6. Sensitivity analysis
Sometimes chain calculations can be so long or complex that it’s hard to track sig fig through them. This is especially challenging when using software to determine results from combined measurements.
One way to get a feel for the accuracy of the result is to vary one or more of the measurements and see their effect on the result, that is, how sensitive the answer is to variation of the input.
Example 1
If rainfall intensity is 1.4 in/hr, what is the volume of water over a 2.3 acre field during a 20 minute storm? Compute in cubic yards to the correct number of sig fig.
Set up as one long computation making sure the units cancel correctly:
Figure out the sig fig in each of the numbers used:
1.4 in/hr 2 sig fig
20 min 1 sig fig ← least number of sig fig
2.3 acres 2 sig fig
all conversion factors are exact so each has infinite number of sig fig.
Since the least number of sig fig for any of the numbers is 1, the answer would also have just 1:
Vol = 100 yd^{3}
If time is good to the nearest 10 minutes, then the volume is only good to the nearest 100 cubic yards.
Let's check this using a sensitivity analysis
20 min has one sig fig, that means that to the nearest 10 minutes the time was 20. In reality the time could have ranged between ~15 and ~25 minutes and still be 20 to the nearest 10 minutes. Notice that the range represents ±half the uncertainty:
±(half 10 minutes) = ±5 minutes
Numbers falling outside this range would not have been recorded as 20:
If real time were 14 minutes, it would have been recorded as 10.
If real time were 26 minutes, it would have been recorded as 30.
So varying time: 20 min
uncertainty is ±5 min
range is 15 to 25 min
time (min) | raw vol (yd^{3}) | raw vol is computed with extra dgiits |
15 | 108.2778 | |
20 | 144.3037 | range is ~72 yd^{3} |
25 | 180.3796 |
Notice at what level the volume is affected by varying the time within the uncertainty range. It’s constant at 100 yd^{3} level but not at the 10 yd^{3} level - the change is >50 yd^{3}. This means you can’t reliably state the volume to the nearest 10 or 1 yd^{3} if time is measured only to the nearest 10 minutes.
What if we instead vary the rainfall intensity or the field area?
Rainfall intensity: 1.4 in/hr
uncertainty is ±0.05
range is 1.35 to 1.45 in/hr
Intensity (in/hr) | raw vol (yd^{3}) | |
1.35 | 139.1500 | |
1.4 | 144.3037 | range is ~10 yd^{3} |
1.45 | 149.4574 |
Area: 2.3 acres
uncertainty is ±0.05 acres
range is 2.25 to 2.35 acres
time (min) | raw vol (yd^{3}) | |
2.25 | 141.1667 | |
2.3 | 144.3037 | range is ~6 yd^{3} |
2.35 | 147.4407 |
Notice that the time uncertainty affects the final volume much more than the uncertainties in intensity or area. That means is if we need a more accurate volume, we should first increase time accuracy.
Example 2
Use a sensitivity analysis to convert 159°10'13" to decimal degrees with the correct number of sig fig.
Using our calculator DMS to Deg conversion button we get 159.17027778°
Since the estimated digit on the angle is in the seconds position, add and subtract 0.5" from the angle and compare those conversions.
DMS | Deg | |
subtract 0.5" | 159°10'12.5" | 159.17013889 |
original angle | 159°10'13" | 159.17027778 |
add 0.5" | 159°10'13.5" | 159.17041667 |
The three converted angles are consistent through the first three decimal places; they vary in the fourth. Within the range of uncertainty of the smallest unit, we would state the decimal equivalent of the angle is 159.170° to 6 sig fig. Notice that this is the same as we determined in the previous section.