The Numbers of Science Data Analysis

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 there are 7 base units some familiar quantities are time, length, mass, and temp

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?

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 Density= mass/volume 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

We also use the Celsius (C) scale 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

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

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.

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: A number between 1 and 10 (only 1 digit to the left of the decimal!) 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 positive When 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 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

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. 8.86g/mL 8.92g/mL 9.00g/mL 8.98g/mL [(8.96 – 8.86)/8.96] x 100% = 1.12% [(8.96 – 8.92)/8.96] x 100% = 0.45% [(9.00 – 8.96)/8.96] x 100% = 0.45% [(8.98-8.96)/8.96] x 100% = 0.22%

Significant Figures Significant figures include all known digits plus one estimated digit Rules Non-zero numbers are always significant Zeros between non-zero numbers are always significant (“trapped zeros”) 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) Zeros that act as place holders are not significant (convert to SN to remove placeholder zeros) (“leading zeros”) 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

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