Significant Figures and Scientific Notation

What is a Significant Figure?

There are 2 kinds of numbers:Exact: the amount of money in your account. Known with certainty.

Approximate: weight, height—anything MEASURED. No measurement is perfect.

What is a Significant Figure?

The numbers reported in a measurement are limited by the measuring tool

Significant figures in a measurement include the known digits plus one estimated digit

Using Significant Figures

When a measurement is recorded only those digits that are dependable are written down.

The numbers reported in a measurement are limited by the measuring tool

1. All non-zero digits in a number are significant

2. Zeros between nonzero numbers are significant 41,026 32.001

Count these

Rules for Significant Figures

How many Significant Figures?7.16251.964843.1042.0003

Learning Check

3. Trailing zeros are not significant8100 3,600,000

4. All numbers after a decimal point are significant except for leading zeros

0.365 0.00972Do not count these

Rules for Significant Figures

How many Significant Figures?34,50028.0771,6000.03934,500.0

Learning Check

5. All numbers in scientific notation are significant. When writing scientific notation, do not write a number that is not meant to be a sig fig.3.2 x 103 2.00 x 10-4

Sig Figs in Scientific Notation

How many Significant Figures?2.62 x 10-2 1.0 x 102

9.7000 x 108

5 x 104

3 x 10-6

Learning Check

11

Sig Figs in CalculationsA calculated answer cannot be more precise than the measuring tool.

A calculated answer must match the least precise measurement.

Significant figures apply to final answers from 1) adding or subtracting2) multiplying or dividing

Adding and Subtracting

The answer has the same number of decimal places as the measurement with the fewest decimal places.

25.2 one decimal place+ 1.34 two decimal places 26.54answer 26.5 one decimal place

What is the answer to this calculation with the correct number of sig figs?

16.309 230.4+ 1.6975 248.4065 248.4Do not count/record these

Learning Check

The answer has the same number of sig figs as the least precise measurement. Round the answer to this number. 4.15 3 sig figsx 20 1 sig fig 83 must round to 1 sig fig 80

Multiplying and Dividing

What is the answer to this calculation with the correct number of sig figs?2.54 x 0.0028 0.0105 x 0.060 Round to 2 sig figs 11

Do not count these

Learning Check

= 11.2888889

Precision is a measurement of how much random error exists in a measurement.

103.1008 is very precise103 is less precise100 is least precise

Accuracy is a measurement of how much systematic error exists in a measurement.

If an instrument is not calibrated correctly, it may give an answer which is not accurate.

Precision vs. Accuracy

Relationship of precision to significant figures.

Precision vs. Accuracy Precision: 103.1 96.2

114.8This set of data shows a lack of

consistency. The margin of error on the measurement is large. The data is not very precise.

Accuracy: 42.443 g 42.441 g42.444 g

This set of data shows a high level of consistency, but the mass being measured is known to be 45.000 g. The balance is giving a systematic error – the reading is consistently off by approximately 2.5 g. The balance may give reliable comparisons of objects, but does not give a reliable value for a single object.

Precision refers to the closeness of two or more measurements to each other. If you weigh a given substance five times, and get 3.2 kg each time, then your measurement is very precise.

Accuracy refers to the closeness of a measured value to a standard or known value. In lab you obtain a weight measurement of 3.2 kg for a given substance, but the actual or known weight is 10 kg. Your measurement is not accurate.

Precision vs. Accuracy

Precision is independent of accuracy. You can be precise but inaccurate: you measure a mass to be almost identical four times, but the measurement is not close to the known value.

You can be accurate but not precise: your measurements are close to the known value, but the measurements are far from each other.

Precision vs. Accuracy

Precision vs. Accuracy

precise and accurate

precise but not accurate

accurate but not precise

neither precise nor accurate

A good analogy for understanding accuracy and precision is to imagine a basketball player shooting baskets. If the player shoots with accuracy, his aim will take the ball close to or into the basket. If the player shoots with precision, his aim will always take the ball to the same location which may or may not be close to the basket. A good player will be both accurate and precise by shooting the ball the same way each time and each time making it in the basket.

Precision vs. Accuracy