LSRL Top Ten Facts and Features

AP Statistics / Mr. Hansen
10/17/2002 [rev. 4/19/2005, 10/9/2014]

Name: ________________________

10.	The point is always on the LSRL, regardless of whether or not that point exists as a data point in the scatterplot.
9.	residuals = 0. (Recall, a residual is defined for each data point. Residual = .)
8.	Corollary of #9: In a LSRL residual plot, there must always be the same total of absolute lengths below the center line as above the center line. This is not true for other types of curve fitting (median-median, exponential, logistic, logarithmic, etc.).
7.	LSRL is not resistant to outliers. The reason is that none of r, s_x, or s_y are resistant, and the LSRL’s slope is b₁ = r (s_y/s_x). What a combination . . .
6.	Don’t extrapolate with LSRL! You can use LSRL for prediction only on the domain of known values. But note, as long as r² is reasonably strong and the linear fit is appropriate (as judged from the resid. plot), it is OK to use LSRL for prediction even if no cause-and-effect relationship exists between explanatory (x_i) and response (y_i).
5.	= b₀ + b₁x (AP notation) = a + bx (textbook and STAT CALC 8 notation)
4.	Random-looking resid. plots are desirable. Common LSRL resid. plot problems: Wedge-shaped or “flange” (not an official AP term): The s.d. of the residuals changes with x. In other words, the amount of vertical “scattering” changes noticeably for different values of x. Bowl-shaped: Try exponential fit instead. The classic example is population growth or other similar growth where the rate of growth feeds upon itself. (Or, if the situation calls for it, try a power or polynomial fit. The classic example is height (x_i) and weight (y_i), which should give a cubic power function.) Dome-shaped: Try log fit instead. S-shaped or wavy: Try logistic fit (if appropriate) or sinusoidal fit (if many waves). Clusters: Separate the clusters and try fitting each cluster separately.
3.	Resid. plot can never be linear. (If it were, we’d just tilt the LSRL to eliminate the tilt in the resid. plot!)
2.	LSRL means Least Squares Regression Line, since the LSRL minimizes (residual²), the sum of the squared residuals.* And now (drum roll, please): The #1 LSRL fact or feature!
1.	The LSRL is unique. Slope = b₁ (see #7 for formula; don’t use the long complicated formula). Intercept = b₀ (can be found if b₁ is known by applying #10 above). * By the way, this is not to be confused with ( residual)², which equals ____ . (You have enough information to figure this out.)