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ACCURACY & PRECISION IN MEASUREMENT

ACCURACY & PRECISION IN MEASUREMENT. GO EAGLES FIGHT !!!!! WIN!!!. ACCURACY: HOW CLOSE YOU ARE TO THE ACTUAL VALUE DEPENDS ON THE PERSON MEASURING CALCULATED BY THE FORMULA: % ERROR = (YV – AV) X 100 ÷ AV WHERE: YV IS YOUR MEASURED VALUE & AV IS THE ACCEPTED VALUE. PRECISION:

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ACCURACY & PRECISION IN MEASUREMENT

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  1. ACCURACY & PRECISIONIN MEASUREMENT GO EAGLES FIGHT !!!!! WIN!!!

  2. ACCURACY: HOW CLOSE YOU ARE TO THE ACTUAL VALUE DEPENDS ON THE PERSON MEASURING CALCULATED BY THE FORMULA: % ERROR = (YV – AV) X 100 ÷ AV WHERE: YV IS YOUR MEASURED VALUE & AV IS THE ACCEPTED VALUE PRECISION: HOW FINELY TUNED YOUR MEASUREMENTS ARE OR HOW CLOSE THEY CAN BE TO EACH OTHER DEPENDS ON THE MEASURING TOOL DETERMINED BY THE NUMBER OF SIGNIFICANT DIGITS ACCURACY & PRECISION

  3. ACCURACY & PRECISION MAY BE DEMONSTRATED BY SHOOTING AT A TARGET. • ACCURACY IS REPRESENTED BY HITTING THE BULLS EYE (THE ACCEPTED VALUE) • PRECISION IS REPRESENTED BY A TIGHT GROUPING OF SHOTS (THEY ARE FINELY TUNED)

  4. Accuracy without Precision Accuracy & Precision No Precision & No Accuracy Precision without Accuracy

  5. SIGNIFICANT DIGITS A MEASUREMENT FOR PRECISION

  6. SIGNIFICANT DIGITS & PRECISION • THE PRECISION OF A MEASUREMENT IS THE SMALLEST POSSIBLE UNIT THAT COULD BE MEASURED. • SIGNIFICANT DIGITS (SD) ARE THE NUMBERS THAT RESULT FROM A MEASUREMENT. • WHEN A MEASUREMENT IS CONVERTED WE NEED TO MAKE SURE WE KNOW WHICH DIGITS ARE SIGNIFICANT AND KEEP THEM IN OUR CONVERSION • ALL DIGITS THAT ARE MEASURED ARE SIGNIFICANT

  7. 0 1 2 3 4 CM SIGNIFICANT DIGITS & PRECISION WHAT IS THE LENGTH OF THE BAR? • HOW MANY DIGITS ARE THERE IN THE MEASUREMENT? • ALL OF THESE DIGITS ARE SIGNIFICANT • THERE ARE 3 SD LENGTH OF BAR = 3.23 CM

  8. SIGNIFICANT DIGITS & PRECISION • IF WE CONVERTED TO THAT MEASUREMENT OF 3.23 CM TO MM WHAT WOULD WE GET? • RIGHT! 32 300 MM • HOW MANY DIGITS IN OUR CONVERTED NUMBER? • ARE THEY ALL SIGNIFICANT DIGITS (MEASURED)? • WHICH ONES WERE MEASURED AND WHICH ONES WERE ADDED BECAUSE WE CONVERTED? • IF WE KNOW THE SIGNIFICANT DIGITS WE CAN KNOW THE PRECISION OF OUR ORIGINAL MEASUREMENT

  9. WHAT IF WE DIDN’T KNOW THE ORIGINAL MEASUREMENT – SUCH AS 0.005670 HM. HOW WOULD WE KNOW THE PRECISION OF OUR MEASUREMENT. • THE RULES SHOWING HOW TO DETERMINE THE NUMBER OF SIGNIFICANT DIGITS IS SHOWN IN YOUR LAB MANUAL ON P. 19. THOUGH YOU CAN HANDLE THEM, THEY ARE SOMEWHAT COMPLEX.

  10. Significant Figures Physical Science

  11. What is a significant figure? • There are 2 kinds of numbers: • Exact: the amount of money in your account. Known with certainty.

  12. WHAT IS A SIGNIFICANT FIGURE? • APPROXIMATE: • WEIGHT, HEIGHT—ANYTHING MEASURED. NO MEASUREMENT IS PERFECT.

  13. WHEN TO USE SIGNIFICANT FIGURES • WHEN A MEASUREMENT IS RECORDED ONLY THOSE DIGITS THAT ARE DEPENDABLE ARE WRITTEN DOWN.

