Mathematics: Measurement Principles

Precision, Accuracy, and Error
Precision: How reliable and repeatable a measurement is.  The more consistent the data is with repeated testing, the more precise it is.  For example, hitting a target consistently in the same spot, which may or may not be the center of the target, is precision.
Accuracy: How close the data is to the correct data. 
For example, hitting a target consistently in the center area of the target, whether or not the hits are all in the same spot, is accuracy.
Note: it is possible for data to be precise without being accurate.  If a scale is off balance, the data will be precise, but will not be accurate.  For data to have precision and accuracy, it must be repeatable and correct.



Approximate error: The amount of error in a physical measurement.  Approximate error is often reported as the measurement, followed by the ± symbol and the amount of the approximate error.
Maximum possible error: Half the magnitude of the smallest unit used in the measurement.  For example, if the unit of measurement is 1 centimeter, the maximum possible error is

cm, written as

following the measurement.  It is important to apply significant figures in reporting maximum possible error.  Do not make the answer appear more accurate than the least accurate of your measurements.

Rounding and Estimation
Rounding is reducing the digits in a number while still trying to keep the value similar. The result will be less accurate, but will be in a simpler form, and will be easier to use. Whole numbers can be rounded to the nearest ten, hundred or thousand.
When you are asked to estimate the solution to a problem, you will need to provide only an approximate figure or estimation for your answer. In this situation, you will need to round each number in the calculation to the level indicated (nearest hundred, nearest thousand, etc.) or to a level that makes sense for the numbers involved. When estimating a sum all numbers must be rounded to the same level. You cannot round one number to the nearest thousand while rounding another to the nearest hundred.

Scientific Notation
Scientific notation is a way of writing large numbers in a shorter form. The form

is used in scientific notation, where a is greater than or equal to 1, but less than 10, and n is the number of places the decimal must move to get from the original number to a. Example: The number 230,400,000 is cumbersome to write.

To write the value in scientific notation, place a decimal point between the first and second numbers, and include all digits through the last non-zero digit ().

To find the appropriate power of 10, count the number of places the decimal point had to move (). The number is positive if the decimal moved to the left, and negative if it moved to the right. We can then write 230,400,000 as . If we look instead at the number 0.00002304, we have the same value for a, but this time the decimal moved 5 places to the right (). Thus, 0.00002304 can be written as . Using this notation makes it simple to compare very large or very small numbers. By comparing exponents, it is easy to see that  
is smaller than
, because 4 is less than 5.
 

P1. Round each number to the indicated degree:
(a) Round to the nearest ten: 11; 47; 118
(b) Round to the nearest hundred: 78; 980; 248
(c) Round each number to the nearest thousand: 302; 1274; 3756
P2. Estimate the solution to  by rounding each number to the nearest ten thousand.
P3. A runner's heart beats 422 times over the course of six minutes. About how many times did the runner's heart beat during each minute?

P1. (a) When rounding to the nearest ten, anything ending in 5 or greater rounds up. So, 11 rounds to 10, 47 rounds to 50, and 118 rounds to 120.
(b) When rounding to the nearest hundred, anything ending in 50 or greater rounds up. So, 78 rounds to 100, 980 rounds to 1000, and 248 rounds to 200.
(c) When rounding to the nearest thousand, anything ending in 500 or greater rounds up. So, 302 rounds to 0, 1274 rounds to 1000, and 3756 rounds to 4000.
P2. Start by rounding each number to the nearest ten thousand: 345,932 becomes 350,000, and 96,369 becomes 100,000. Then, add the rounded numbers:
. So, the answer is approximately 450,000. The exact answer would be
.
So, the estimate of 450,000 is a similar value to the exact answer.
P3. 'About how many' indicates that you need to estimate the solution. In this case, look at the numbers you are given. 422 can be rounded down to 420, which is easily divisible by 6. A good estimate is

beats per minute. More accurately, the patient's heart rate was just over 70 beats per minute since his heart actually beat a little more than 420 times in six minutes.