UNIT 5 • CURVE FITTING & STATISTICS

Fitting a Straight Line
Linear Least Squares Regression

Linear Least Squares Regression finds the best-fit line $y = a + bx$ by minimizing the sum of squared vertical distance errors (residuals) between raw sample data points and the regression line!

LINEAR SYSTEM OF NORMAL EQUATIONSMINIMUM RESIDUAL SQUARES

y = a + b x

\begin{aligned} \sum y &= n a + b \sum x \\ \sum x y &= a \sum x + b \sum x^2 \end{aligned}

1. Intercept (a)The vertical y-axis baseline elevation:

a = \bar{y} - b \bar{x}

2. Slope (b)Rate of change coefficient:

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

3. Normal DeterminantsDirect 2×2 linear system solver using sample sums.

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Fit a straight line $y = a + bx$ to the 5 observation pairs:

x	1	2	3	4	5
y	2	3	5	4	6

STEP 01 • TABULATE STATISTICAL SUMS

x	y	x²	x · y
1	2	1	2
2	3	4	6
3	5	9	15
4	4	16	16
5	6	25	30
Sum = 15	Sum = 20	Sum = 55	Sum = 69

STEP 02 • SOLVE NORMAL EQUATIONS

Plug sums into system with $n = 5$ :

5 a + 15 b = 20

15 a + 55 b = 69

b = \frac{5(69) - 15(20)}{5(55) - (15)^2} = \frac{345 - 300}{275 - 225} = \frac{45}{50} = 0.9

a = \frac{20 - 15(0.9)}{5} = \frac{20 - 13.5}{5} = \frac{6.5}{5} = 1.3

FINAL RESULT

Linear Regression Line: y = 1.3 + 0.9 x

\boxed{y = 1.3 + 0.9 x}

The straight line of best fit calculated by linear least squares is y = 1.3 + 0.9x!

LABORATORY

Interactive Live Calculator & Visualizer

CURVE FITTING LABORATORY

Fitting Straight Lines (Linear Regression)

PreviousRunge-Kutta Method

Unit 5•22 of 28

NextFitting Parabola

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 5 • CURVE FITTING & STATISTICS

Fitting a Straight Line
Linear Least Squares Regression

Linear Least Squares Regression finds the best-fit line $y = a + bx$ by minimizing the sum of squared vertical distance errors (residuals) between raw sample data points and the regression line!

LINEAR SYSTEM OF NORMAL EQUATIONSMINIMUM RESIDUAL SQUARES

y = a + b x

\begin{aligned} \sum y &= n a + b \sum x \\ \sum x y &= a \sum x + b \sum x^2 \end{aligned}

1. Intercept (a)The vertical y-axis baseline elevation:

a = \bar{y} - b \bar{x}

2. Slope (b)Rate of change coefficient:

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

3. Normal DeterminantsDirect 2×2 linear system solver using sample sums.

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Fit a straight line $y = a + bx$ to the 5 observation pairs:

x	1	2	3	4	5
y	2	3	5	4	6

STEP 01 • TABULATE STATISTICAL SUMS

x	y	x²	x · y
1	2	1	2
2	3	4	6
3	5	9	15
4	4	16	16
5	6	25	30
Sum = 15	Sum = 20	Sum = 55	Sum = 69

STEP 02 • SOLVE NORMAL EQUATIONS

Plug sums into system with $n = 5$ :

5 a + 15 b = 20

15 a + 55 b = 69

b = \frac{5(69) - 15(20)}{5(55) - (15)^2} = \frac{345 - 300}{275 - 225} = \frac{45}{50} = 0.9

a = \frac{20 - 15(0.9)}{5} = \frac{20 - 13.5}{5} = \frac{6.5}{5} = 1.3

FINAL RESULT

Linear Regression Line: y = 1.3 + 0.9 x

\boxed{y = 1.3 + 0.9 x}

The straight line of best fit calculated by linear least squares is y = 1.3 + 0.9x!

LABORATORY

Interactive Live Calculator & Visualizer

CURVE FITTING LABORATORY

Fitting Straight Lines (Linear Regression)

PreviousRunge-Kutta Method

Unit 5•22 of 28

NextFitting Parabola

Fitting a Straight Line Linear Least Squares Regression

Step-by-Step Worked Example

Linear Regression Line: y = 1.3 + 0.9 x

Interactive Live Calculator & Visualizer

Fitting Straight Lines (Linear Regression)

NETZ Engine

Fitting a Straight Line Linear Least Squares Regression

Step-by-Step Worked Example

Linear Regression Line: y = 1.3 + 0.9 x

Interactive Live Calculator & Visualizer

Fitting Straight Lines (Linear Regression)

Fitting a Straight Line
Linear Least Squares Regression

Fitting a Straight Line
Linear Least Squares Regression