# Error Analysis

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1. Jun 17, 2017

### Nikhil Rajagopalan

1. The problem statement, all variables and given/known data
The internal and external diameter of a hollow cylinder are measured with the help of a vernier calipers. Their values are (3.87 ± 0.01) cm and (4.23 ± 0.01) cm respectively. The thickness of the wall of the cylinder is ?

2. Relevant equations
Thickness of cylinder wall= 1/2 (Outer Diameter - Inner Diameter)
t = 1/2 (Do - Di)

3. The attempt at a solution

for value of t,

t = 1/2 (Do - Di)
t = 1/2(4.23 - 3.87)
t = 1/2(0.36)
t = 0.18

for finding the error in t,
by differentiating on both sides,

Δt = 1/2 (ΔDo - ΔDi)
Δt = 1/2 (0.01 + 0.01)
Δt = 0.01

thickness t ± Δt = 0.18 ± 0.01

most sources show the answer as 0.18 ± 0.02. Kindly help me to figure out the mistake that i made in calculating the error.

2. Jun 17, 2017

### haruspex

Sources you regard as reliable?
Many would take a statistical approach. This allows that the two errors will often cancel out somewhat, and rarely be at opposite extremes. So they would divide the error by √2. But when you need to be sure that engineering tolerances are met, the simple method you used is appropriate.

3. Jun 19, 2017

### Jamison Lahman

The formula for finding error in the case of $t=au+bv$ where a and b are constants is:
$$\sigma_t = \sqrt{(a\sigma_u)^2+(b\sigma_v)^2}$$
since a and b both equal 1/2 and both $\sigma_u$ and $\sigma_v$ equal 0.01, this nicely simplifies to:
$$\sigma_t = \sqrt{2(1/2(0.01))^2} = \frac{0.01}{\sqrt{2}}$$
So I would agree with @haruspex

4. Jun 19, 2017

### haruspex

As I wrote, the approach should depend on the purpose. In manufacturing, the engineer sets tolerances for each component. Each machinist works to those specifications. If the resulting components don't fit the engineer is in trouble.

5. Jun 19, 2017

### Jamison Lahman

True, but if you'd like to take the statistical approach as you suggested many would in your first post, that's the rationale behind the root 2. The equation is directly out of Bevington and Robinson (2003).