Classic Graphing 84 manual · Chapter 14 of 25

Regression: fitting curves to data

Fit lines, polynomials, exponentials, powers, logarithms, logistic and sine curves to data, store the equation in Y= and read r and R².

14.1The regression commands#

Regression finds the curve of a chosen type that fits the data best. All the commands are in STAT, CALC. They need an Xlist and a Ylist of equal length (L1 and L2 by default), an optional frequency list and an optional place to store the equation.

The CALC tab of the STAT menu. The display shows: EDIT [CALC] TESTS / 1:1-Var Stats / 2:2-Var Stats / 3:Med-Med / 4:LinReg(ax+b) / 5:QuadReg / 6:CubicReg / 7↓QuartReg. — The CALC tab of the STAT menu.

Item	Command	Fits
3	Med-Med	$y=ax+b$ through medians (resists outliers)
4	LinReg(ax+b)	$y=ax+b$
5	QuadReg	$y=ax^2+bx+c$
6	CubicReg	$y=ax^3+bx^2+cx+d$
7	QuartReg	$y=ax^4+bx^3+cx^2+dx+e$
8	LinReg(a+bx)	$y=a+bx$
9	LnReg	$y=a+b\ln x$
0	ExpReg	$y=a\,b^{x}$
A	PwrReg	$y=a\,x^{b}$
B	Logistic	$y=\dfrac{c}{1+a\,e^{-bx}}$
C	SinReg	$y=a\sin(bx+c)+d$

14.2Fit a straight line#

a=\frac{n\sum xy-\sum x\sum y}{n\sum x^2-(\sum x)^2},\qquad b=\bar{y}-a\bar{x}

Least-squares slope and intercept.

The LinReg(ax+b) form: lists, frequency and where to store the equation. The display shows: LinReg(ax+b) / Xlist:L1 / Ylist:L2 / FreqList: / Store RegEQ: / Calculate. — The LinReg(ax+b) form: lists, frequency and where to store the equation.

Example 14.1

A line through exact data

y=2x+1

L1 = 1, 2, 3, 4 and L2 = 3, 5, 7, 9.
stat11enter2enter3enter4enter▶3enter5enter7enter9enter
Open STAT, CALC, 4 (LinReg(ax+b)), move down four times to Calculate and press enter.
stat▶4▼▼▼▼enter

Result of the example "A line through exact data" The display shows: LinReg / y=ax+b / a=2 / b=1 / r²=1 / r=1. — You should see`y=ax+b a=2 b=1 r²=1 r=1`

Example 14.2

A line through scattered data

L1 = 1 to 5, L2 = 2, 4, 5, 4, 5.
stat11enter2enter3enter4enter5enter▶2enter4enter5enter4enter5enter
Run LinReg(ax+b).
stat▶4▼▼▼▼enter

r is the correlation coefficient: close to 1 or −1 means a strong straight-line pattern, close to 0 means none. r² is the share of the variation the line explains. These lines appear when STAT DIAGNOSTICS is ON in MODE.

Example 14.3

The other form y = a + bx

Use the same L1 and L2 as above (exact data), and choose item 8.
stat11enter2enter3enter4enter▶3enter5enter7enter9enterstat▶8▼▼▼▼enter

Example 14.4

A resistant line (Med-Med)

L1 = 1 to 6, L2 = 2, 4, 6, 8, 10, 12, then item 3.
stat11enter2enter3enter4enter5enter6enter▶2enter4enter6enter8enter10enter12enterstat▶3▼▼▼▼enter

14.3Store the regression equation in Y=#

The Store RegEQ field puts the equation into a Y= slot so you can graph it. Move down to that field, press vars, ▶, 1, 1 to paste Y1, then move down and run the command. Even without storing, the last equation is available as RegEQ in vars, 5 (Statistics…), EQ tab.

Example 14.5

Put the line into Y1

Enter the data L1 = 1…4 and L2 = 3, 5, 7, 9. Open LinReg(ax+b) and move down to Store RegEQ.
stat11enter2enter3enter4enter▶3enter5enter7enter9enterstat▶4▼▼▼
Paste Y1 with vars, right, 1, 1, then down and enter to calculate.
vars▶11▼enter
Open Y=.
y=

Example 14.6

Predict with RegEQ

Run LinReg(ax+b) on the same data and leave with clear.
stat11enter2enter3enter4enter▶3enter5enter7enter9enterstat▶4▼▼▼▼enterclear
Store 10 in X. Press vars, 5, right, right, 1 (RegEQ) and enter.
10sto→X,T,θ,nentervars5▶▶1enter

14.4Fit a parabola, cubic, exponential, power or logarithm#

QuadReg results. The display shows: QuadReg / y=ax²+bx+c / a=1 / b=0 / c=1 / R²=1. — QuadReg results.

Example 14.7

Quadratic regression

y=x^2+1

L1 = 0, 1, 2, 3 and L2 = 1, 2, 5, 10, then CALC item 5.
stat10enter1enter2enter3enter▶1enter2enter5enter10enterstat▶5▼▼▼▼enter

Result of the example "Quadratic regression" The display shows: QuadReg / y=ax²+bx+c / a=1 / b=0 / c=1 / R²=1. — You should see`y=ax²+bx+c a=1 b=0 c=1 R²=1`

Example 14.8

Cubic regression

y=x^3

L1 = 0 to 4 and L2 = 0, 1, 8, 27, 64, then item 6.
stat10enter1enter2enter3enter4enter▶0enter1enter8enter27enter64enterstat▶6▼▼▼▼enter

Result of the example "Cubic regression" The display shows: CubicReg / y=ax³+bx²+cx+d / a=1 / b=0 / c=0 / d=0 / R²=1. — You should see`a=1 b=0 c=0 d=0`

Example 14.9

Exponential regression

y=3\cdot2^{x}

L1 = 0 to 3 and L2 = 3, 6, 12, 24, then item 0.
stat10enter1enter2enter3enter▶3enter6enter12enter24enterstat▶0▼▼▼▼enter

Result of the example "Exponential regression" The display shows: ExpReg / y=a*b^x / a=3 / b=2 / r²=1 / r=1. — You should see`y=a*b^x a=3 b=2`

Example 14.10

Power regression

y=2x^{2}

L1 = 1 to 4 and L2 = 2, 8, 18, 32, then alpha math (item A).
stat11enter2enter3enter4enter▶2enter8enter18enter32enterstat▶alphaAmath▼▼▼▼enter

Result of the example "Power regression" The display shows: PwrReg / y=a*x^b / a=2 / b=2 / r²=1 / r=1. — You should see`y=a*x^b a=2 b=2`

Example 14.11

Logarithmic regression

y=1+2.885\ln x

L1 = 1, 2, 4, 8 and L2 = 1, 3, 5, 7, then item 9.
stat11enter2enter4enter8enter▶1enter3enter5enter7enterstat▶9▼▼▼▼enter

Result of the example "Logarithmic regression" The display shows: LnReg / y=a+blnx / a=1 / b=2.885390082 / r²=1 / r=1. — You should see`y=a+blnx a=1 b=2.885390082`

Example 14.12

Lists of unequal length

L1 has three values but L2 only two. The regression refuses.
stat11enter2enter3enter▶1enter2enterstat▶4▼▼▼▼enter

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