Cha 2: Analyzing Data

Types of Measurements

When we measure things, we can do it in two main ways: digital measurements and analog measurements.

Digital Measurements

• What are they?

  Digital measurements are like using a calculator or a digital clock. They show you the exact number without any guessing.

• How do they work?

  When you see a number like 28.653 g, it means:

  • The 28 is the whole number part.

  • The .653 is the decimal part, which gives us even more detail.

  • The last digit (3 in this case) tells us that there might be a little bit of uncertainty. It means we are pretty sure about the number, but it might not be 100% perfect.

• Example:

  If you weigh something and it says 28.653 grams, you can trust that number is very precise!

Analog Measurements

• What are they?

  Analog measurements are like using a ruler or a thermometer with a needle. They show you a number, but you have to guess a little bit.

• How do they work?

  When you measure something and get a number like 4.72 cm, it means:

  • You can see the 4 and the 7 clearly on the ruler.

  • The 2 is a guess. You look at where the needle or line is pointing and estimate what comes after the 7.

• Example:

  If you measure a paperclip and it shows 4.72 cm, you can see the 4 and 7 clearly, but you had to guess a little for the last number (2).

Summary

• Digital Measurements give you exact numbers with no guessing.

• Analog Measurements show you numbers, but you have to estimate a little bit for the last part.

Both types of measurements are important, and they help us understand the world around us better!

a) You need quick results during multiple trials. → digital

b) Precision is paramount while working on highly sensitive chemical analysis. → analog

Common Fundamental SI Units

IMPORTANT:

Freezing temperature of water?   0°C

Boiling temperature of water? 100°C

Room temperature? 20°C

Body temperature? 37°C

SI Units

• Mass: kilogram (kg)

• Length: meter (m)

• Time: second (s)

• Temperature: Kelvin (K)

SI Prefixes

Density and Precision

Density Calculation

• Density is mass per unit volume, a physical property of matter.

• Formula: Density = mass / volume

  • V = m/d

  • M= dv

• Units: g/cm^3 for solids, g/mL for liquids

• 1 cc = 1 mL = 1 cm^3 for water

Precision and Accuracy

• Precision refers to how exact or repeatable a measurement is.

• Accuracy indicates how close a measurement is to its true value.

• The scientific goal is to achieve both accuracy and precision.

Scientific Notation and Data Presentation

Scientific Notation

• Represents a number between 1 and 10 multiplied by a power of 10.

• Useful for very large or very small numbers.

• Example: 3.0 x 10^8 m/s = 300,000,000 m/s

Data Presentation

Dimensional Analysis and Error Calculation

Dimensional Analysis

• Dimensional analysis is a systematic problem-solving method using conversion factors.

• Conversion Factor: A ratio of equivalent values with different units.

• Steps: Start with a given value, use conversion factors to cancel units, multiply, and validate the result.

• Common metric-metric conversion factors

  • Length 1 km=1000 m1 km=1000 mLength:

    • 1 kilometer (km) = 1,000 meters (m)

    • 1 meter (m) = 100 centimeters (cm)

    • 1 centimeter (cm) = 10 millimeters (mm)

  • Mass:

    • 1 kilogram (kg) = 1,000 grams (g)

    • 1 gram (g) = 1,000 milligrams (mg)

  • Volume:

    • 1 liter (L) = 1,000 milliliters (mL)

    • 1 milliliter (mL) = 1 cubic centimeter (cm³)

  • Area:

    • 1 hectare (ha) = 10,000 square meters (m²)

    • 1 square kilometer (km²) = 1,000,000 square meters (m²)

  Common Metric-English Conversions

  1. Length:

     • 1 inch = 2.54 centimeters (cm)

     • 1 foot = 30.48 centimeters (cm)

     • 1 yard = 0.9144 meters (m)

     • 1 mile = 1.60934 kilometers (km)

  2. Mass:

     • 1 ounce = 28.3495 grams (g)

     • 1 pound = 0.453592 kilograms (kg)

     • 1 ton (US) = 907.185 kilograms (kg)

  3. Volume:

     • 1 fluid ounce (fl oz) = 29.5735 milliliters (mL)

     • 1 pint (US) = 473.176 milliliters (mL)

     • 1 quart (US) = 0.946353 liters (L)

     • 1 gallon (US) = 3.78541 liters (L)

  4. Temperature:

     • Celsius to Fahrenheit: F = (C * 9/5) + 32

     • Fahrenheit to Celsius: C = (F - 32) * 5/9

  5. Area:

     • 1 acre = 4,046.86 square meters (m²)

     • 1 square mile = 2.58999 square kilometers (km²)

  Quick Reference:

  Metric Unit	Equiv in Other Metric Units	Equiv in English Units
  1 km	1,000 m	0.6214 miles
  1 m	100 cm	3.2808 feet
  1 kg	1,000 g	2.2046 pounds
  1 L	1,000 mL	0.2642 gallons (US)

Error Calculation

• Error is the variance between an experimental and accepted value.

  • Formula: Error = Experimental Value - Accepted Value

Percent Error expresses error as a percentage of the accepted value.

