# UnitMath FAQ ( Precision ) Last Modified: 2/3/2000

### What is Precision? TOP

Precision is the exactness ( uncertainty ) of a measurement.

All measurements have uncertainties that range from exact to the really rough "ball park figure". For example there is exactly one item being measured below, but its length is not exact. Due to the unevenness of the object and the courseness of the ruler, the best we can say is the length is about 4.8 inches. It would be misleading to say the object has a length of 4.81950 inches. Precision is often represented in terms of significant digits.

### What are significant digits (figures)? TOP

Significant digits are the number of meaningful digits in a number. More significant digits means the value is know more precisely. For example "5." has one significant digit and implies a value is between 4.5 and 5.5; while, "4.8" has 2 significant digits and implies a value is between 4.75 and 4.85 and is more precise. 4.81950 inches has six significant digits and implies a range of 4.819495 to 4.819505 inches. The following table should clarify this.

Value Low High % { ( high - low ) / low }
5. 4.5 5.5 22.2%
4.8 4.75 5.85 2.1%
4.81950 4.819495 4.819505 2.1/10,000 %

Significant digits are very useful in estimating the precision of a calculation. The rule of thumb for multiplication and division is: the precision of the answer has no more significant digits than the smallest number of significant digits in the equation. Therefore 8.95 * 0.53 ~ 4.744 would have at best 2 significant digits in the answer so 8.95 * 0.53 ~ 4.7.

### Is precision in UnitMath just significant digits? TOP

No, significant digits are just one of many ways to describe precision in UnitMath. This makes UnitMath easier to use, and more precise than just using significant digits.

### Should I care about Precision? TOP

Yes, in fact you already do.

If you were told the cost of an item is "\$100 give or take", you would be very interested in finding out what give or take meant. Is that give or take \$1 or \$10,000? Give or take is the precision of the \$100.

### Is more precision always better? TOP

No, the acceptable precision depends on the question being asked. Increasing precision beyond what is needed to answer the question wastes time and effort and can be very expensive.

• Finding your age to the nearest second is difficult/ imposible and doesn't tell you anything useful.
• Finding your weight to the nearest ( 1/100 ounce ) requires very careful measurements, will only be valid for a few moments, and doesn't really tell you anything useful.

### Is anything exact? TOP

Yes, anything that is counted can be known exactly. For example the number of apples in a basket or people in a room. On the other hand nearly anything that is measured is not exact. For example the weight, volume, and temperature of the apples.

### What does precision look like in UnitMath? ( I'm not sure I need precision. ) TOP

Perhaps the best way to show you how to use precision in UnitMath and to perhaps convince you that precision is useful is through examples.

### Example: Gold Bar

You find a gold bar weighing 15 pounds based on an accurate scale that's precise to the nearest pound. You are then offered \$40,000 for the bar is that a good deal?

The price of gold changes, but recently has traded between \$280 and \$350 a troy ounce, so your bar's worth:

(15 pm 1/2) pounds * \$( 280 to 350 )/ounce_troy ~ ( 59,208.3 to 68,906.3 to 79,114.6 ) \$

Note: "pm" means plus or minus.

Even based on this imprecise data the \$40,000 is to low.

Now, if you are offered \$80,000 don't hesitate because that's more than the best case based on the precision of your current data. If you insist on more precision you will know your bars worth much more precisely ( within the above range ), but the increased precision will take time, during which the \$80,000 offer will likely be retracted.

An offer of \$70,000 may still be a good deal, but you would want to get more precise weight and gold exchange rates before you make the decision.

15.123 pounds * \$( 301 to 315 )/ounce_troy ~ ( 66,381. to 67,927. to 69,474. ) \$

As you can see getting better precision and losing \$10,000+ ( \$80,000 - 69,000 ) is expensive.

### Example: Driving from Washington D.C. to New York

According to the 1988 Ran McNally Road Atlas the distance from Washington D.C. to New York is 237 miles, and by definition: speed = distance / time so time = distance / speed.

Ridiculous Precision
Assume that the distance is exact, and that you drive exactly 55 mph the entire way

driveTime: 237 miles / ( 60 mph );
driveTime as hours & minutes & seconds = 3 hours + 57 minutes

The mileage may change ( detours ), and the average speed can't be exactly known before hand.

driveTime: ( 230 to 260 ) miles / ( ( 55 to 65 ) mph );
driveTime as hours ~ ( 3.5 to 4.7 ) hours

So if you give yourself 5 hours you should have no problem, but if you only give yourself 4 hours you will likely be late.

