Module 6: Precision and GPE

Precision

The precision of a number is the place value of the rightmost significant figure. For example,

100.45

is precise to the hundredths place, and 3,840 is precise to the tens place.

Exercises

Identify the precision (i.e., the place value of the rightmost significant figure) of each number.

1.

63,400

2.

63,040

3.

63,004

4.

8,000

5.

8,\overline{0}00

6.

8,0\overline{0}0

7.

8,00\overline{0}

8.

0.085

9.

0.0805

10.

0.08050

Precision-Based Rounding

In a previous module about decimals, we used precision-based rounding because we were rounding to a specified place value; for example, rounding to the nearest tenth. Let’s practice this with overbars and trailing zeros.

Precision-based rounding:

  1. Locate the rounding digit in the place to which you are rounding.

  2. Look at the test digit directly to the right of the rounding digit.

  3. If the test digit is 5 or greater, increase the rounding digit by 1 and drop all digits to its right. If the test digit is less than 5, keep the rounding digit the same and drop all digits to its right.

Remember, when the rounding digit of a whole number is a

9

that gets rounded up to a 0, we must write an overbar above that 0.

Also, when the rounding digit of a decimal number is a

9

that gets rounded up to a 0, we must include the 0 in that decimal place.

Exercises

Round each number to the indicated place value. Be sure to include trailing zeros or an overbar if necessary.

11. 

13,997

(thousands)

12. 

13,997

(hundreds)

13. 

13,997

(tens)

14. 

0.5996

(tenths)

15. 

0.5996

(hundredths)

16. 

0.5996

(thousandths)

Precision when Adding and Subtracting

Suppose the attendance at a large event is estimated at

25,000

people, but then you see 3 people leave. Is the new estimate 24,997? No, because the original estimate was precise only to the nearest thousand. We can’t start with an imprecise number and finish with a more precise number. If we estimated that 1,000 people had left, then we could revise our attendance estimate to 24,000 because this estimate maintains the same level of precision as our original estimate.

When adding or subtracting numbers with different levels of precision, the answer must be rounded to the same precision as the least precise of the original numbers.

Don’t round off the original numbers; do the necessary calculations first, then round the answer as your last step.

Exercises

Add or subtract as indicated. Round to the appropriate level of precision.

17. Find the combined weight of four packages with the following weights:

9.7

lb, 13 lb, 10.5 lb, 6.1 lb.

18. Find the combined weight of four packages with the following weights:

9.7

lb, 13.0 lb, 10.5 lb, 6.1 lb.

19. While purchasing renter’s insurance, Chandra estimates the value of her insurable possessions at $

10,200

. After selling some items valued at $ 375, what would be the revised estimate?

20. Chandra knows that she has roughly $

840

in her checking account. After using her debit card to make two purchases of $ 25.95 and $ 16.38, how much would she have left in her account?

If you are multiplying by an exact number, you can consider this a repeated addition. For example, suppose you measure the weight of an object to be 4.37 ounces and you want to know the weight of three of these objects; multiplying 4.37 times 3 is the same as adding 4.37 + 4.37 + 4.37 = 13.11 ounces. The precision is still to the hundredths place. The issue of significant figures doesn’t apply to exact numbers, so it would be wrong to treat 3 as having only one sig fig. (Treat exact numbers like royalty; their precision is perfect and it would be an insult to even question it.)

Greatest Possible Measurement Error (GPE)

Suppose you are weighing a dog with a scale that displays the weight rounded to the nearest pound. If the scale says Sir Barks-A-Lot weighs

23

pounds, he could weigh anywhere from 22.5 pounds to almost 23.5 pounds. The true weight could be as much as 0.5 pounds above or below the measured weight, which we could write as 23\pm0.5.

Now suppose you are weighing Sir Barks-A-Lot with a scale that displays the weight rounded to the nearest tenth of a pound. If the scale says Sir Barks-A-Lot weighs

23.0

pounds, we now know that he could weigh anywhere from 22.95 pounds to almost 23.05 pounds. The true weight could be as much as 0.05 pounds above or below the measured weight, which we could write as 23.0\pm0.05.

As we increase the level of precision in our measurement, we decrease the greatest possible measurement error or GPE. The GPE is always one half the precision; if the precision is to the nearest tenth,

0.1

, the GPE is half of one tenth, or five hundredths, 0.05. The GPE will always be a 5 in the place to the right of the place value of the number’s precision.

Another way to think about the GPE is that it gives the range of values that would round off to the number in question. Back to weighing Sir Barks-A-Lot:

23\pm0.5

tells us a lower value and an upper value. 23-0.5=22.5 is the lowest weight that would round up to 23. Similarly, 23+0.5=23.5 is the highest weight that would round down to 23. Yes, perhaps we should say 23.49 or 23.499, etc., for the upper limit here, but it is easier to just say 23.5 and agree that 23.5 is the upper limit even though it would round up instead of down. Using inequalities, we could represent 23\pm0.5 as the range of values 22.5\leq\text{weight}<23.5 instead.

When you are asked to identify the GPE, it may help to think “What are the minimum and maximum numbers that would round to the given number?” For example, suppose the attendance at a Portland Thorns match is estimated to be 14,000 people. This number is precise to the nearest thousand. The minimum number that would round up to 14,000 would be 13,500 (because 13,449 would round down to 13,000), and the maximum number that would round down to 14,000 would be just below 14,500 (because 14,500 would round up to 15,000). Because these numbers are each 500 away from 14,000, the GPE is 500. If the estimate of 14,000 is correct to the nearest thousand, we know that the actual attendance is within \pm500 of 14,000.

Exercises

21. A package weighs

3.76

pounds. What is the GPE?

22. A roll of plastic sheeting is

0.00031

inches thick. What is the GPE in inches?

23. Plastic sheeting

0.00031

inches thick is referred to as 0.31 mil. What is the GPE in mils?

Recall from the previous module that the accuracy of a measurement is the number of significant figures. Let’s put together the ideas of accuracy, precision, and greatest possible measurement error.

Exercises

Google Map showing a 300-mile route from Clackamas Community College north to the Canadian border

Google Maps says that the driving distance from CCC’s main campus to the Canadian border is

300

miles. (Note: this is rounded to the nearest mile.)

24. What is the accuracy?

25.  What is the precision?

26.  What is the GPE?

A new stadium is expected to have around

23,000

seats.

27. What is the accuracy?

28.  What is the precision?

29.  What is the GPE?

The capacity of a car’s gas tank is

14.2

gallons.

30. What is the accuracy?

31. What is the precision?

32. What is the GPE?

Here is a summary of the important terms from these two modules. It is easy to get them mixed up, but remembering that “precision” and “place value” both start with “p” can be helpful.

Summary of Terms

Significant figures: the digits in a number that we trust to be correct

Accuracy: the number of significant digits

Precision: the place value of the rightmost significant digit

Greatest possible measurement error (GPE): one half the precision

 

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Technical Mathematics by Morgan Chase is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.