# Examples of Uncertainty calculations

#### Uncertainty in a single measurement

Bob weighs himself on his bathroom scale. The smallest divisions on the scale are 1-pound marks, so the least count of the instrument is 1 pound.

Bob reads his weight as closest to the 142-pound mark. He knows his weight must be larger than 141.5 pounds (or else it would be closer to the 141-pound mark), but smaller than 142.5 pounds (or else it would be closer to the 143-pound mark). So Bob's weight must be

```          weight = 142 +/- 0.5 pounds
```
In general, the uncertainty in a single measurement from a single instrument is half the least count of the instrument.

#### Fractional and percentage uncertainty

What is the fractional uncertainty in Bob's weight?

```                               uncertainty in weight
fractional uncertainty = ------------------------
value for weight

0.5 pounds
= -------------  =  0.0035
142 pounds

```

What is the uncertainty in Bob's weight, expressed as a percentage of his weight?

```                                uncertainty in weight
percentage uncertainty =  ----------------------- * 100%
value for weight

0.5 pounds
=  ------------ * 100%  =  0.35%
142 pounds
```

#### Combining uncertainties in several quantities: adding or subtracting

When one adds or subtracts several measurements together, one simply adds together the uncertainties to find the uncertainty in the sum.

Dick and Jane are acrobats. Dick is 186 +/- 2 cm tall, and Jane is 147 +/- 3 cm tall. If Jane stands on top of Dick's head, how far is her head above the ground?

```                        combined height  =  186 cm  +  147 cm

=  333 cm

uncertainty in combined height       =    2 cm  +    3 cm

=    5 cm

combined height  =  333 cm  +/-  5 cm
```

Now, if all the quantities have roughly the same magnitude and uncertainty -- as in the example above -- the result makes perfect sense. But if one tries to add together very different quantities, one ends up with a funny-looking uncertainty. For example, suppose that Dick balances on his head a flea (ick!) instead of Jane. Using a pair of calipers, Dick measures the flea to have a height of 0.020 cm +/- 0.003 cm. If we follow the rules, we find

```                        combined height  =  186 cm  +  0.020 cm

=  186.020 cm

uncertainty in combined height       =    2 cm  +    0.003 cm

=    2.003 cm

???   combined height  =  186.020 cm  +/-  2.003 cm  ???
```

But wait a minute! This doesn't make any sense! If we can't tell exactly where the top of Dick's head is to within a couple of cm, what difference does it make if the flea is 0.020 cm or 0.021 cm tall? In technical terms, the number of significant figures required to express the sum of the two heights is far more than either measurement justifies. In plain English, the uncertainty in Dick's height swamps the uncertainty in the flea's height; in fact, it swamps the flea's own height completely. A good scientist would say

```                        combined height  =  186 cm   +/-  2 cm
```

because anything else is unjustified.

#### Combining uncertainties in several quantities: multiplying and dividing

When one multiplies or divides several measurements together, one can often determine the fractional (or percentage) uncertainty in the final result simply by adding the uncertainties in the several quantities.

Jane needs to calculate the volume of her pool, so that she knows how much water she'll need to fill it. She measures the length, width, and height:

```
length  L  =  5.56    +/-  0.14 meters
=  5.56 m  +/-  2.5%

width   W  =  3.12    +/-  0.08 meters
=  3.12 m  +/-  2.6%

depth   D  =  2.94    +/-  0.11 meters
=  2.94 m  +/-  3.7%

```

To calculate the volume, she multiplies together the length, width and depth:

```              volume =  L * W * D =  (5.56 m) * (3.12 m) * (2.94 m)

=  51.00 m^3
```

In this situation, since each measurement enters the calculation as a multiple to the first power (not squared or cubed), one can find the percentage uncertainty in the result by adding together the percentage uncertainties in each individual measurement:

```
percentage uncertainty in volume =   (percentage uncertainty in L) +
(percentage uncertainty in W) +
(percentage uncertainty in D)

=  2.5% + 2.6% + 3.7%

=  8.8%
```

Therefore, the uncertainty in the volume (expressed in cubic meters, rather than a percentage) is

```
uncertainty in volume  = (volume) * (percentage uncertainty in volume)

= (51.00 m^3) * (8.8%)

= 4.49 m^3
```

Therefore,

```       volume  =  51.00    +/-  4.49  m^3
=  51.00 m  +/-  8.8%
```

If one quantity appears in a calculation raised to a power p, it's the same as multiplying the quantity p times; one can use the same rule, like so:

Fred's pool is a perfect cube. He measures the length of one side to be

```           length  L  =  8.03    +/-  0.25 meters
=  8.03 m  +/-  3.1%
```

The volume of Fred's cubical pool is simply

```                         3
volume =  L

volume =  L * L * L =  (8.03 m) * (8.03 m) * (8.03 m)

=  517.8 m^3
```
Just as before, one can calculate the uncertainty in the volume by adding the percentage uncertainties in each quantity:
```
percentage uncertainty in volume =   (percentage uncertainty in L) +
(percentage uncertainty in L) +
(percentage uncertainty in L)

=  3.1% + 3.1% + 3.1%

=  9.3%
```

But another way to write this is using the power p = 3 times the uncertainty in the length:

```
percentage uncertainty in volume =  3 * (percentage uncertainty in L)

=  3 * 3.1%

=  9.3%
```

When the power is not an integer, you must use this technique of multiplying the percentage uncertainty in a quantity by the power to which it is raised. If the power is negative, discard the negative sign for uncertainty calculations only.

