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Measuring and Significant Digits
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Measuring and Significant Digits

Jan 03, 2016

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Page 1: Measuring and Significant Digits

Measuring and Significant Digits

Page 2: Measuring and Significant Digits

Parallax Error• Parallax is the apparent shift in position of an object

caused by the observer’s movement relative to a fixed background. When the observer moves, it looks like the object is at a different place (star is against the blue when viewed from viewpoint A).

Page 3: Measuring and Significant Digits

Parallax When Measuring • If the eye is not directly across from what is being

measured, the measuring scale will give an inaccurate reading. The line is 91.50 cm long, not 92.10 or 90.90 cm.

• incorrect correct incorrect• best eye position

Page 4: Measuring and Significant Digits

Beware of Using the End of a Metre Stick • Many metre sticks when mass produced are cut too long or

short at the end. Check the end before using it or measure starting with the 1 cm or 10 cm mark and then deduct 1 cm or 10 cm from your measurement, depending on if you started with 1 or 10.

Page 5: Measuring and Significant Digits

Estimate Beyond the Smallest Marked Unit• The black object being measured is 41.63 cm long. Note

that the measurement has been estimated 3/10 past the 6 mm mark. It would be less accurate to record the measurement as either 41.6 or 41.7 cm. The estimated 41.63 is more accurate even though the last 3 is estimated.

Page 6: Measuring and Significant Digits

Reading Liquid Volumes : The Meniscus • Some liquids like water

have molecules with strong attractive forces for other substances. These so called adhesive forces cause water to attract and cling to other materials. Water typically climbs up the edges of containers it is in because its molecules attract to the container’s molecules. The curved water surface is called a meniscus.

Page 7: Measuring and Significant Digits

Avoid Parallax When Reading a Volume• For accurate liquid measurement, read directly across from

the liquid’s surface and read the bottom of the meniscus for concave liquid surfaces (liquid rising at the container’s edges). The liquid to the right has a volume of 43.0 mL, not 43.5 ml – the top of the meniscus.

Page 8: Measuring and Significant Digits

What Should be the Reading for This Volume? • Answer is on the next slide. (The units are mL)

Page 9: Measuring and Significant Digits

What Should be the Reading for This Volume?

• The answer is approximately 6.72 mL.

Page 10: Measuring and Significant Digits

The Mercury Meniscus • Some liquids like Mercury have less adhesive forces for

other substances and much stronger cohesive forces (attractions between its own molecules). Mercury in a glass container has a meniscus that bulges upward, a convex surface. Reading a mercury column in a thermometer, a person should record the value at the top of the meniscus. The thermometer reading at left is 37.08 Celsius (Normal human body temperature is 37.0 Celsius.

Page 11: Measuring and Significant Digits

Uncertainty in a Measurement • In any measurement, there is a degree of uncertainty due

to the degree of accuracy of the measurement, the estimation of the last digit, inaccuracies in measuring instruments and errors made by persons using measuring instruments.

Page 12: Measuring and Significant Digits

Precision • When something is repeatedly measured a number of

times, how close the several measurements are is a measure of precision. Precision is the degree of exactness to which the measurement of a quantity can be reproduced.

Page 13: Measuring and Significant Digits

An Example of Precision • A person measures the speed of light three times as 3.000

x 108 m/s, 3.002 x 108 m/s and 3.001 x 108 m/s. Considering the average of these measurements (3.001 x 108 m/s), the person reports his measurement of light’s speed as 3.001 x 108 m/s ± 0.001 m/s. The last ± 0.001 is his measurement’s precision. In any measurement, precision is limited by the finest markings on the measurement instrument as well as the care taken when measuring.

Page 14: Measuring and Significant Digits

Accuracy • Accuracy is the extent to which a person’s measurement

agrees with the standard value determined by scientists worldwide.

Page 15: Measuring and Significant Digits

An Example of Accuracy • The accepted value for the speed of light (worldwide) is

2.998 x 108 m/s. So the earlier measurement of 3.001 x 108 ± 0.001 m/s has a difference of (3.001 - 2.998) x 108 m/s = .003 x 108 m/s . The degree of difference from an accepted value reflects the accuracy of a measurement.

• Example:• My density determination is .879 g/mL

A reference book gives .886 g/mL• The difference is .007 .

• .879 g/mL is called the Observed value• .886 g/mL is called the Accepted value

Page 16: Measuring and Significant Digits

Percent Lab Error • The accuracy of a person’s measuring is often expressed

as a % lab error.• Percent Lab Error is the absolute value of the difference

between a person’s measurement and a the accepted standard divided by the standard, multiplied times 100% as shown below.

• % Lab Error = |O – A|/A x 100% where O is the observed measurement and A is the accepted standard value.

• If .879 g/mL is the Observed value and .886 is the Accepted value, the % lab error is = |.879 - .886 g/mL|/.886 g/mL x 100% = .007/.886 x 100% = .790 %

Page 17: Measuring and Significant Digits

Distinguishing Accuracy and Precision • A series of repeated measurements may be very precise but

inaccurate if the measuring instrument is uncalibrated or consistently used improperly. Likewise a series of measurements may be imprecise but their average might by chance be extremely close to the accepted value – very accurate.

Page 18: Measuring and Significant Digits

The Math of Precision • An instrument is precise to ½ of the smallest unit of

measure. In the measurement, 13.57 cm, the 7 is imprecise (uncertain) and the length could be anywhere between 13.565 cm and 13.574 cm. This is usually expressed as 13.57 ± .005 cm. The ± .005 is referred to as the absolute error of the measurement.

Page 19: Measuring and Significant Digits

Completely Precise Numbers • Measurements coming from counting are always

completely precise. If a bell rings 20 times, it rings exactly 20 times, not 20.1 or 19.8 – there is no imprecision or uncertainty.

Page 20: Measuring and Significant Digits

Percent Error (Relative Error) of a Measurement • The percent error (relative error) in a measurement is its

uncertainty (absolute error) divided by the measurement itself. In the measurement 13.57 ± .005 cm, the percent error is 0.005/13.57 x 100% = .04% (Use significant digits in rounding percent error)

Page 21: Measuring and Significant Digits
Page 22: Measuring and Significant Digits

Determining Significant Digits • All measured and estimated numbers are referred to as

significant digits. All non-zero digits (1-9) are significant digits. Zeros in measurements can be numbers or place holders. When they are place holders, zeros are NOT significant.

• In 216 000 m, the zeros indicate the decimal point (are place holders) so they are NOT significant. This number has 3 significant digits (sig. dig.). Zeros following non zero digits and before a decimal are NOT significant.

• In 0.0023 g, the zeros indicate the decimal point (are place holders) so they are NOT significant. This number has 2 significant digits (sig. dig.). Zeros before non zero digits are NOT significant.

Page 23: Measuring and Significant Digits

When Zeros Are Significant Digits • In the measurement, 230.05 hm, the zeros ARE

significant and have been measured since the non zero digits ahead and behind have been measured. This number has 5 significant digits. Zeros in between non zero digits ARE significant.

• In the measurement, 125.00 mL, the zeros ARE considered to be significant. In a Math class thinking of numbers purely as concepts, 3 = 3.0 = 3.00, but when measuring 3 ≠ 3.0 ≠ 3.00 . In 125.00 mL the zeros indicate that a person has actually measured with greater precision than with 125 mL, finding zero values in the tenths and hundreths places. This number has 5 significant digits. Zeros after non zero digits AND after a decimal ARE significant.

Page 24: Measuring and Significant Digits

What if Zeros Look Like Placeholders but HAVE been Measured?

• In the measurement, 2000 mm, as written there is 1 significant digit but what if the zeros have been measured and really are significant. To handle this and other problems, standard form (or scientific notation) was invented. The above measurement should not be written

as 2000 mm but as 2.000 x 103 mm which has 4 significant digits because the zeros are after a non zero number AND after a decimal.

Page 25: Measuring and Significant Digits

Measurements and Significant Digits • 13.57 cm 4 sig figs (sig digs) 1.357 x 101 cm• 22 000 km 2 sig figs 2.2 x 104 km• 100 000.0 g 7 sig figs 1.000 000 x 105 g• 0.000 002 kg1 sig fig 2 x 10-6 kg• 4000 mm 1 sig fig 4 x 103 mm

• Note that in standard form (scientific notation), the number part before the power has all significant digits (no exceptions).

• Some textbooks assume that all whole numbers are accurate to the ones place. This is confusing and wrong.

Page 26: Measuring and Significant Digits

Addition and Subtraction of Measured Numbers• When adding and subtracting, the leftmost place ABOVE

determines the rightmost place BELOW.• In the example below, the red numbers are uncertain

(estimated). Note that for the sum, this produces uncertainty in all the places after the decimal. If the largest place (the tenths place) has uncertainty then any place after it is meaningless and should be dropped.

Example : 2.4 cm

7.836 cm

+ 4213.2 cm ------------------------------

Sum: 4223.436 cm

Answer: 4223.4 cm

Page 27: Measuring and Significant Digits

Addition and Subtraction of Measured Numbers• When adding and subtracting, the leftmost place ABOVE

determines the rightmost place BELOW.• In the example below, the red numbers are uncertain

(estimated). Note that for the difference, this produces uncertainty in all the places after the decimal. If the largest place (the tenths place) has uncertainty then any place after it is meaningless and should be dropped.

