Experimental Error

Experimental Error Significant Figures

“The Rules for Zeros”

“The last significant digit in a measured quantity always has associated uncertainty”

•Zeros are significant when they occur in the middle of a number

KNOWN

Example:

1.2034 x 105

e.g. 9.63 mL

•Zeros are significant when they occur at the end of a number on the right-hand side of a decimal point

* Interpolation = Estimation of readings to the nearest tenth of distance between scale readings

5

Example:

1.2340 x 105

NOTE: Assumes that zero is accurate estimate.

Experimental Error

Experimental Error Arithmetic and Significant Figures

Significant Figures

1. Addition/Subtraction

“Some numbers are exact - with an infinite number of unwritten significant digits”

Express numbers with the same exponent Significant figures are limited to the leastcertain number

Example:

5 people 1.234 x 10-5 + 6.78 x 10-9

= 5.0 people = 5.00 people

1.234 x 10-5 + 0.000678 x 10-5 1.234678 x 10-5

= 5.000 people

Significant

Significant Figures………… Number of all certain digits plus the first uncertain digit

31

30.5 mL

30.67 mL

30.6 mL

30.68 mL

30.7 mL

30.69 mL

3 sig figs

31

Significant Figures………… Rules for determining the number of sig figs 1) Disregard all initial zeros

30

30

Answer: 1.235 x 10-5

4 sig figs 0.03068 L

2) Disregard all final (terminal) zeros unless they follow a decimal point 3) All remaining digits including zeros between nonzero digits are significant Tip: Express data in scientific notation to avoid confusion in determining whether terminal zeros are significant

2.0 L or 2000 mL best expressed as 2.0 X 103 mL 2 sig figs ? sig figs

2 sig figs

1

Significant Figures………… Numerical Computation Conventions When adding and subtracting, express the numbers to the same power of ten. 3.4

2.432 X 106 = 2.432 X 106

0.020

6.512 X 104 = 0.06512 X 106

7.31

-1.227 X 105 = -0.1227 X 106

10.73

= 2.37442 X 106

Answer: 10.7

Answer: 2.374 X106

For Multiplication or Division, round the answer to contain the same # of sig figs as the original number with the smallest # of sig figs.

Chapter 4 - Lecture # 3 Overview

(next 3 lectures)

Basic Tools (continued) •Microsoft Excel - Analysis of Data Statistics •Descriptive Statistics •Statistical Tests •Least Squares and Calibration

Introduction to Statistics “Statistics give us tools to accept conclusions that have a high probability of being correct and to reject conclusions that do not.” 1. Descriptive Statistics - Describes estimate of an actual value, and the uncertainty associated with estimate 2. Statistical Tests - Allow us to compare estimates and uncertainties, and make conclusions about these comparisons.

Arithmetic and Significant Figures 1. Addition/Subtraction The number of significant figures in the answer may exceed or be less than that in the original data. If the numbers being added do not have the same number of significant figures, we are limited by the least-certain one. 5.345 (4 sig figs) + 6.728 (4 sig figs) 12.073 (5 sig figs)

7.26 X 1014 - 6.69 X 1014 0.57 X 1014

Good Laboratory Practice (GLP) “embodies a set of principles that provides a framework within which laboratory studies are planned, performed, monitored, recorded, reported and archived…GLP helps assure regulatory authorities that the data submitted are true reflection of the results obtained during the study and can therefore by relied upon when making…assessments.”

Introduction to Statistics Statistics deals only with RANDOM ERROR, not systematic (determinate) error Measurements affected by random error will approach a Gaussian Distribution as the number of measurements increases.

2

Class Experiment………..

