Uncertainty (not just a feeling)

A property that can be measured and described by a number and a unit

Numbers without units are useless!

A Quantity is…

Descriptions of quantities such as length, mass, or temperature

Ex. 3.54cm

Cannot be exact values

Exact numbers = no uncertainty

Ex. 4 people (there cannot be 4.25 people)

What are measurements?

Accuracy: how close the measured value is to what it should be (the actual value)

Precision: how reproducible the measurement is

can you get the same value over and over again?

Accuracy vs. Precision

Dartboard Analogy:

if a measurement is precise, but not accurate there is a systematic error or bias

Ex. you get the same wrong answer over and over

if a measurement is accurate but not precise, experimenter is being inconsistent

indicate the precision and accuracy of a measured value

the last significant figure in any measurement is uncertain (it was estimated/rounded/etc.)

all other numbers are certain

Ex. 8.658m

8.65 are certain numbers

last 8 is uncertain

could be 8.657m or 8.659m

Significant Figures (aka. Sig figs):

accuracy is shown by the number of significant figures

Ex. 526 is more accurate than 530

precision is shown by the place value of the number

the less uncertainty, the greater the precision

the more sig.figs you have, the less uncertainty and greater precision

Ex. 526.45789563 is more precise than 526.5

Ex.

Student A 34.94 g, 35.01 g, 35.07 g

Student B 34.98 g, 34.96 g, 34.97 g

Student C 35.01 g, 34.99 g, 35.03 g

Exact mass of the beaker is 35.02 grams

Who was the most accurate? C Who was the most precise? B

Who was the least accurate? B

Who was the least precise? A

1. All non-zero digits are significant regardless of the position of the decimal point

Ex. 8.658m = 4 sig figs

2. In the absence of a decimal point, zeros at the end of a number are not significant

Ex. 1990g = 3 sig figs

3. Zeros are significant at the end of a number with a decimal place

Ex. 1990.g = 4 sig figs

Sig Fig Rules:

4. All zeros between significant figures are significant

Ex. 1205 = 4 sig figs

5. Zeros at the beginning of a number are not significant

Ex. 0.0023 = 2 sig figs

How many significant figures are in each of the following:

a) 21.35 4

b) 8.005 4

c) 121.2000 7

d) 0.000823 3

e) 0.0980 3

f) 38,020 4

Reminder: starts with a non-zero digit which may be followed by a decimal and other digits

first # 1< x < 10

all digits in scientific notation are significant

Ex. 1990g = 1.99x103g

Ex. Write in scientific notation:

1. 863,000,000 8.63 x 108

2. 0.000009357 9.357 x 10-6

Convert to decimal form:

1. 4.261 x 105 426100

2. 8.47 x 10-6 0.00000847

Scientific Notation and Sig Figs

Math with Significant Figures:

Addition/ Subtraction Rule: round to the least precise place of the

starting values

Ex. 3.05 + 1.0004 = 4.05

Multiplication/ Division Rule: round to the least number of significant

figures of the starting values

Ex. (9.00005 x 3.1) = 7

4

When addition / subtraction and multiplication / division appear in the same calculation…

1.Follow the order of operations rules (BEDMAS) finding the correct sig figs at each step

2.Round your final answer to the correct number of significant figures

Mixed calculations?

NOTE : DO NOT ROUND until your final answer, just keep track of how many significant figures your answer should have…or else it could change your answer

WARNING:

Example:

1(0.093 )

0.00320

(9.0361 - 8.943)

0.0032029

Not significant, but

don’t round off yet!

Reading a Scale!

Remember in all measurements the last number is uncertain…that is because…

the last digit is always estimated (between the smallest division)

Ex. 42.6mL would have 2 certain digits (4, 2) and 1 uncertain digit (6)

Example:

What number

would this be?

6.237

More Examples:

6.214 6.372

Uncertainty in Measurements: The uncertainty is always in the last digit

(the one that was estimated)

This can be expressed as part of the number

Uncertainty is 1/10 of the smallest division on the instrument

Ex. 42.6 + 0.1mL

uncertainty term

Range = 42.5mL to 42.7mL

Homework Time: Uncertainty worksheet