Error Propagation in Arithmetic Calculations

[Pages:1]Error Propagation in Arithmetic Calculations

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Type of Calculation Addition or Subtraction

Example* x pqr

Standard Deviation of x

sx

s

2 p

sq2

sr2

Multiplication or Division

x

pq r

sx x

sp p

2

sq q

2

sr r

2

Exponentiation

x py

sx

y

x

sp p

Logarithm

x log10 p

sx

0.434

sp p

Natural Logarithm

x ln p

sx

sp p

Antilogarithm

x anti log10 p

sx 2.303 x s p

Natural Antilogarithm

x ep

sx x sp

* p, q, and r are experimental variables whose standard deviations are sp, sq, and sr, respectively: y is a constant.

Least Squares Error Analysis

With a set of data points, the error for a linear fit can be found following these equations: (x and y are the data points and N is the number of data points)

y m x b : line equation

N x2 x2 :

b x2 y x xy

: y-intercept

m N xy x y

: slope

y

1N N 2 i1

yi b mxi

2

: standard deviation of the y's

x2

b y

: standard deviation of the y-intercept

m y

N

: standard deviation of the slope

y m m x b b : line equation with error

This system of equations is what Microsoft Excel uses in its LINEST function.

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