Gaussian Distribution

Flip a coin 10 times and tabulate the results as follows:

“Bell-Shaped Curve”

Number of Tails

|||| |

Number of Measurements

Number of Heads

||||

14 12 10

6 4 2 0 0

1

2

3

4

5

6

7

8

9

Systematic Error (Less Accurate)

Number of Measurements

8

Number of Measurements

Frequency (Number of Students)

Quantitative Analysis Class Coin Flip Experiment

More Random Error (Less Precise)

10

Number of Heads (out of 10)

Normal Distribution

Introduction to Statistics

Number of Standard Deviations from Mean

Population vs. Sample

Frequency

A population is the whole set of points measurable. A sample is a subset of the population that typically gets measured. The population mean is denoted as μ and the standard deviation as σ. The sample mean is denoted as x and the standard deviation as s. …“Population” is an infinite number of the same measurement

-4

-3

-2

-1

0

1

2

3

4

Number of SD’s

By definition, 68% of all measurements will fall within 1 SD of the Mean

95 % of measurements fall within 2 SD’s of Mean Number of Standard Deviations from Mean

Frequency

Frequency

Number of Standard Deviations from Mean

68 % 16 % -4

-3

-2

-1

0

1

Number of SD’s

2

95 %

2.5 %

16 % 3

4

-4

-3

-2

-1

0

2.5 % 1

2

3

4

Number of SD’s

3

Descriptive Statistics

Descriptive Statistics Mean (Average) x

=

Σi xi n

Σi xi = the sum of the measurements n = number of measurements

10 9 8 7 6 5 4 3 2 1

Mode (most common measurement)

“Outlier” 24.0 24.1 24.2 24.3 24.4 24.5 24.6 24.7 24.8 24.9 25.0 25.1 25.2 25.3 25.4 25.5 25.6 25.7 25.8 25.9 26.0 26.1 26.2 26.3 26.4 26.5 26.6 26.7 26.8 26.9 27.0

Number

Example: You calibrate a 25-mL Class A pipet by dispensing 25 mL of water into a pre-weighed volumetric flask, and “weigh by difference” to measure the mass of water and subsequently determine the actual volume of water. You do this thirty (30) times, and obtain the data shown graphically below.

Descriptive Statistics Example: You calibrate a 25-mL Class A pipet by dispensing 25 mL of water into a pre-weighed volumetric flask, and “weigh by difference” to measure the mass of water and subsequently determine the actual volume of water. You do this thirty (30) times, and obtain the data shown graphically below.

24.7, 24.9, 25.0, 25.0, 25.1, 25.1, 25.1, 25.2, 25.2, 25.2, 25.2, 25.2, 25.3, 25.3, 25.3, 25.3, 25.3, 25.3, 25.3, 25.4, 25.4, 25.4, 25.4, 25.5, 25.5, 25.6, 25.6, 25.7, 25.8, 26.5 n = 30 x = 24.7+24.9+25.0+25.0+25.1+25.1+25.1+25.2 ...+26.5 30 =25.3266667 =25.3

4

Descriptive Statistics

10 9 8 7 6 5 4 3 2 1

Mean ( x ) = 25.3

Approximates µ µ is the mean for the populatoin (“an infinite set of data”)

Descriptive Statistics

Mean /Average (x) Not as precise? How do we quantify?

24.0 24.1 24.2 24.3 24.4 24.5 24.6 24.7 24.8 24.9 25.0 25.1 25.2 25.3 25.4 25.5 25.6 25.7 25.8 25.9 26.0 26.1 26.2 26.3 26.4 26.5 26.6 26.7 26.8 26.9 27.0

Number

Example: You calibrate a 25-mL Class A pipet by dispensing 25 mL of water into a pre-weighed volumetric flask, and “weigh by difference” to measure the mass of water and subsequently determine the actual volume of water. You do this thirty (30) times, and obtain the data shown graphically below.

