Accuracy vs Precision

Accuracy: how close a set of measurements is to the actual value.Precision: how close a set of measurements are to one another.

• Just because a device works does not mean it is accurate. It must be CALIBRATED.

Calibrate: (1) mark an instrument with a standard scale of readings.(2) correlate the readings of an instrument with those of a standard in order to check the instrument's accuracy.

• Kitchen microwaves baffle Australian space scientists

Mr. Sapone, do I round this?

All Measurements in Science have Error associated with them.1) Systematic 2) Random 3) Screw-Ups (e.g. parallax)

Calculators result in errors since they assume all numbers are known.Measured Number: the length of a desk.Known Number: the number of desks in this room.

.

4.54 × 109 years ± 1%4.54 ± 0.05 billion years

Age of the Earth via Radiometric Dating

Uncertainty % Uncertainty

Suppose I ask you to measure the volume of a rectangular solid in cm3 with a ruler and you come up with the following dimensions.

2.74cm

2.23cm

2.68cm

2.74cm x 2.23cm x2.68cm = 16.365cm3

Your three measurements have three significant figures but your result has five.

Your measurements are only given to a hundredth of a centimeter cubed but your answer has the precision of a thousandth of a centimeter cubed, How can this be?

A measurement cannot be more accurate than your measuring devices!

Reading a Meter Stick / Ruler

One Uncertain Digit

17.6 mlCertain Numbers

Uncertain Digit

We know for certain the water lies between 17 and 18ml.

You should report one uncertain Digit

1 2 3 4 cm

What is the length?

1 2 3 4 cm

• We can see the markings between 1.6-1.7cm• We can’t see the markings between the .6-.7• We must guess between the .6 and .7 mark• We record 1.67 cm as our measurement• The last digit 7 is estimated...stop there

Report these measurements!

Sigfig Rules:1. ALL non-zero numbers (1,2,3,4,5,6,7,8,9)

are ALWAYS significant2. ALL zeroes between non-zero numbers

are ALWAYS significant.3. ALL zeroes which are SIMULTANEOUSLY

to the right of the decimal point AND at the end of the number are ALWAYS significant.

4. ALL zeroes which are to the left of a written decimal point and are in a number ≥ 10 are ALWAYS significant.

Sometimes writing the number in scientific notation helps determine the number of significant digits.

Number # of SigFigs Rule(s)

48,923

3.967

900.06

0.0004

8.1000

501.040

3,000,000

10.0

Sigfig Rules:1. ALL non-zero numbers (1,2,3,4,5,6,7,8,9)

are ALWAYS significant2. ALL zeroes between non-zero numbers

are ALWAYS significant.3. ALL zeroes which are SIMULTANEOUSLY

to the right of the decimal point AND at the end of the number are ALWAYS significant.

4. ALL zeroes which are to the left of a written decimal point and are in a number ≥ 10 are ALWAYS significant.

Sometimes writing the number in scientific notation helps determine the number of significant digits.

Number # of SigFigs Rule(s)

48,923 5 1

3.967 4 1

900.06 5 1,2,4

0.0004 1 1,4

8.1000 5 1,3

501.040 6 1,2,3,4

3,000,000 1 1

10.0 3 1,3,4

Sig Fig Arithmetic

Products and Quotients• When multiplying or dividing measured

numbers, the answer cannot have more significant figures than the term with the least number of significant figures.

Example: Calculator: 25.2 x 2.543 = 64.0836 Corrected Answer: 64.1

Addition and Subtraction• In addition and subtraction the number

of decimal places is what is important. The answer cannot have more decimal places than the term with the least number.

Example: Calculator: 25.331 + 1.33 = 26.661Corrected Answer: 26.66

SigFig Problems

1247 + 134.5 + 450 + 78 =Calculator: Answer:

13.214 + 234.6 + 7.0350 + 6.38 = Calculator:Answer:

2.5 x 3.42Calculator:Answer:

3.10 x 4.520Calculator:Answer:

(4.52 x 10¯4) ÷ (3.980 x 10¯6)Calculator:Answer:

The number of tables in this room times the width of my desk:

Calculator:Answer:

Sig Fig Arithmetic

Products and Quotients• When multiplying or dividing measured

numbers, the answer cannot have more significant figures than the term with the least number of significant figures.

Example: Calculator: 25.2 x 2.543 = 64.0836 Corrected Answer: 64.1

Addition and Subtraction• In addition and subtraction the number

of decimal places is what is important. The answer cannot have more decimal places than the term with the least number.

Example: Calculator: 25.331 + 1.33 = 26.661Corrected Answer: 26.66

SigFig Problems

1247 + 134.5 + 450 + 78 =Calculator: 1909.5Answer: 1910

13.214 + 234.6 + 7.0350 + 6.38 = Calculator:Answer:

2.5 x 3.42Calculator:Answer:

3.10 x 4.520Calculator:Answer:

(4.52 x 10¯4) ÷ (3.980 x 10¯6)Calculator:Answer:

The number of tables in this room times the width of my desk:

Calculator:Answer:

Statistical Analysis

• In science there is no such thing as a perfect measurement.

• ALL MEASUREMENTS HAVE ERROR.

• Any result given without a consideration of error is useless!

• Standard Deviation is a measure of how spread out numbers are.

• Its symbol is σ (the Greek letter sigma)

• The formula is easy: it is the square root of the Variance divided by the mean minus one. The variance is the average of the squared differences from the Mean.

Standard Deviation Curve

• The average height for adult men in the United States is about 70 inches, with a standard deviation of around 3 inches.

• 68% have a height within 3 inches of the mean (67–73 inches)

• 95% have a height within 6 inches of the mean (64–76 inches

Standard Deviation (sample)1. Calculate the mean or average of each

data set. To do this, add up all the numbers in a data set and divide by the total number of pieces of data.

2. Get the deviance of each piece of data by subtracting the mean from each number.

3. Square each of the deviations. 4. Add up all of the squared deviations.5. Divide this number by one less than the

number of items in the data set.6. Calculate the square root of the

resulting value. This is the sample standard deviation.

xi are individual valuesu is the meann is total number of data points

Standard Deviation Problem

Show in Excel

Percent Error

How to Calculate it.

ERROR!

Error = Measured Value – Accepted Value

• Science references list the density of aluminum as being 2.7g/cm3.

• Suppose you perform an experiment and your results show aluminum having a density of 2.8g/cm3

Aluminum 2.7g/cm3

Aluminum 2.8g/cm3

Accepted

Measured

Error = 2.8g/cm3 – 2.7g/cm3 = 0.1g/cm3

Error can be positive or negative but sometimes it isn’t very useful!

• If you have a 10ft long table (accepted value) and your results state it is 5ft long (measured value) your error is 5ft.

• Goal line to Goal line, a football field is 300 feet (known value). If you measure its length and come up with 290ft your error is 10ft.

• Even though the first measurement was off by only 5ft, the second measurement which is off by 10ft is much better.

• In terms of percentages, the first measurement was off by 50% of the value whereas the second was only off by 3.3%.

Error is 5ft out of 10

Error is 10ft but out of 300ft

You can have an error of a million miles if you were measuring the distance to the sun but it would only be a 1-2% error because the sun is 93 million miles away!

• Aluminum has an accepted value of 2.7g/cm3.Suppose you perform an experiment and your results show aluminum as having a density of 2.8g/cm3. What is your percent error?

Get a difference.

Compare to Accepted value.

Multiply by 100.

A student measured the temperature of boiling water and got an experimental reading of 97.5°C. Calculate the % error.

Get a difference.

Compare to Accepted value.

Multiply by 100.