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).

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 cm long, not or cm. incorrect correct incorrect best eye position

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.

Estimate Beyond the Smallest Marked Unit The black object being measured is 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 is more accurate even though the last 3 is estimated.

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.

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.

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

What Should be the Reading for This Volume? The answer is approximately 6.72 mL.

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 Celsius (Normal human body temperature is 37.0 Celsius.

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.

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.

An Example of Precision A person measures the speed of light three times as x 10 8 m/s, x 10 8 m/s and x 10 8 m/s. Considering the average of these measurements (3.001 x 10 8 m/s), the person reports his measurement of light’s speed as x 10 8 m/s ± m/s. The last ± 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.

Accuracy Accuracy is the extent to which a person’s measurement agrees with the standard value determined by scientists worldwide.

An Example of Accuracy The accepted value for the speed of light (worldwide) is x 10 8 m/s. So the earlier measurement of x 10 8 ± m/s has a difference of ( ) x 10 8 m/s =.003 x 10 8 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 g/mL is called the Observed value.886 g/mL is called the Accepted value

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 = | g/mL|/.886 g/mL x 100% =.007/.886 x 100% =.790 %

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.

The Math of Precision An instrument is precise to ½ of the smallest unit of measure. In the measurement, cm, the 7 is imprecise (uncertain) and the length could be anywhere between cm and cm. This is usually expressed as ±.005 cm. The ±.005 is referred to as the absolute error of the measurement.

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.

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 ±.005 cm, the percent error is 0.005/13.57 x 100% =.04% (Use significant digits in rounding percent error)

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 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 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.

When Zeros Are Significant Digits In the measurement, 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, 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 ≠ In 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.

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 x 10 3 mm which has 4 significant digits because the zeros are after a non zero number AND after a decimal.

Measurements and Significant Digits cm4 sig figs (sig digs)1.357 x 10 1 cm km2 sig figs2.2 x 10 4 km g7 sig figs x 10 5 g kg1 sig fig2 x kg 4000 mm1 sig fig4 x 10 3 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.

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 cm cm Sum: cm  Answer: cm

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 :  cm (4 sig figs) cm (5 sig figs) Difference: cm  Answer: cm (4 sig figs) * Note that when subtracting, the answer may have less sig figs than one or either of the numbers subtracted.

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 _______________ _______________ 8576 = = 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 : mm 2 (mm x mm = mm 2 ) Answer: 86 mm 2 (Rounded to the least # of sig figs).

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: rounded to the nearest tenth is 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

Rounding With 5s (Check After The 5) Example : 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) (over halfway)……..99 Answer: rounded to the hundreths place is 36.77

Rounding With 5s: (Check Ahead of 5) Example : or 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: or rounded to the hundredths place is since 6 is EVEN. Example 2: or to be rounded to hundredths place. Check after (its zero or blank). Check ahead (odd). Round up. Answer: (When rounding with 5s, answers are always even)

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 %)

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 ± % (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)

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