10 9 8 7 6 5 4 3 2 1

Descriptive Statistics

~ 68% of the measurements -s +s

Approximates σ σ is the standard deviation for the population (“an infinite set of data”)

24.0 24.1 24.2 24.3 24.4 24.5 24.6 24.7 24.8 24.9 25.0 25.1 25.2 25.3 25.4 25.5 25.6 25.7 25.8 25.9 26.0 26.1 26.2 26.3 26.4 26.5 26.6 26.7 26.8 26.9 27.0

Number

Standard Deviation (s) “measures how closely the data are clustered about the mean” Small s = Precise 10 9 8 7 6 5 4 3 2 1

Example: You calibrate a 25-mL Class A pipet by dispensing 25 mL of water into a pre-weighed volumetric flask, and “weigh by difference” to measure the mass of water and subsequently determine the actual volume of water. You do this thirty (30) times, and obtain the data shown graphically below.

Median = middle number in a series of measurements (ordered low to high); less prone to outliers than mean. 24.7, 24.9, 25.0, 25.0, 25.1, 25.1, 25.1, 25.2, 25.2, 25.2, 25.2, 25.2, 25.3, 25.3, 25.3, 25.3, 25.3, 25.3, 25.3, 25.4, 25.4, 25.4, 25.4, 25.5, 25.5, 25.6, 25.6, 25.7, 25.8, 26.5

24.0 24.1 24.2 24.3 24.4 24.5 24.6 24.7 24.8 24.9 25.0 25.1 25.2 25.3 25.4 25.5 25.6 25.7 25.8 25.9 26.0 26.1 26.2 26.3 26.4 26.5 26.6 26.7 26.8 26.9 27.0

Number

Example: You calibrate a 25-mL Class A pipet by dispensing 25 mL of water into a pre-weighed volumetric flask, and “weigh by difference” to measure the mass of water and subsequently determine the actual volume of water. You do this thirty (30) times, and obtain the data shown graphically below.

Descriptive Statistics

= 25.3 + 25.3 2

= 25.3

Descriptive Statistics Range = difference between the lowest and highest Data Set 1 24.7, 24.9, 25.0, 25.0, 25.1, 25.1, 25.1, 25.2, 25.2, 25.2, 25.2, 25.2, 25.3, 25.3, 25.3, 25.3, 25.3, 25.3, 25.3, 25.4, 25.4, 25.4, 25.4, 25.5, 25.5, 25.6, 25.6, 25.7, 25.8, 26.5 Range = 26.5 - 24.7 = 1.8 Range = 25.8 - 24.7 = 1.1

Outlier?

Data Set 2 24.2, 24.3, 24.4, 24.5, 24.6, 24.7, 24.8, 24.9, 25.0, 25.1, 25.1, 25.2, 25.2, 25.3, 25.3, 25.3, 25.4, 25.4, 25.5, 25.5, 25.6, 25.7, 25.8, 25.9, 26.0, 26.1, 26.2, 26.3, 26.4, 26.5 Range = 26.5 - 24.2 = 2.3

Descriptive Statistics Standard Deviation (s) “measures how closely the data are clustered about the mean” s

=

Σi(xi - x)2 n-1

*Note: “n-1” is referred to as the “degrees of freedom”

“Degrees of Freedom”

=

The pieces of independent information available.

…we know average (so not “available”), so only n-1 available.

5

Descriptive Statistics

Descriptive Statistics Standard Deviation (s) “measures how closely the data are clustered about the mean”

Example: You calibrate a 25-mL Class A pipet by dispensing 25 mL of water into a pre-weighed volumetric flask, and “weigh by difference” to measure the mass of water and subsequently determine the actual volume of water. You do this thirty (30) times, and obtain the data shown graphically below.

(30)-1 =0.32156228 =0.3

Number

s = (24.7-11.3)2+(24.9-11.3)2+(25.0-11.3)2+(25.0-11.3)2+…

10 9 8 7 6 5 4 3 2 1

The mean and standard deviation end at the same decimal place!

25.3 ± 0.3 mg

Descriptive Statistics

Descriptive Statistics

s

Σi(xi - x)2 n-1 = 0.63278258 = 0.6

Variance “square of the standard deviation”

=

25.3 ± 0.6 mg

s2 Not as precise!

