Contents

# Uncertainty on the slope and they-intercept

### 1. Maximum values for two points

From the graph, we consider the highest point 2(x2,y2) and the lowest point 1(x1,y1), and the rectangle (2Δx) (2Δy ) around each point.

Therefore:

y2 max = y2 + Δy and y2 min = y2 - Δy
y1 max = y1 + Δy and y1 min = y1 - Δy

x2 max = x2 + Δx and x2 min = x2 - Δx
x1 max = x1 + Δx and x1 min = x1 - Δx

### 2. The expression of the slope: m

mmax = [(y2 + Δy) - (y1 - Δy)]/[(x2 - Δx) - (x1 + Δx) ]

mmax = [(y2 + Δy) - (y1 - Δy)]/[(x2 - Δx) - (x1 + Δx) ]

Here:
Δx = xc because we have constraint in the point C.

Similarly,

mmin = [(y2 - Δy) - (y1 + Δy)]/[(x2 + Δx) - (x1 - Δx) ]

mmin = [(y2 - Δy) - (y1 + Δy)]/[(x2 + Δx) - (x1 - Δx)]

Now, we determine Mm nad Δm:

Mm = (mmax + mmin)/2 = [(y2 - y1)(x2 - x1) + (4Δx Δy)]/[(x2 - x1)2 - 4 Δx2]
And
Δm = (mmax - mmin)/2 = [2(x2 - x1)Δy + 2(y2 - y1)Δx ]/[(x2 - x1)2 - 4 Δx2]

mmin = [(y2 - Δy) - (y1 + Δy)]/[(x2 + Δx) - (x1 - Δx)]
mmin = [(y2 - Δy) - (y1 + Δy)]/[(x2 + Δx) - (x1 - Δx)]
Mm = (mmax + mmin)/2
Δm = (mmax - mmin)/2

### 3. The expression of the y-intercept: b

2.1. Using the slope m max:

[(y2 + Δy) - bmin ]/[(x2 - Δx) - 0] = slope = mmax

That is:
[(y2 + Δy) - bmin ] = (x2 - Δx) mmax, or
bmin = (y2 + Δy) - (x2 - Δx) mmax

Similarly,

2.2. Using the slope m min:

[(y2 - Δy) - bmax ]/[(x2 + Δx) - 0] = slope = mmin

That is:

[(y2 - Δy) - bmax ] = (x2 + Δx) mmin, or
bmax = (y2 - Δy) - (x2 + Δx) mmin

bmin = (y2 + Δy) - (x2 - Δx) mmax
bmax = (y2 - Δy) - (x2 + Δx) mmin

Now, we determine Mb and Δb:

Mb = (bmax + bmin)/2 = y2 - x2 (Mm) + Δx Δm ,
Δb = (bmax - bmin)/2 = - Δy + x2Δm - Δx (Mm)

Mb = (bmax + bmin)/2 = y2 - x2 (Mm) + Δx Δm
Δb = (bmax - bmin)/2 = - Δy + x2Δm - Δx (Mm)

The y-intercept is written as:

b = Mb ± Δb

### 4. The equqtion y = (slope) x + Y-interecept

Using the above results, the equation of the related line is:
y = b + m t = (Mb ± Δb) + (Mm ± Δm)t

y = b + m t = (Mb ± Δb) + (Mm ± Δm)t

Get your related calculations here: Uncertainty calculator .

 chimie labs | Physics and Measurements | Probability & Statistics | Combinatorics - Probability | Chimie | Optics | contact |