  14. IF YOU MEASURED THE WIDTH OF A PAPER WITH YOUR RULER YOU MIGHT RECORD 21.7CM. TO A MATHEMATICIAN 21.70, OR 21.700 IS THE SAME, BUT, TO A SCIENTIST 21.7CM AND 21.70CM IS NOT THE SAME

  15. GRADE SCHOOL PLACEHOLDERS AND NUMBER LINE • 21.700CM TO A SCIENTIST MEANS THE MEASUREMENT IS ACCURATE TO WITHIN ONE THOUSANDTH OF A CM. • BUT, TO A SCIENTIST 21.7CM AND 21.70CM IS NOT THE SAME

  16. IF YOU USED AN ORDINARY RULER, THE SMALLEST MARKING IS THE MM, SO YOUR MEASUREMENT HAS TO BE RECORDED AS 21.7CM.

  17. HOW DO I KNOW HOW MANY SIG FIGS? • RULE: ALL DIGITS ARE SIGNIFICANT STARTING WITH THE FIRST NON-ZERO DIGIT ON THE LEFT. • EXCEPTION TO RULE: IN WHOLE NUMBERS THAT END IN ZERO, THE ZEROS AT THE END ARE NOT SIGNIFICANT. GO EAGLES • 2ND EXCEPTION TO RULE: IF ZEROS ARE SANDWICHED BETWEEN NON-ZERO DIGITS, THE ZEROS BECOME SIGNIFICANT. • 3RD EXCEPTION TO RULE: IF ZEROS ARE AT THE END OF A NUMBER THAT HAS A DECIMAL, THE ZEROS ARE SIGNIFICANT.

  18. 7 40 0.5 0.00003 7 x 105 7,000,000 1 1 1 1 1 1 HOW MANY SIG FIGS?

  19. 3RD EXCEPTION TO RULE: THESE ZEROS ARE SHOWING HOW ACCURATE THE MEASUREMENT OR CALCULATION ARE.

  20. 1.2 2100 56.76 4.00 0.0792 7,083,000,000 2 2 4 3 3 4

  21. 3401 2100 2100.0 5.00 0.00412 8,000,050,000 4 2 5 3 3 6

  22. RULES FOR SIGNIFICANT FIGURESRULE #1 • ALL NON ZERO DIGITS ARE ALWAYS SIGNIFICANT • HOW MANY SIGNIFICANT DIGITS ARE IN THE FOLLOWING NUMBERS? • 3 SIGNIFICANT FIGURES • 5 SIGNIFICANT DIGITS • 4 SIGNIFICANT FIGURES • 274 • 25.632 • 8.987

  23. RULE #2 • ALL ZEROS BETWEEN SIGNIFICANT DIGITS ARE ALWAYS SIGNIFICANT • HOW MANY SIGNIFICANT DIGITS ARE IN THE FOLLOWING NUMBERS? 3 SIGNIFICANT FIGURES 5 SIGNIFICANT DIGITS 4 SIGNIFICANT FIGURES 504 60002 9.077

  24. RULE #3 • ALL FINAL ZEROS TO THE RIGHT OF THE DECIMAL ARE SIGNIFICANT • HOW MANY SIGNIFICANT DIGITS ARE IN THE FOLLOWING NUMBERS? 3 Significant Figures 5 Significant Digits 7 Significant Figures 32.0 19.000 105.0020

  25. RULE #4 • ALL ZEROS THAT ACT AS PLACE HOLDERS ARE NOT SIGNIFICANT • ANOTHER WAY TO SAY THIS IS: ZEROS ARE ONLY SIGNIFICANT IF THEY ARE BETWEEN SIGNIFICANT DIGITS OR ARE THE VERY FINAL THING AT THE END OF A DECIMAL

  26. 0.0002 6.02 x 1023 100.000 150000 800 1 Significant Digit 3 Significant Digits 6 Significant Digits 2 Significant Digits 1 Significant Digit FOR EXAMPLE HOW MANY SIGNIFICANT DIGITS ARE IN THE FOLLOWING NUMBERS?

  27. RULE #5 • ALL COUNTING NUMBERS AND CONSTANTS HAVE AN INFINITE NUMBER OF SIGNIFICANT DIGITS • FOR EXAMPLE: 1 HOUR = 60 MINUTES 12 INCHES = 1 FOOT 24 HOURS = 1 DAY

  28. 0.0073 100.020 2500 7.90 x 10-3 670.0 0.00001 18.84 2 SIGNIFICANT DIGITS 6 SIGNIFICANT DIGITS 2 SIGNIFICANT DIGITS 3 SIGNIFICANT DIGITS 4 SIGNIFICANT DIGITS 1 SIGNIFICANT DIGIT 4 SIGNIFICANT DIGITS HOW MANY SIGNIFICANT DIGITS ARE IN THE FOLLOWING NUMBERS?

  29. Rules Rounding Significant DigitsRule #1 • If the digit to the immediate right of the last significant digit is less that 5, do not round up the last significant digit. • For example, let’s say you have the number 43.82 and you want 3 significant digits • The last number that you want is the 8 – 43.82 • The number to the right of the 8 is a 2 • Therefore, you would not round up & the number would be 43.8

  30. Rounding Rule #2 • If the digit to the immediate right of the last significant digit is greater that a 5, you round up the last significant figure • Let’s say you have the number 234.87 and you want 4 significant digits • 234.87 – The last number you want is the 8 and the number to the right is a 7 • Therefore, you would round up & get 234.9

  31. Rounding Rule #3 • If the number to the immediate right of the last significant is a 5, and that 5 is followed by a non zero digit, round up • 78.657 (you want 3 significant digits) • The number you want is the 6 • The 6 is followed by a 5 and the 5 is followed by a non zero number • Therefore, you round up • 78.7

  32. Rounding Rule #4 • If the number to the immediate right of the last significant is a 5, and that 5 is followed by a zero, you look at the last significant digit and make it even. • 2.5350 (want 3 significant digits) • The number to the right of the digit you want is a 5 followed by a 0 • Therefore you want the final digit to be even • 2.54

  33. Say you have this number • 2.5250 (want 3 significant digits) • The number to the right of the digit you want is a 5 followed by a 0 • Therefore you want the final digit to be even and it already is • 2.52

  34. 200.99 (want 3 SF) 18.22 (want 2 SF) 135.50 (want 3 SF) 0.00299 (want 1 SF) 98.59 (want 2 SF) 201 18 136 0.003 99 Let’s try these examples…

  35. Scientific Notation • Scientific notation is used to express very large or very small numbers • I consists of a number between 1 & 10 followed by x 10 to an exponent • The exponent can be determined by the number of decimal places you have to move to get only 1 number in front of the decimal

  36. Large Numbers • If the number you start with is greater than 1, the exponent will be positive • Write the number 39923 in scientific notation • First move the decimal until 1 number is in front – 3.9923 • Now at x 10 – 3.9923 x 10 • Now count the number of decimal places that you moved (4) • Since the number you started with was greater than 1, the exponent will be positive • 3.9923 x 10 4

  37. Small Numbers • If the number you start with is less than 1, the exponent will be negative • Write the number 0.0052 in scientific notation • First move the decimal until 1 number is in front – 5.2 • Now at x 10 – 5.2 x 10 • Now count the number of decimal places that you moved (3) • Since the number you started with was less than 1, the exponent will be negative • 5.2 x 10 -3

  38. 99.343 4000.1 0.000375 0.0234 94577.1 9.9343 x 101 4.0001 x 103 3.75 x 10-4 2.34 x 10-2 9.45771 x 104 Scientific Notation Examples Place the following numbers in scientific notation:

  39. Going from Scientific Notation to Ordinary Notation • You start with the number and move the decimal the same number of spaces as the exponent. • If the exponent is positive, the number will be greater than 1 • If the exponent is negative, the number will be less than 1

  40. 3 x 106 6.26x 109 5 x 10-4 8.45 x 10-7 2.25 x 103 3000000 6260000000 0.0005 0.000000845 2250 Going to Ordinary Notation Examples Place the following numbers in ordinary notation:

  41. Accuracy - Calculating % Error How Close Are You to the Accepted Value (Bull’s Eye)

  42. Accuracy - Calculating % Error • If a student measured the room width at 8.46 m and the accepted value was 9.45 m what was their accuracy? • Using the formula:% error = (YV – AV) x 100 ÷ AV • Where YV is the student’s measured value & AV is the accepted value

  43. Accuracy - Calculating % Error • Since YV = 8.46 m, AV = 9.45 m • % Error = (8.46 m – 9.45 m) x 100 ÷9.45 m • = -0.99 m x 100 ÷ 9.45 m • = -99 m÷9.45 m • = -10.5 % • Note that the meter unit cancels during the division & the unit is %. The (-) shows that YV was low • The student was off by almost 11% & must remeasure • Acceptable % error is within 5%

  44. Acceptable error is +/- 5% • Values from –5% up to 5% are acceptable • Values less than –5% or greater than 5% must be remeasured remeasure -5% 5% remeasure

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