Data Analysis & Presentation How We Use and Communicate Data

Data Analysis We call the process of using and interpreting experimental data as data analysis Raw data from measurements is often converted to other units or used in a calculation before it is presented in tables and graphs We want to preserve all the precision of the data without claiming more than there is The method of significant figures is a powerful and convenient way of tracking precision

Significant Figures Significant figures (sig-figs) are digits that carry information and contribute to precision – The more sig-figs, the greater the precision When making measurement we maximize precision by estimating the least significant digit When doing calculations we keep track of precision by keeping track of sig-figs 1.We throw away extra digits that carry no meaning 2.We keep all digits that do have meaning 3.The method of sig-figs is a balance between 1 & 2

Rules of Significant Figures Never round a measurement: – The sig-figs of a measurement are the digits clearly measured plus one estimated digit Communication: (Rules 1-5) – Non-zero digits are always significant Example: m has four sig-figs – Zeros may or may not be significant Example: g has three sig-figs Calculation: (Rules 6 & 7) – One rule for multiplication and division – Another for addition and subtraction

Rule #1: All non-zero digits are significant Rule #2: Zeroes between non-zero digits are significant Rule #3: Leading zeroes are not significant Rule #4: Trailing zeroes to the right of the decimal point are significant Rule #5: Trailing zeroes without a decimal point are ambiguous and assumed to not be significant Communicating Sig-Figs

How Many Sig Fig?  (See Rule 1)  7.89 (See Rule 1)  7.09 (See Rule 2)  (See Rule 3)  (See Rule 4)  (See Rule 5)   10 3 o Answers: 5, 3, 3, 3, 4, 3-5 (assumed to be just 3), and 4 (ignore the “  10 3 ”)

Sig-Figs after Calculations Rule #6: When multiplying and dividing values, the result should be rounded to as many significant figures as the value with the smallest number of significant figures Rule #7: When adding and subtracting values, the result should be rounded so that its absolute precision (indicated by which power- of-ten column its least significant digit falls in) does not exceed the absolute precision of any of the values added and subtracted

How Many Sig-Fig?  (1.23 m)(4.3 m) = ____ m 2 (See Rule 6)  (1.23 m)/(4.3 s) = ____ m/s (See Rule 6)  (1.23 kg)(4.312 m)/(7.090 s) = ____ kg  m/s (See Rule 6)  (1.23  10 3 km)/(7.090  10 7 s) = ____ km/s (See Rule 6)  12.1 m m m = ____ m (See Rule 7)  s s s = ____ s (See Rule 7) o Answers: 5.3, 0.29, 0.748, 1.73  10 -5, 27.6, 74000

We often use tables to collect and organize our data Ball Drop Experiment: – Table gives units – Measurements are not rounded – All sig-fig are shown – Units are only shown at the top of columns Tables can also organize conversions/calculations Time (s)Distance (cm) Data Tables

The Ten Rules of Graphing 1.Understand what you are graphing and why 2.Provide a large space for the graph (half a page) 3.Be neat: always use a straight-edge to draw the axes 4.Choose horizontal and vertical scales such that the data will spread across the entire space 5.If possible, have point (0,0) in the lower left corner 6.Label your axes with titles and units 7.Plot the data with small dots to at least two sig. fig. 8.Double-check the accuracy of the plotted points 9.Draw the "best fit" line (or curve) if appropriate 10.Give your graph a meaningful title

Example Graph Voltage (V) Current (mA) Applied Voltage vs. Current same Title of Graph Vertical Axis Title (with units) Horizontal Axis Title (with units) Scale is consistent Graph includes the origin (0,0) Best Fit Line (if appropriate)

Summary We use the rules of significant figures to track and communicate the precision of values We always try to get the most precision (greatest number of sig-figs) from direct measurements We round the results of calculations to remain faithful to the precision of the values used Well-made tables help us organize, communicate, and do calculations with our data Well-made graphs are powerful tools for analyzing and communicating results