Notes
Center and spread
- Mean $\displaystyle \bar{x} = \frac{\sum x}{n}$, median (middle of the ordered list), mode (most frequent value).
- Range $=$ max $-$ min. IQR $= Q_3 - Q_1$. Standard deviation $\displaystyle s = \sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}}$ is the typical distance from the mean.
- Adding a constant shifts the mean but not the spread. Multiplying by $k$ multiplies the mean and $s$ by $|k|$.
Outliers and box plots
- 1.5 IQR rule: fences are $Q_1 - 1.5\,\text{IQR}$ and $Q_3 + 1.5\,\text{IQR}$. Values beyond them are outliers.
- A box plot shows the five-number summary: min, $Q_1$, median, $Q_3$, max. The box length is the IQR.
Choosing and comparing
- Skew pulls the mean toward the tail. Use median and IQR for skewed data or outliers.
- Compare distributions using shape, center, spread and outliers, in context.
Topics in this unit
Each topic has its own short notes and practice.
Practice
The sum is 65 and $\displaystyle \frac{65}{5} = 13$. The median is 12.
Median 7. $Q_1 = 4$ (middle of 2, 4, 5) and $Q_3 = 10$ (middle of 8, 10, 11), so IQR $= 6$. The range is 9.
A larger standard deviation means the scores are more spread out around the mean.
IQR $= 20$, $1.5 \times 20 = 30$, and $50 + 30 = 80$.
The box runs from $Q_1$ to $Q_3$.
Multiplying by $k$ multiplies both the center and the spread by $|k|$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
Ten students recorded the minutes they spent on homework one night: 4, 6, 7, 8, 9, 10, 11, 12, 13, 20.
(a) Find the mean and the median.
Show the answer and scoring guide
Sum $= 100$, so the mean is $\displaystyle \frac{100}{10} = 10$. The middle two values are 9 and 10, so the median is $9.5$.
Check off the points you earned:
(b) Find $Q_1$, $Q_3$ and the IQR.
Show the answer and scoring guide
Lower half: 4, 6, 7, 8, 9 gives $Q_1 = 7$. Upper half: 10, 11, 12, 13, 20 gives $Q_3 = 12$. IQR $= 12 - 7 = 5$.
Check off the points you earned:
(c) Use the 1.5 IQR rule to decide whether 20 is an outlier.
Show the answer and scoring guide
$1.5 \times 5 = 7.5$. Upper fence $= 12 + 7.5 = 19.5$. Since $20 > 19.5$, 20 is an outlier. (Lower fence $7 - 7.5 = -0.5$, so there are no low outliers.)
Check off the points you earned: