# Workshop 1X: Propagation of errors, and some practice with graphs

See Chapter 1 of your textbook

You know what to do if you have several measurements of the same quantity -- say, the distance travelled by a cart over a 1-second interval. You can calculate the mean value and the standard deviation from the mean. Fine.

But what if you then use that result to compute a further quantity? You know that

```
displacement
average velocity   =     ---------------
time interval
```

but what if the displacement has a known uncertainty? That must lead to some uncertainty in the computed velocity -- but how?

It turns out that this subject is very important in many branches of experimental sciences, including physics. You will be using quantities with uncertainties to compute results over and over and OVER in this course, and in the next course, and in almost all your physics courses. The topic of propagation of errors is a big one. I'm not going to spend a great deal of time explaining the theory behind the formulae we'll use; if you're interested, you should take a good mathematics course.

Professor Lindberg has written a very nice guide to this subject:

As you will see if you read his guide, there are at least two ways to deal with uncertainties as you go through some set of calculations: the simple approach and the sophisticated approach using standard deviations. The two methods will often yield similar results, so in this case, we're going to stick with the simple method. You should be aware, however, that later on in your physics career, you may need to switch to the more sophisticated approach.

#### Ultra-condensed description of propagation of errors

Remember, the methods described here are the simplified versions of what are really more complicated expressions.

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
```

#### Using measurements of position and time to derive acceleration

I placed a cart onto a tilted track and let it roll for a set of different time intervals. At each time interval, I made three trials. My results are shown in the table below, which you can download as an ASCII text file if you wish.

The goal is to use these measurements to determine the average acceleration of the cart, together with the uncertainty in that value.

```
#
#  Time (seconds)              Distance rolled (cm)
#                        1       2       3        avg      uncert
#-------------------------------------------------------------------

1.0 +/- 0.005      25.0    25.5    24.9        25.1 +/-  0.3

2.0 +/- 0.005     100.9   101.3   101.1       101.1 +/-  0.2

3.0 +/- 0.005     230.1   230.0   230.1       230.2 +/-  0.3

4.0 +/- 0.005     412.1   413.2   411.8       412.3 +/-  0.7

#-------------------------------------------------------------------

```

1. There are three time intervals in this table: 1.0 - 2.0 seconds, 2.0 - 3.0 seconds, and so on. Compute the average velocity of the cart during each time interval, and the uncertainty in that velocity. Make a table showing your three average velocities.

You should find that the velocity increases with time.

2. Make a graph showing the average velocity as a function of time. (Hint: use limits of 50 - 200 cm/s on the vertical axis, and 1.0 to 4.0 seconds on the horizontal axis). Use your graph to determine the average accleration of the cart, together with the uncertainty in that acceleration.
3. (if you have time) How precise are these measurements? One way to answer is to calculate the percentage uncertainty in a typical measurement. For example, the third measurement of position is 230.2 +/- 0.3 cm. The percentage uncertainty in that value is
```

0.3 cm
--------- *  (100 %)   =  0.1 percent
230.2 cm

```
• What is the typical percentage uncertainty in one value of position?
• What is the typical percentage uncertainty in one value of average velocity?
• What is the percentage uncertainty in the value of average acceleration?

You should see a trend here. Can you explain it?