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# Data and How to Study It Help (page 3)

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#### Example 1

40 lbs potatoes x \$0.45 per lb = \$18.00 or \$18 since the first number is only measured to two places

#### Example 2

0.5 ounce of perfume x \$25.00 per ounce = \$12.50 for 0.5 ounces of perfume. (Note: the zero is only written because you cannot divide coins further.

#### Example 3

6.23 of wood x \$2.00 per linear foot = \$12.50 per linear foot.

Imagine you are trying to prove a theory based on a specific property, like boiling point. Unless the boiling point temperature was recorded with precision by other chemists, you will have trouble repeating the experiment, let alone proving a new theory. Since theories become laws by repeated experimentation, it is important to record measurements precisely.

Scientific knowledge moves forward by building upon results and experiments done by earlier scientists. If measurements are taken with little care or precision, a researcher doesn’t know if the observed results are new and exciting or just plain wrong.

Precision is the closeness of two sets of measured groups of values.

Precision is directly related to the amount of reproducibility of a measurement. Closely related to the topic of precision is that of accuracy . Some people use the two interchangeably, but there is a difference.

Accuracy is linked to how close a single measurement is to its true value.

In baseball, when Player A throws balls at a target’s center, it represents high precision and accuracy. Player B’s aim, with balls high and low missing the target, represents low precision and low accuracy. Player C’s hits, clumped together at the bottom left side of the target, define high precision (since they all landed in the same place), but low accuracy (since the object is to hit the target’s center). Player C, then, has to work on hitting the target’s center, if he wants to win games and improve accuracy.

## Rounding

Rounding is the way to drop (or leave off) non-significant numbers in a calculation and adjusting the last number up or down. There are three basic rules to remember when rounding numbers:

(1) If a digit is ≥ 5 followed by non-zeros, then add 1 to the last digit. (Note: 3.2151 would be rounded to 3.22.)

(2) If a digit is < 5 then the digits would be dropped. (Note: 7.12132 would be rounded to 7.12.)

(3) If the number is 5 (or 5 and a bunch of zeros), round to the least certain number of digits. (Note: 4.825, 4.82500, and 4.81500 all round to 4.82.)

Rounding reduces accuracy, but increases precision. The numbers get closer, but are not necessarily on target.

#### Examples

Try rounding the numbers below for practice.

(a) 2.2751 to 3 significant digits

(b) 4.114 to 3 significant digits

(c) 3.177 to 2 significant digits

(d) 5.99 to 1 significant digit

(e) 2.213 to 2 significant digits

(f) 0.0639 to 2 significant digits

Did you get (a) 2.28, (b) 4.11, (c) 3.2, (d) 6, (e) 2.2, and (f) 0.064?

When multiplying or dividing measurements, the number of significant digits of the measurement with the least number of significant digits determines the number of significant digits of the answer.

#### Examples

Do you see how significant digits are figured out?

(1) 1.8 pounds of oranges × \$3.99 per pound = \$7.182 = \$7.18 = \$7.2 (Note: 1.8 pounds of oranges has two significant digits)

(2) 15.2 ounces of olive oil × \$1.35 per ounce = \$20.50

(3) 25 linear feet of rope × \$3.60 per linear feet = \$90.00

Measurements can be calculated to a high precision. Calculators give between 8 and 10 numbers in response to the numbers entered for a calculation, but most measurements require far less accuracy.

Rounding makes numbers easier to work with and remember.

Think of how out-of-town friends would react if you said to drive 3.4793561 miles west on Main Street; turn right and go 14.1379257 miles straight until Union Street; turn left and travel 1.24900023 miles around the curve until the red brick house on the right. They might never arrive! But by rounding to 3.5 miles, 14 miles, 1.2 miles, and watching the car’s odometer (the instrument that measures distance), they would arrive with a lot less trouble and confusion.

Practice problems for these concepts can be found at - Science Experiment Data Practice Test

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