LOWER PERCENT ERROR = HIGHER ACCURACY.

.

                   |Experimental ­- Accepted|  x 100
PERCENT ERROR =
                          _______________________
                            
                                  Accepted

Significant Figures and Arithmetic Operations

• Significant figures indicate the precision of a measurement.

• Rules: HOW TO IDENTIFY SIG FIGS?

  1. Non-Zero digits = significant

     → EX: any digits b/w 1-9

  2. Trapped Zeros: zeros “trapped/sandwiched” b/w non-zero digits = significant

     → EX: 509 has 3 sig figs bc it has non-zero digits & a trapped zero = significant

  3. Leading Zeros [beginning]: zeros that come before a non-zero digit = NOT significant (it serves as a placeholder)

     → EX: 0.0045 has only 2 sig figs (4 & 5) bc the 3 zeros before the first non-zero digit serves as placeholders; therefore they are NOT significant.

  4. Trailing Zeros [end]: zeros @ the end of a number are ONLY significant if there is a DECIMAL point ; NO decimal point = NOT significant

     → EX: 0.500 has 3 sig figs (significant) & 20,000 has 1 sig fig (2) (not significant)

In scientific notation, only significant digits are placed before the exponential part. For example: 3.20 * 10^4 has 3 sig figs

Calculations w/Sig Figs

How do we know how many sig figs we should leave our final answers in?

When performing calculations, the number of significant figures in your result should reflect the precision of your measurements.

Addition + Subtraction Rule:

When you add or subtract numbers, your result should have the same number of decimal places as the measurement with the fewest decimal places. It’s not about counting significant figures—just the number of digits after the decimal point.

Example: add

Step 1: Line up the numbers by the decimal point to see how many decimal places each number has:

12.345 (3 decimal places)

+ 1.2 (1 decimal place)

-----------

Step 2: Identify the fewest decimal places. Here, 1.2 has 1 decimal place.

Step 3: Add the numbers together:

12.345

+ 1.2

-----------

13.545

Step 4: Round the result to match the fewest decimal places. Since 1.2 has 1 decimal place, your answer should also have 1 decimal place:

• 13.5 (Rounded to 1 decimal place)

• Multiplication + Division Rule:

  When you multiply or divide numbers, your result should have the same number of sig figs as the measurement with the fewest sig figs.

Example: subtract

Let’s try 56.78 - 4.1:

• Line up the numbers:

56.78 (2 decimal places)

- 4.1 (1 decimal place)

-----------

• Identify the fewest decimal places: 4.1 has 1 decimal place.

• Subtract the numbers:

  56.78

  - 4.1

  -----------

  52.68

  
  

  Round the result to 1 decimal place:

• 52.7 (Rounded to 1 decimal place)

Example: multiply

Let’s say we want to multiply:

• 3.24 (3 significant figures)

• 1.2 (2 significant figures)

Step 1: Multiply the numbers:

• 3.24×1.2= 3.888

Step 2: Identify the fewest significant figures:

• 3.24 has 3 significant figures.

• 1.21 has 2 significant figures.

• The fewest is 2 significant figures.

Step 3: Round the result to match the fewest significant figures:

• 3.888 rounded to 2 significant figures is 3.9.

So, 3.24×1.2=3.9

Example: divide

Let’s try division:

• 150.0 (4 significant figures)

• 3.45 (3 significant figures)

Step 1: Divide the numbers:

• 150.0÷3.45=43.47826087

Step 2: Identify the fewest significant figures:

• 150.0 has 4 significant figures.

• 3.45 has 3 significant figures.

• The fewest is 3 significant figures.

Step 3: Round the result to match the fewest significant figures:

• 43.47826087 rounded to 3 significant figures is 43.5.

So, 150.0÷3.45=43.5

Rounding Rules

• > 5: Round up.

• < 5: Stay the same.

• = 5:

  • Followed by non-zero: Round up.

  • Not followed by non-zero:

    • Odd last digit: Round up.

    • Even last digit: Round down.

• 
  

Rounding Examples:

1. Digit Greater than 5:

   • Example: Round 4.67 to 2 significant figures.

     • The digit after the 6 (7) is greater than 5.

     • Result: 4.7.

2. Digit Less than 5:

   • Example: Round 3.21 to 2 significant figures.

     • The digit after the 2 (1) is less than 5.

     • Result: 3.2.

3. Digit Exactly 5 (followed by non-zero digit):

   • Example: Round 2.75 to 2 significant figures.

     • The digit to be removed is 5 (followed by a non-zero digit, 7).

     • Result: 2.8.

4. Digit Exactly 5 (not followed by any digit, preceded by an odd digit):

   • Example: Round 5.5 to 1 significant figure.

     • The digit to be removed is 5 (preceded by an odd digit, 5).

     • Result: 6.

5. Digit Exactly 5 (not followed by any digit, preceded by an even digit):

   • Example: Round 4.50 to 2 significant figures.

     • The digit to be removed is 5 (preceded by an even digit, 4).

     • Result: 4.5.