Chapter 2

Analyzing Data

2.1 - Measurements and Units

Units of Measure• SI units: Systeme Internationale d’ Unites• standard units of measurement to be understood by all scientists

• Base Units: defined unit of measurement that is based on an object or event in the physical world

Table 2.1 – The Base UnitsQuantity Base Unit

Time Second (s)

Length Meter (m)

Mass Kilogram (kg)

Temperature Kelvin (K)

Amount of a substance Mole (mol)

Electric current Ampere (A)

Luminous intensity Candela (cd)

Time• second (s)• Many chemical reactions take

place in less than a second so scientist often add prefixes, based on multiples of ten, to the base units.

• ex. Millisecond

Length• meter (m)• A meter is the distance that light

travels though a vacuum in 1/299 792 458 of a second.

• What is a vacuum?• Close in length to a yard.• Prefixes also apply…ex. millimeter

Mass• mass is a measurement of matter• kilogram (kg)• about 2.2 pounds• Masses measured in most

laboratories are much smaller than a kilogram, so scientists use grams (g) or milligrams (mg).

• How many grams are in a kilogram?– 1000

• How many milligrams are in a gram?– 1000

Derived Units

• Not all quantities are measured in base units• A unit that is defined by a combination of base

units is called a derived unit.• Volume and Density are measured in derived

units.

Volume• The space occupied by

an object• Unit = cm3 = mL• Liters are used to

measure the amount of liquid in a container (about the same volume as a quart)

• Prefixes also applied…ex. milliliter

Density• The ratio that compares the mass of an

object to its volume is called density.• Units are g/cm3

• You can calculate density by the following equation:

Density= mass/volume Ex: What is the density of a sample of

aluminum that has a mass of 13.5 g and a volume of 5.0 cm3?

Density= 13.5g/5.0cm3

=2.7g/cm3

• Temperature

• A measurement of how hot or cold an object is relative to other objects

• The kelvin (K) scale water freezes at 273K

water boils at 373K

• We also use the Celsius (C) scale water freezes at 0oC water boils at 100oC

To Convert Celsius to Kelvin…

• Add 273!!

ex: -39oC + 273= 234 K

To Convert Kelvin to Celsius…• Subtract 273!! ex: 234K- 273= -39°C

2.2 - Scientific Notation

Scientific Notation

• Numbers that are extremely large can be difficult to deal with…sooo

• Scientists convert these numbers into scientific notation

• Scientific notation expresses numbers as a multiple of two factors:

1. A number between 1 and 10 (only 1 digit to the left of the decimal!)2. Ten raised to a power

For example:A proton’s mass =0.0000000000000000000000000017262 kg

If you put it in scientific notation, the mass of a proton is expressed as 1.7262 x 10-27 kg

Remember:When numbers larger than 1 are expressed in scientific

notation, the power of ten is positiveWhen numbers smaller than 1 are expressed in

scientific notation, the power of ten is negative

Try these:

Convert 1,392,000 to scientific notation. = 1.392 x 106

Convert 0.000,000,028 to scientific notation. = 2.8 x 10-8

Adding and Subtracting using Scientific Notation

• Make sure the exponents are the same!! 7.35 x 102 + 2.43 x 102 = 9.78 x 102

• If the exponents are not the same, you have to make them the same!!

• Tip: if you increase the exponent, you decrease the decimal ----- if you decrease the exponent, you increase the decimal

• Example:Tokyo pop: 2.70 x 107

Mexico City pop: 15.6 x 106 = 1.56 x 107

Sao Paolo pop: 0.165 x 108 = 1.65 x 107

NOW you can add them together and carry thru the exponent Total= 5.91 x 107

Multiplying and Dividing using Scientific Notation

• Multiplication:– Multiply decimals and ADD exponents

• Ex : (1.2 x 106) x (3.0 x 104) = 3.6 x 1010 6 + 4 = 10• * Ex: (1.2 x 106) x (3.0 x 10-4) = 3.6 x 102 6 + (-4) = 2

• Division:– Divide decimals and SUBTRACT exponents

• Ex: (5.0 x 108) ÷ (2.5 x 104) = 2.0 x 104 8 – 4 = 4• *Ex: (5.0 x 108) ÷ (2.5 x 10-4) = 2.0 x 1012 8 – (-4) = 12

More 2.2 - Dimensional Analysis

Dimensional Analysis• Conversion factor:

– A numerical factor used to multiply or divide a quantity when converting from one system of units to another.

