Measurements: Accuracy, Precision, & Error

August 7 & 8, 2014

How well can I measure this object?

Accuracy vs PrecisionAccuracy the extent to which a reported measurement approaches the true value of the quantity measured – how close is the measurement to the reality.

Precisionthe degree of exactness of a measurement (results from limitations of measuring device used).

Accuracy vs. PrecisionExample: game of darts

precise, not accurate

accurate, not precise

neither accurate nor

precise

accurate and precise

Which ruler will allow the most precise measurements? Why?

A

B

C

Accuracy vs. PrecisionExample: game of darts

precise, not accurate

accurate, not precise

neither accurate nor

precise

accurate and precise

Which ruler will allow the most accurate measurements? Why?A

B

C

Is the most precise instrument always the most accurate instrument? Why or why not?

Accuracy vs. PrecisionAnother example:

Discuss in pairs

Errors in MeasurementRandom ErrorsMeasured value can be above OR below the true value with equal probability.Example: normal user error

Systematic Errors• Due to the system or apparatus• Errors are consistently in one direction

(always high or always low)Examples:

– Apparatus calibrated incorrectly– Scale not zeroed – User making the same error

Errors in MeasurementTurn & Talk with table partnerYounger partner …

Which type of error would be more common when using a ruler?Describe an example of each type of error with a ruler.

Older partner – Which type of error would be more common when using a digital scale?Describe an example of each type of error with a digital scale.

Significant FiguresCan measurements ever be exact? No!

Significant figures = reliably known measurements + one estimate

52 mL – reliably known0.8 – estimate

Measurement = 52.8 mL

How many significant figures?

What is the precision of the measurement?

3

+ 0.2 mL

Significant FiguresIn table groups … What are the known measurements?What is estimated?What is overall measurement? How many sig figs?

2.6 cm

0.04 cm

2.64 cm

3

Significant FiguresWhich numbers in a measurement are significant?

The simple answer: all measured & estimated digits are significant

all ‘place holders’ are not

Significant FiguresWhich numbers in a measurement are significant?• All non-zero numbers are significant

Significant FiguresWhich numbers in a measurement are significant?• All non-zero numbers are significant• All zeros between other non-zero digits are

significant. (e.g. 503 km)

Significant FiguresWhich numbers in a measurement are significant?• All non-zero numbers are significant• All zeros between other non-zero digits are

significant. (e.g. 503 km)• Zeros to the left of non-zero digits are not significant

(e.g 0.0087 L)

Significant FiguresWhich numbers in a measurement are significant?• All non-zero numbers are significant• All zeros between other non-zero digits are

significant. (e.g. 503 km)• Zeros to the left of non-zero digits are not significant

(e.g 0.0087 L)• Zeros to the right of a decimal are significant. (e.g.

23.50 g)

Significant FiguresWhich numbers in a measurement are significant?• All non-zero numbers are significant• All zeros between other non-zero digits are

significant. (e.g. 503 km)• Zeros to the left of non-zero digits are not significant

(e.g 0.0087 L)• Zeros to the right of a decimal are significant. (e.g.

23.50 g)• Zeros to the right of a non-decimal are ambiguous.

Without other info, assume not significant. (e.g. 5200 m)

Significant FiguresHow can you make it obvious whether zeros at the end are significant or not?

Use scientific notation!

3000 km Sig figs are ambiguous. 1, 2, 3, or 4?3.0 X 103 km Sig figs = 2

Alternatively, you can put a line over / under the last significant digit (e.g. 3000 km)

Significant FiguresHow many significant figures?

4509.0 g

0.0087 kg

0.0908 mm

13000 mL

Significant FiguresHow many significant figures?

4509.0 g 5 sig figs

0.0087 kg 2 sig figs

0.0908 mm 3 sig figs

13000 mL 2 sig figs

Significant FiguresIndividually, identify the number of significant figures

5000.0 g

3008 L

0.0090 m

5080 cm

Significant FiguresIndividually, identify the number of significant figures

5000.0 g 5 sig figs

3008 L 4 sig figs

0.0090 m 2 sig figs

5080 cm ambiguous – without further info, assume 3 sig figs

Calculations with Sig FigsWhen making calculations with measurements, the least precise measurement determines the precision of the final answer.

Calculations with Sig FigsWhen making calculations with measurements, the least precise measurement determines the precision of the final answer.

Example:If a 5.6 meter flag is placed on top of a 3000 mmountain, how high is the of the flag?

Calculations with Sig FigsWhen making calculations with measurements, the least precise measurement determines the precision of the final answer.

Example:If a 5.6 meter flag is placed on top of a 3000 mmountain, how high is the of the flag?

IT DOESN’T MAKE SENSE TO SAY 3005.6 m.

Calculations with Sig FigsWhen adding or subtracting The final answer has the same number of decimals as the least precise measurement.

Calculations with Sig FigsWhen adding or subtracting The final answer has the same number of decimals as the least precise measurement.

Example: 2.2 + 1.25 + 23.894 = 27.164 → 27.2

2.2?? 1.25? 23.894

27.164 → 27.2

you don’t know second decimal in the first measurement and third decimal in second measurement, so the result can not have reliably known second and third decimal.

Calculations with Sig FigsWhen adding or subtracting The final answer has the same number of decimals as the least precise measurement.

Example: 2.2 + 1.25 + 23.894 = 27.164 → 27.2

2.2?? 1.25? 23.894

27.164 → 27.2

IMPORTANT: ROUND AT THE END OF CALCULATIONS

Calculations with Sig FigsWhen multiplying or dividingThe final answer has the same number of significant figures as the least precise measurement.

Calculations with Sig FigsWhen multiplying or dividingThe final answer has the same number of significant figures as the least precise measurement.

Example: 121.30 x 5.35 = (648.955) = 649 (5 SF) x (3 SF) = = (3SF)

Answer should be rounded up to 3 SF only

Calculations with Sig FigsDo these individually.

4.3 km + 2.567 km + 6 km =

8.23 g – 1.04 g - 5.1 g =

45 mL X 5000 mL =

0.00085 mg ÷ 0.0090 mg =

Calculations with Sig FigsDo these individually.

4.3 km + 2.567 km + 6 km = 13 km (1s digit)

8.23 g – 1.04 g - 5.1 g = 2.1 g (1 past decimal)

45 mL X 5000 mL = 300000 mL (1 sig fig)

0.00085 mg ÷ 0.0090 mg = 0.094 mg (2 sig figs)

Exit Ticket!HW and HW Quiz

ClosureWhat were our objectives today,

and how well did we accomplish them?

How did we address our unit statement today?

What was our LP trait and how did we demonstrate it?