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.

Week map

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.

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

Deep study

Statistics I · Week 3 — Numerical summaries

Week map

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.

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)

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

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

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)

1. Find mean and median of [8, 3, 11, 3, 9].
2. 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).
3. Two datasets same median 50. A has IQR 10, B has IQR 40. Which is more spread in the middle 50%?
4. Add outlier 500 to [10, 12, 11, 13, 12]. Which changes more: mean or median?
5. 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

Pattern families

Easy — Mean and median

Medium — Five-number summary

Hard — Outlier effect

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

1. When prefer median over mean?
2. What is IQR?
3. Five numbers in summary?

Practice loop

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

Related

• Full weeks 1–8 prep guide
• Pattern atlas note
• Formula chain desk