Chapter 2

Analyzing Data

Scientific Notation & Dimensional Analysis

• Scientific notation – way to write very big or very small numbers using powers of 10

3 x 108

Coefficient

Superscript

Superscript Rules

• Numbers greater than 10 = – Ex. 257000000000000

• Numbers less than 10 = – Ex. 0.0000000000000257

Rules for Scientific Notation• The coefficient must be between 1.0 and 9.99.• Your coefficient must contain all significant digits.• Move the decimal point as many places as necessary until

you create a coefficient between 1.0 and 9.99.• The exponent will be the number of places you move

your decimal point.• Moving the decimal to the left makes the number larger

= POSITIVE EXPONENT– Numbers greater than 10 always have exponents that

are positive.• Moving the decimal to the right makes the number smaller

= NEGATIVE EXPONENT– Numbers less than 1.0 always have exponents that are negative

• Write the following in scientific notation– 1,392,000 km

– 0.0000000028

– 1176.9

– 0.0123

• Write the following in regular notation– 3.6 x 105

– 5.4 x 10-5

– 5.060 x 103

– 8.9 x 10-7

Uncertainty in Measurement

Uncertainty in Measurement

• A measurement always has some degree of uncertainty.

• Different people estimate differently.

• Record all certain numbers and one estimated number.

Significant Figures

• Numbers recorded in a measurement. – All the certain numbers plus first estimated number

Rules for Significant Digits

1. Every nonzero digit is significantEx. 24.7 m

2. Zeros appearing between nonzero digits are significant

Ex. 24.07 m

3. Zeros after significant digits are only significant if there is a decimal point

Ex. 2470

Ex. 2470.0

4. Zeros in front of numbers are NOT significant, even after a decimal point

Ex. 0.0000247

Ex. 0.247

5. When a number is in scientific notation, all numbers in the coefficient are significant

Ex. 2.470 x 103

Significant Digits in Calculations

• An answer cannot be more precise than the least precise measurement from which it was calculated.

• To round off an answer you must first decide how many significant digits the answer should have.

• Your calculator DOES NOT keep track of significant digits, you have to do it!

Addition & Subtraction

• Answer can have no more decimal places than the number in the problem with the fewest decimal places.

– Ex: 4.5 + 6.007 + 13.39 = 23.897• Correct sig figs = 23.9

Multiplication & Division

• Answer can have no more significant digits than the number in the problem with the fewest significant digits

– Ex: 3.24 x 7.689 x 12.0 = 298.94832• Correct Sig. Figs = 299

Units and Measurement

• Systeme Internationale d’Unites (SI Units) – standard units of measure used by all scientists.– Why?

Base Units and SI Prefixes

• Base unit – measurements that can be taken with one instrument– Time– Length– Mass– Temperature– Amount

• Prefixes are added to base units to indicate very large or very small quantities.

• Second – determined by the frequency of radiation given of by cesium – 133

• Meter – distance light travels in a vacuum in 1/299,792,458 of a second

• Kilogram – defined by a platinum and iridium cylinder kept in France

• Temperature – quantitative measurement of the average kinetic energy of the particles that make up an object

Temperature Scales

• Fahrenheit– Water freezes at – Water boils at– oF = 1.8(oC) + 32

• Celsius– Water freezes at– Water boils at – oC = (oF – 32)/1.8

• Which is warmer, 25 oF or 25 oC?

• What is 98 oF in oC?

• What is 20 oC in oF?

• Kelvin – Water freezes at 273– Water boils at 373– Theoretically molecule movement completely

stops at 0 K (absolute zero)– K = C + 273

– What is 25 oC in K?

– What is 300 K in oC?

– What is 35 oF in K?

Derived Units

• Derived unit – unit that is made by combining two or more base units– m/s– g/mL– cm3

• Volume – space an object takes up– L x w x h– SI unit – m3

– More useful unit = L• 1 L = 1 dm3

• 1 mL = 1 cm3

• Volumes of irregular objects can be found by placing them into a graduated cylinder and measuring the amount of water that is displaced– What is the volume of the dinosaur?

• Density = amount of mass per unit volume– g/cm3

– g/mL– kg/L

• Always the same for a given substance– D = M/V

• What is the density of a cube that has a mass of 20 g and a volume of 5 cm3?

• When a piece of aluminum is placed in a 25 mL graduated cylinder that contains 10.5 mL of water, the water level rises to 13.5 mL. The density of aluminum is 2.7 g/mL. What is the mass of the piece of aluminum?

• What is the volume of an object with a mass of 13.5 g and a density of 1.4 g/mL?

Dimensional Analysis

• A systematic approach to problem solving that uses conversion factors to move from one unit to another– Conversion factor is a ratio of equivalent values

with different units– 1 km = 1000 m

– 12 inches = 1 foot

Tools for Problem Solving

• Be systematic • Ask yourself these questions

– Where do we want to go? – What do we know? – How do we get there?– Does it make sense?

Tools for Problem Solving Tools for Converting from One Unit to Another

Step 1 Find an equivalence statement that relates the 2 units.Step 2 Choose the conversion factor by looking at the direction of the required change (cancel the unwanted units).Step 3 Multiply the original quantity by the conversion factor. Step 4 Make sure you have the correct number of significant figures.

1 step conversions

Ex 1: A roll of wire is 15m long, what is the length in cm?

Ex 2: convert 8.96L to milliliters

• Convert 100 yards to feet

• Convert 5 kilometers to miles

Multi step conversions

• Convert 525 km to cm

• Convert 10000 in to miles

• Convert 3,000,000 s to years

Conversions with derived units

• Convert 365 mm3 to m3

• Convert 15.9 cm3/s to L/h

• Convert 25 miles/hour to ft/second

• Convert 1.004 g/cm3 to kg/mL

Uncertainty in Data

• All measurements contain uncertainties

Accuracy vs. precision

• Accuracy is how close a single measurement comes to the actual dimension or true value of what is measured– 4.555555 vs. 4.56 – More decimal places make a measurement more

accurate. – Depends on quality of measuring device

• Precision is how close several measurements are to the same value– Depends on more than one measurement– Depends on the skill of the person making the

measurement

Precise Accurate

Error and percent error

• Experimental value – value measured during experiment

• Accepted value – true or known value

• Error = experimental value – accepted value

• Percent error:

• You calculate the density of sucrose to be 1.40 g/mL. The accepted value for the density of sucrose is 1.59 g/mL. What is your % error?

2.4 Representing Data

• Graphs are a visual representation of data which make it easier to see patterns and trends

Circle graphs

• Aka –

• Show parts of a fixed whole

Bar Graphs

• Show how a quantity varies across categories

• Y axis –

• X axis –

Line Graphs

• Points on line = intersection of data for independent and dependent variable

• Y axis –

• X axis –

• Relationship between variables can be analyzed by the slope of the line.– Slope =

– + slope =

– - slope =

Interpreting Graphs

1. What is the independent variable2. What is the dependent variable3. Is the relationship linear?4. Is the slope positive or negative

• Interpolation – reading data from any point that falls between recorded data points

• Extrapolation – extending line beyond data points to estimate future values– Be careful! Can easily lead to errors