Example : 312.1 cm (4 sig figs)

- 7.8926 cm (5 sig figs) ------------------------------

Difference: 304.2074 cm

Answer: 304.2 cm (4 sig figs)* Note that when subtracting, the answer may have less sig figs

than one or either of the numbers subtracted.

Page 28: Measuring and Significant Digits

Multiplying and Dividing Measured Numbers • When two measurements are multiplied or divided, the

answer can have only as many sig figs as the least number of sig figs in either measurement. The maximum sig figs in the answer is determined by the least sig figs in either of the measurements. (Red numbers below are uncertain)

For example, 26.8 mm (3 sig figs) x 3.2 mm (2 sig figs) The LEAST SIG FIGS _______________

536 8040 _______________

8576 = 85.76 = 86 (Rounded to 2 sig figs)* Note that if there is uncertainty in the 5 above (ones place)

then all numbers following it (tenths and hundreths places) need to be dropped because they are meaningless.

Raw Product : 85.76 mm2 (mm x mm = mm2)Answer: 86 mm2 (Rounded to the least # of sig figs).

Page 29: Measuring and Significant Digits

Rounding Numbers • When rounding to a given place, check the number

behind the place. If it is 4 or lower, round down. If it is 6 or higher, round up.

Example: 3.742 rounded to the nearest tenth is 3.7 3.761 rounded to the nearest tenth is 3.8

• If it is a 5, special considerations apply because 5 is in the middle of 1-9 and therefore should be rounded up 50% of the time and down the other 50% of the time.

1 2 3 4 5 6 7 8 9

Page 30: Measuring and Significant Digits

Rounding With 5s (Check After The 5)Example : 36.7651 to be rounded to the hundredths place

1. When a 5 is to be rounded, check next place after the 5. If the next place has a NONZERO number ROUND UP because 51, 52, 53 etc are more than half.

1 ………50 (exactly halfway) 51 52 53 (over halfway)……..99

Answer: 36.7651 rounded to the hundreths place is 36.77

Page 31: Measuring and Significant Digits

Rounding With 5s: (Check Ahead of 5) Example : 36.7650 or 36.765 to be rounded to the

hundredths place

2. When the number after a 5 is zero or there is no number after the 5, check the number AHEAD of the 5. If this number (the red one above) is EVEN ROUND DOWN. If this number is ODD ROUND UP. This made-up rule ensures that 5s are rounded up 50% and down 50% since half of the numbers are even and the other half are odd.

Answer: 36.7650 or 36.765 rounded to the hundredths place is 36.76 since 6 is EVEN.

Example 2: 36.735 or 36.7350 to be rounded to hundredths place. Check after (its zero or blank). Check ahead (odd). Round up.

Answer: 36.74 (When rounding with 5s, answers are always even)

Page 32: Measuring and Significant Digits

Calculating With Measurements Expressed With Error • When adding or subtracting numbers expressed with

measurement error, add the absolute errors of the numbers being added or subtracted.

• Examples:(4.2 ± 0.5 g) + (0.275 ± 0.5 g) = 4.5 ± 1.0 g (Also use sig fig rules)

(280 ± 5 g) – (6.75 ± 0.5 g) = 270 ± 5.5 g

(18.5 m ± 4.2%) + (28.7m ± 3.5%) = (18.5 ± 0.78 m) + (28.7 ± 1.0 m) = 47.2 ± 1.8 m (To go from % error to absolute error, divide by 100% and multiply times the measurement – this is the reverse process to changing absolute error into %)

Page 33: Measuring and Significant Digits

Calculating With Measurements Expressed With Error • When multiplying or dividing numbers expressed with

measurement error, add the relative or percent errors of the numbers being multiplied or divided.

• Examples:

(14.3 ± 2.5 %) x (19.7 ± 3.7 %) = 282 ± 6.2 % (0.001 ± 3.75 %) x (45.24 ± 5 %) = 0.04 ± 8.75 % (Use sig fig rules)

(854 ± .001%) ÷ (15 ± 13.5 %) = 57 ± 13.501 %

(11 ± 1.0 mm) ÷ (3.0 ± 1.0 mm) = (11 mm ± 9.1 %) ÷ (3.0 mm ± 33 %) = 3.7 mm ± 42 % = 3.7 ± 1.5 mm

(To go from absolute error to % error, divide the absolute error by the measurement and multiply by 100%)

(To go from % error to absolute error, divide by 100% and then multiply by the measurement)

Page 34: Measuring and Significant Digits

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