5

. Mean =

(3+7+8+12)/4 = 7.5

here coincidentally.

Spread notation

Range → max − min → sensitive to outliers.
Q1 → median of lower half (convention varies — follow problem).
Q3 → median of upper half.
IQR → $Q3 - Q1$ → middle 50% spread.
Five-number summary → min, Q1, median, Q3, max.

Boxplot read

Box from Q1 to Q3.
Line inside box at median.
Whiskers to min/max or to fences (course rule).
Outliers sometimes plotted as points beyond fences.

Mean vs median (when to use)

Situation	Prefer
symmetric, no outliers	mean
skewed or outliers	median
ordinal data	neither mean (usually)

Mini-example: salaries [30k, 32k, 35k, 200k]. Mean pulled high; median ~32.5k more typical.

Pattern families

Easy — Mean and median

Compute on small list.
Sort first for median.
Even vs odd $n$ .

Medium — Five-number summary

Split halves for Q1, Q3 with stated convention.
IQR from quartiles.
Compare two datasets via medians and IQRs.

Hard — Outlier effect

Add extreme point; track mean shift vs median.
Which measure changes more for one outlier?
Interpret boxplot: skew direction from median position in box.

Worked mini-examples

Example 1 — Median odd.

[4, 1, 9, 2, 7] → sort [1,2,4,7,9] → median 4.

Example 2 — Mean.

[10, 20, 30] → mean 20.

Example 3 — Five-number (simple convention).

[2, 4, 5, 7, 9, 11, 15] (

n=7

min 2, max 15, median 7 (middle)
Lower half [2,4,5] Q1=4; upper [9,11,15] Q3=11
IQR = 11 − 4 = 7

(If problem uses inclusive/exclusive split differently, follow their rule — note it.)

Example 4 — Outlier.

[5,6,6,7,7,8,100]. Mean ≈ 18.9, median 7 — median better center tag.

Example 5 — Boxplot skew.

Median near Q1 with long whisker to max → right skew suggested.

Traps

Median without sorting.
Q1/Q3 by wrong half split (include median in both halves vs exclude).
Mean on ordinal codes.
Range as only spread measure when outliers present.
Confusing IQR with full range.

Diagnostic (try yourself)

Find mean and median of [8, 3, 11, 3, 9].
Sorted data [1, 3, 4, 6, 8, 10, 12]. Report min, max, median, Q1, Q3, IQR (use lower/upper half excluding overall median for quartiles).
Two datasets same median 50. A has IQR 10, B has IQR 40. Which is more spread in the middle 50%?
Add outlier 500 to [10, 12, 11, 13, 12]. Which changes more: mean or median?
A boxplot shows median line near the top of the box and longer whisker below. Describe skew direction.

ChatGPT prep archive

Chatty reference archive

Archived import for extra depth — complements the notes above, not official IITM material.

Core concepts

Mean: sum/n; sensitive to outliers.
Median: middle after sort; resists outliers.
Spread: range, IQR = Q3−Q1; five-number summary: min, Q1, median, Q3, max.
Boxplot: box Q1–Q3, line at median, whiskers to extremes or fences.

Notation & vocabulary

Measure	Definition
mean	$\bar{x} = \sum x_i / n$
IQR	Q3 − Q1
median	middle of sorted list

Pattern families

Easy — Mean and median

Sort data for median. Mean uses all values. Even n: median average of two middle values.

Medium — Five-number summary

Find Q1 median of lower half, Q3 median of upper half (exclude median overall for odd n per course rule—follow problem convention).

Hard — Outlier effect

Compare mean vs median shift when extreme added. IQR often stable; range changes. State which center matches context.

Drill these on the pattern atlas — filter to week 3.

Traps

Median without sorting.
Q1/Q3 by wrong half split.
Mean of ordinal data inappropriately.
Range as only spread measure ignoring outliers.

Retrieval prompts

When prefer median over mean?
What is IQR?
Five numbers in summary?

Practice loop

Read Deep study (if present) or core concepts once.
Recite the formula chain without looking.
Open one easy pattern on the interactive atlas for week 3.
Attempt without solutions; mark studied after an honest try.
Say one trap aloud before closing the tab.