• Conversion factors are always equal to 1

• Dimensional analysis: – A fancy way of saying “converting units” by using

conversion factors

Table 2.2 – SI PrefixesPrefix Symbol Numerical Value in

Base UnitsPower of 10 Equivalent

Giga G 1,000,000,000 109

Mega M 1,000,000 106

Kilo K 1000 103

-- -- 1 100

Deci d 0.1 10-1

Centi c 0.01 10-2

Milli m 0.001 10-3

Micro µ 0.000001 10-6

Nano n 0.000000001 10-9

Pico p 0.000000000001 10-12

Dimensional analysis often uses conversion factors

Suppose you want to know how many meters are in 48 km. You have to choose a conversion factor that relates kilometers to

meters.You know that for every 1 kilometer there is 1000 meters.What will your conversion factor be?

1000m/1km Now that you know your conversion factor, you can multiply it

by your known…BUT you want to make sure you set it up so that kilometers cancels out. How would you do this?

48km x 1000m 1km

=48,000 m

TIP: Put the units you already have on the bottom of the conversion factor and the units you want

on top.

2.3 - Accuracy vs. Precision

Significant Figures

Accuracy and Precision

Accuracy: How close measurements are to the actual value

Precision: How close measurements are to each other

Percent Error

• An error is the difference between an experimental value and an accepted value

• Percent error=

Percent error = accepted - experimental x 100 accepted value• A tolerance is a very narrow range of error

Example: The accepted density

for copper is 8.96g/mL. Calculate the percent error for each of these measurements.

A. 8.86g/mLB. 8.92g/mLC. 9.00g/mLD. 8.98g/mL

A. [(8.96 – 8.86)/8.96] x 100% = 1.12%

B. [(8.96 – 8.92)/8.96] x 100% = 0.45%

C. [(9.00 – 8.96)/8.96] x 100% = 0.45%

D. [(8.98-8.96)/8.96] x 100% = 0.22%

Significant Figures Significant figures include all

known digits plus one estimated digit

Rules1. Non-zero numbers are always

significant2. Zeros between non-zero numbers are

always significant (“trapped zeros”)3. All final zeros to the right of the

decimal place are significant (“trailing zeros”) (but trailing zeros don’t count if there is no decimal in the number)

4. Zeros that act as place holders are not significant (convert to SN to remove placeholder zeros) (“leading zeros”)

5. Counting numbers and defined constants have an infinite number of sig figs

Rounding numbers

• An answer should have no more significant figures than the data with the fewest significant figures

• Example:Density of a given object = m = 22.44g = 1.5802817g/cm3

V 14.2cm3

How should the answer be rounded? 1.58 g/cm3

Addition & Subtraction

• How do you add or subtract numbers that contain decimal point?

• The easiest way (which you learned in third grade) is to line up the decimal points then perform the math

• Then round according to the previous rule, rounding to the least numbers after the decimal! (ex: 5.25 + 10.3 = 15.55 15.6)

Multiplication & Division

• When you multiply or divide, your answer must have the same number of significant figures as the measurement with the fewest significant figures…just like adding or subtracting!

• Ex: 38736km 4784km = 8.096989967 8.097

2.4 – Representing Data

Representing Data

• A goal of many experiments is to discover whether a pattern exists in a certain situation…when data are listed in a table the patterns may not be obvious

• Soooo, scientists often use graphs, which are visual displays of data– X-axis independent variable– Y-axis dependent variable

Graphing Types

• Line Graphs – most graphs you complete will be line graphs

Temperature and Elevation Relationship

100 300 500 7000

5

10

15

20

25

Tem

pera

ture

(°

C)

Elevation (m)

Graphing Types

• Circle Graphs – used for graphing parts of a whole (percentages)

Sales

1st Qtr2nd Qtr3rd Qtr4th Qtr

Graphing Types

• Bar Graphs – shows how a quantity varies across categories

Dietary sources of magnesium

Halibut (3oz) Cashews (1 oz)

Baked potato (medium)

Raisins (1/4 cup)

0102030405060708090

100

Mag

nesi

um C

onte

nt

(mg)