stats1-week3

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.