#### Is one result consistent with another?

Jane's measurements of her pool's volume yield the result

```       volume  =  51.00    +/-  4.49  m^3
```
When she asks her neighbor to guess the volume, he replies "54 cubic meters." Are the two estimates consistent with each other?

In order for two values to be consistent within the uncertainties, one should lie within the range of the other. Jane's measurements yield a range

```            51.00 - 4.49  m^3   <  volume  <  51.00 + 4.49  m^3

46.51    m^3   <  volume  <      55.49     m^3
```

The neighbor's value of 54 cubic meters lies within this range, so Jane's estimate and her neighbor's are consistent within the estimated uncertainty.

#### What if there are several measurements of the same quantity?

Joe is making banana cream pie. The recipe calls for exactly 16 ounces of mashed banana. Joe mashes three bananas, then puts the bowl of pulp onto a scale. After subtracting the weight of the bowl, he finds a value of 15.5 ounces.

Not satisified with this answer, he makes several more measurements, removing the bowl from the scale and replacing it between each measurement. Strangely enough, the values he reads from the scale are slightly different each time:

```              15.5, 16.4, 16.1, 15.9, 16.6  ounces
```

Joe can calculate the average weight of the bananas:

```                        15.5 + 16.4 + 16.1 + 15.9 + 16.6   ounces
average = -------------------------------------------
5

=   80.4 ounces / 5  =  16.08 ounces
```

Now, Joe wants to know just how flaky his scale is. There are two ways he can describe the scatter in his measurements.

• The mean deviation from the mean is the sum of the absolute values of the differences between each measurement and the average, divided by the number of measurements:
```                                   0.5 + 0.4 + 0.1 + 0.1 + 0.6   ounces
mean dev from mean = --------------------------------------
5

=  1.6 ounces / 5  =  0.32 ounces
```

• The standard deviation from the mean is the square root of the sum of the squares of the differences between each measurement and the average, divided by one less than the number of measurements:
```                            [  (0.5)^2 + (0.4)^2 + (0.1)^2 + (0.1)^2 + 0.6)^2 ]
stdev from mean = sqrt [ ----------------------------------------------- ]
[                    5 - 1                        ]

[   0.79 ounces^2  ]
= sqrt [  --------------  ]
[        4         ]

= 0.44 ounces
```

Either the mean deviation from the mean, or the standard deviation from the mean, gives a reasonable description of the scatter of data around its mean value.

Can Joe use his mashed banana to make the pie? Well, based on his measurements, he estimates that the true weight of his bowlful is (using mean deviation from the mean)

```            16.08 - 0.32 ounces  < true weight <  16.08 + 0.32 ounces

15.76 ounces     < true weight <      16.40  ounces
```
The recipe's requirement of 16.0 ounces falls within this range, so Joe is justified in using his bowlful to make the recipe.

#### How can one estimate the uncertainty of a slope on a graph?

If one has more than a few points on a graph, one should calculate the uncertainty in the slope as follows. In the picture below, the data points are shown by small, filled, black circles; each datum has error bars to indicate the uncertainty in each measurement. It appears that current is measured to +/- 2.5 milliamps, and voltage to about +/- 0.1 volts. The hollow triangles represent points used to calculate slopes. Notice how I picked points near the ends of the lines to calculate the slopes! 1. Draw the "best" line through all the points, taking into account the error bars. Measure the slope of this line.

2. Draw the "min" line -- the one with as small a slope as you think reasonable (taking into account error bars), while still doing a fair job of representing all the data. Measure the slope of this line.

3. Draw the "max" line -- the one with as large a slope as you think reasonable (taking into account error bars), while still doing a fair job of representing all the data. Measure the slope of this line.

4. Calculate the uncertainty in the slope as one-half of the difference between max and min slopes.

In the example above, I find

```
147 mA - 107 mA             mA
"best" slope   =   ------------------  =  7.27 ----
10 V  - 4.5 V               V

145 mA - 115 mA             mA
"min"  slope   =   ------------------  =  5.45 ----
10.5 V - 5.0 V               V

152 mA - 106 mA             mA
"max"  slope   =   ------------------  =  9.20 ----
10 V  - 5.0 V               V

mA
Uncertainty in slope is  0.5 * (9.20 - 5.45) =  1.875  ----
V
```

There are at most two significant digits in the slope, based on the uncertainty. So, I would say the graph shows

```
mA
slope  =   7.3  +/-  1.9   ----
V
```