Notes
One variable
Describe center (mean or median), spread (range or IQR) and shape (symmetric or skewed). Use the median and IQR when there are outliers or skew, because they barely move.
Displays
Dot plots show each value. Histograms show counts in equal bins. Box plots show the five-number summary: minimum, $Q_1$, median, $Q_3$, maximum.
Two categorical variables
Two-way tables show joint and marginal frequencies. Compare conditional relative frequencies to look for an association.
Two numerical variables
Scatter plots show correlation (positive, negative, none); $r$ near $\pm 1$ is strong. A line of fit $\hat{y} = mx + b$ predicts values. Residual = actual $-$ predicted. Correlation is not causation, and extrapolating far beyond the data is risky.
Topics in this unit
Each topic has its own short notes and practice.
Practice
The outlier pulls the mean much more than the median.
In order: 14, 18, 21, 25, 30.
$65 - 40 = 25$.
Positive residual: actual is above predicted.
Strong negative correlation.
$\displaystyle \frac{45}{75} = 0.6$ and $\displaystyle \frac{20}{50} = 0.4$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A teacher records hours of sleep $x$ and test scores $y$ for 8 students. The line of best fit is $\hat{y} = 6x + 38$, with $r = 0.82$.
(a) Interpret the slope and the $y$-intercept in context. Is the intercept meaningful?
Show the answer and scoring guide
Slope: each extra hour of sleep is linked to about 6 more points. Intercept: a student with 0 hours of sleep is predicted to score 38, which isn't meaningful because no one in the data slept 0 hours.
Check off the points you earned:
(b) A student slept 7 hours and scored 77. Find the residual.
Show the answer and scoring guide
Predicted: $6(7) + 38 = 80$. Residual: $77 - 80 = -3$.
Check off the points you earned:
(c) Does this data prove that more sleep causes higher scores? Explain.
Show the answer and scoring guide
No. $r = 0.82$ shows a strong positive association, but it's an observational study, so other factors (like study habits) could explain both. Only a controlled experiment could show cause.
Check off the points you earned: