accuracy & precision in measurement

ACCURACY & PRECISIONIN MEASUREMENT

GO EAGLES

FIGHT !!!!! WIN!!!

ACCURACY & PRECISION

• ACCURACY: – HOW CLOSE YOU ARE TO THE

ACTUAL VALUE – DEPENDS ON THE PERSON

MEASURING – CALCULATED BY THE FORMULA:

% ERROR = (YV – AV) X 100 ÷ AVWHERE: 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 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)

Accuracy & Precision

Precision without Accuracy

No Precision & No Accuracy

Accuracy without Precision

SIGNIFICANT DIGITS

A MEASUREMENT FOR PRECISION

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


• HOW MANY DIGITS ARE THERE IN THE MEASUREMENT?

• ALL OF THESE DIGITS ARE SIGNIFICANT

• THERE ARE 3 SD

LENGTH OF BAR = 3.23 CM

CM1 2 3 40

WHAT IS THE LENGTH OF THE BAR?


• 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

• 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.

Significant Figures

Physical Science

What is a significant figure?

• There are 2 kinds of numbers:

–Exact: the amount of money in your account. Known with certainty.

WHAT IS A SIGNIFICANT FIGURE?

– APPROXIMATE:

– WEIGHT, HEIGHT—ANYTHING MEASURED. NO MEASUREMENT IS PERFECT.

WHEN TO USE SIGNIFICANT FIGURES

• WHEN A MEASUREMENT IS RECORDED ONLY THOSE DIGITS THAT ARE DEPENDABLE ARE WRITTEN DOWN.

– 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

• 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

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

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.

HOW MANY SIG FIGS?

• 7• 40• 0.5• 0.00003• 7 x 105

• 7,000,000

• 1• 1• 1• 1• 1• 1

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

• 1.2• 2100• 56.76• 4.00• 0.0792• 7,083,000,000

• 2• 2• 4• 3• 3• 4

• 3401• 2100• 2100.0• 5.00• 0.00412• 8,000,050,000

• 4• 2• 5• 3• 3• 6

RULES FOR SIGNIFICANT FIGURESRULE #1

• ALL NON ZERO DIGITS ARE ALWAYS SIGNIFICANT• HOW MANY SIGNIFICANT DIGITS ARE IN THE

FOLLOWING NUMBERS?

•274274

•25.63225.632

•8.9878.987

•3 SIGNIFICANT FIGURES3 SIGNIFICANT FIGURES

•5 SIGNIFICANT DIGITS5 SIGNIFICANT DIGITS

•4 SIGNIFICANT FIGURES4 SIGNIFICANT FIGURES

RULE #2

• ALL ZEROS BETWEEN SIGNIFICANT DIGITS ARE ALWAYS SIGNIFICANT

• HOW MANY SIGNIFICANT DIGITS ARE IN THE FOLLOWING NUMBERS?

504

60002

9.077

3 SIGNIFICANT FIGURES

5 SIGNIFICANT DIGITS

4 SIGNIFICANT FIGURES

RULE #3

• ALL FINAL ZEROS TO THE RIGHT OF THE DECIMAL ARE SIGNIFICANT

• HOW MANY SIGNIFICANT DIGITS ARE IN THE FOLLOWING NUMBERS?

32.0

19.000

105.0020

3 Significant Figures

5 Significant Digits

7 Significant Figures

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

FOR EXAMPLE

0.0002

6.02 x 1023

100.000

150000

800

1 Significant Digit




1 Significant Digit

HOW MANY SIGNIFICANT DIGITS ARE IN THE FOLLOWING NUMBERS?

RULE #5

• ALL COUNTING NUMBERS AND CONSTANTS HAVE AN INFINITE NUMBER OF SIGNIFICANT DIGITS

• FOR EXAMPLE: 1 HOUR = 60 MINUTES

12 INCHES = 1 FOOT24 HOURS = 1 DAY

HOW MANY SIGNIFICANT DIGITS ARE IN THE FOLLOWING NUMBERS?

0.0073

100.02025007.90 x 10-3

670.00.0000118.84






1 SIGNIFICANT DIGIT


Rules Rounding Significant Digits

Rule #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

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

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

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

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

Let’s try these examples…

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

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

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

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

Scientific Notation Examples

99.343

4000.10.000375

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

Place the following numbers in scientific notation:

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

Going to Ordinary Notation Examples

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

Place the following numbers in ordinary notation:

Accuracy - Calculating % Error

How Close Are You to the Accepted Value (Bull’s Eye)

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

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%

remeasure -5%5% remeasure

Acceptable error is +/- 5%

Values from –5% up to 5% are acceptable

Values less than –5% or greater than 5% must be remeasured

accuracy & precision in measurement

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