### How can I indicate precision in UnitMath? TOP

There are five ways to indicate precision in UnitMath:

1. Integers are exact
5 + 1 = 6
36 / 6 = 6

2. Decimals indicate significant digits
2. ~ 1.5 to 2 to 2.5
2.00 ~ 1.995 to 2 to 2.005

3. ± or pm indicates a range
{ pm stands for Plus or Minus and is the same as ± }
5 pm 2 ~ 3 to 5 to 7
100 * ( 1 pm 5 % ) ~ 95 to 100 to 105

4. "to" indicates a range
5 to 15 ~ 5 to 10 to 15

5. "to to" indicates a range and a best estimate. Note the best estimate doesn't have to be centered between the extremes.
5 to 6 to 15 ~ 5 to 6 to 15

### How does UnitMath calculate precision? TOP

1. Your data is converted into the form minimum, best estimate, maximum.
2. At each step of your calculation UnitMath calculates the minimum, best estimate, maximum.
3. UnitMath displays the result based on the current preferences and the final minimum, best estimate, maximum.

### Aren't significant digits good enough for precision calculations? TOP

No.

Significant digits are a crude measure of the uncertainty. Even when done correctly in simple calculations significant digits are crude. For example if we solve the simple equation 1.1 * 1.2 * 1.3 * 14. Significant digits misses the true uncertainty by nearly a factor of ten. The following table shows the different results.

Answer Low High % Error Comment
24.024 24.024 24.024 0% Assume exact numbers
24. 23.5 24.5 4.2% Assume 2 significant digits
20.38 to 28.14 20.38 28.14 38.1% Using UnitMath

The low estimate is 1.05 * 1.15 * 1.25 * 13.5 ~ 20.38 , and the high estimate is 1.15 * 1.25 * 1.35 * 14.5 ~ 28.14 . For information on making decimals exact see How can I make \$49.95 exact?

To put the above numbers into perspective, assume that you were calculating the cost of a project in millions of dollars. If you used significant digits you'd say that 25 million will cover the project with no problem. When the actual project is nearing completion you could find that you are several million short, and nobody likes cost overruns.

With UnitMath you would know that the cost could run as high as 28.2 million, and could budget appropriately, or you could get better data to reduce the uncertainty.

The following examples show that when calculations involve other than multiplication and division significant digits don't apply, or are often very misleading.

Equation Using S.D. The Correct Answer Comment
12345.2 + 0.4 1.e4 ~ 12,346 Significant digits don't apply.
2.2^5 51. ~ 45.9 to 51.5 to 57.7 Significant digits are too precise.
tan( 89.9 deg ) 573. ~ 382. to 573. to 1,146 Significant digits are too precise.
cos( 0.3 deg ) 1. ~ 1.0000 Significant digits are not precise enough.

UnitMath correctly handles precision with all operators { +, -, *, ... } and all functions { abs(), ... tanh() } in all calculations.

### How does UnitMath decide which precision form to use on output? TOP

UnitMath uses the most compact format that legitimately reflects the precision of your output, based on the preferences you have selected. UnitMath's preferences are set in the "Preferences" window; which, can be opened from UnitMath's Windows menu.

There are four preferences that effect the output:

• Show Range
• Show Exact Fractions
• Scientific Notation
• Decimal Digits
There are four formats for the output:
• significant digits
• ±
• to
• to to

### How do I find the results precision? TOP

The function precision() is used to find the precision of the result.

precision( year_Sidereal ) as seconds ~ 1.037 seconds
precision(year_Tropical) as milliseconds ~ 86.400 milliseconds
precision( density_Of_Mercury ) as gram/cc = 0.09 gram/cc
precision( meter ) = 0

Example: For a tank with sides 12.4 m, 24.5 inches, and 15.3 feet in length, find the variability of the volume.

volume: 12.4 m * 24.5 inches * 15.3 feet;
volume ~ ( 9,418. to 9,595. ) gal
precision( volume ) ~ 177.6 gal
177.6 gallon /volume as % ~ 1.9 %

The above calculations show:
volume range is ~ ( 9,418. to 9,595. ) gal
precision of the volume is ~ 177.6 gal
relative precision is ~ 1.9 %