# Ultra-condensed description of propagation of errors

Suppose that you measure two quantities, x +/- dx cm and y +/- dy cm. What happens to their uncertainties if you need to combine these values?

```

total length  =  (x + y)    cm

uncertainty in total length  =  (dx + dy)  cm

```

• If you subtract the quantities, again, their uncertainties ADD (not subtract).
```

difference in  length  =  (x - y)    cm

uncertainty in difference    =  (dx + dy)  cm

```

• If you multiply the quantities, you must add their fractional (or percentage) uncertainties to find the fractional (or percentage) uncertainty in the product.
```

area of rectangle    =  (x * y)                 square cm

uncertainty in area              dx   dy
-------------------          = ( -- + -- )              (pure fraction)
area                        x    y

dx   dy
uncertainty in area          = ( -- + -- ) * ( area )   square cm
x    y
```

• If you divide the quantities, you must again add their fractional (or percentage) uncertainties to find the fractional (or percentage) uncertainty in the ratio.
```
x
ratio of length to width  =    ---                   (pure fraction)
y

uncertainty in ratio             dx   dy
-------------------          = ( -- + -- )              (pure fraction)
ratio                       x    y

dx   dy
uncertainty in ratio         = ( -- + -- ) * ( ratio )  (pure fraction)
x    y
```

• If you raise a value to a power N, you multiply its fractional (or percentage) uncertainty by N to find the fractional (or percentage) uncertainty in the result.
```
3                              3
length cubed           =   x                             cm

uncertainty in length cubed        ( dx )
----------------------------  =    (----) * 3            (pure fraction)
length cubed                ( x  )

dx                            3
uncertainty in length cubed   =    (----) * 3 * (length cubed)   cm
x
```