%CV =

x 100

x s

μ

y =

-σ +σ

~ 95% of the measurements

1

e-(x-μ)

2/2σ2

σsqrt2pie x-μ σ (Table 4-1: Area)

z

+2 s

y 24.0 24.1 24.2 24.3 24.4 24.5 24.6 24.7 24.8 24.9 25.0 25.1 25.2 25.3 25.4 25.5 25.6 25.7 25.8 25.9 26.0 26.1 26.2 26.3 26.4 26.5 26.6 26.7 26.8 26.9 27.0

Number

s/x

Gaussian Curve and Probability

Standard Deviation (s) “measures how closely the data are clustered about the mean” 25.3 ± 0.6 mg ~95% of the data

-2 s

Σi(xi - x)2 n-1

(“Relative Standard Deviation”)

Descriptive Statistics

10 9 8 7 6 5 4 3 2 1

=

Coefficient of Variation (%CV)

68% of the data

24.0 24.1 24.2 24.3 24.4 24.5 24.6 24.7 24.8 24.9 25.0 25.1 25.2 25.3 25.4 25.5 25.6 25.7 25.8 25.9 26.0 26.1 26.2 26.3 26.4 26.5 26.6 26.7 26.8 26.9 27.0

Number

Standard Deviation (s) “measures how closely the data are clustered about the mean” 10 9 8 7 6 5 4 3 2 1

~68% of the data

0.3 0.3

24.0 24.1 24.2 24.3 24.4 24.5 24.6 24.7 24.8 24.9 25.0 25.1 25.2 25.3 25.4 25.5 25.6 25.7 25.8 25.9 26.0 26.1 26.2 26.3 26.4 26.5 26.6 26.7 26.8 26.9 27.0

24.7, 24.9, 25.0, 25.0, 25.1, 25.1, 25.1, 25.2, 25.2, 25.2, 25.2, 25.2, 25.3, 25.3, 25.3, 25.3, 25.3, 25.3, 25.3, 25.4, 25.4, 25.4, 25.4, 25.5, 25.5, 25.6, 25.6, 25.7, 25.8, 26.5

=

≈

x-x s

-∞

x

+∞

6

Descriptive Statistics z ≈ 24.7-25.3 0.3 z≈2 Area from μ to z=2 = 0.4773

-s = -0.3

*Note: Area from μ to - ∞ = 0.5000

+s = +0.3

Area from z=2 to - ∞ = 0.5000-0.4773 =0.0227 =2.27%

Probability of ≤ 24.7?

Statistical Tests

Rejection of Measurements….

Q-Test for “Bad Data”

In the case where you suspect something went wrong, you use the Q-test. (Minimum of 3 measurements)

Qcalculated

Q-test = |suspect value - nearest value| / total range Calculate Qexp. If Qexp ≥ Qcritical then reject.

=

gap/range

Range = “total spread of the data” Gap =

d X1 X2 X3 X4 X5

(from Table 4.1)

24.0 24.1 24.2 24.3 24.4 24.5 24.6 24.7 24.8 24.9 25.0 25.1 25.2 25.3 25.4 25.5 25.6 25.7 25.8 25.9 26.0 26.1 26.2 26.3 26.4 26.5 26.6 26.7 26.8 26.9 27.0

Number

Mean ( x ) = 25.3 10 9 8 7 6 5 4 3 2 1

“difference between questionable point and nearest value”

X6

Qexp = d/w

If Qcalculated > Qtable then data point should be discarded.

w

Q-Test for “Bad Data”

10 9 8 7 6 5 4 3 2 1

“Bad” data? “Outlier”

Qcalculated

=

gap/range

24.7, 24.9, 25.0, 25.0, 25.1, 25.1, 25.1, 25.2, 25.2, 25.2, 25.2, 25.2, 25.3, 25.3, 25.3, 25.3, 25.3, 25.3, 25.3, 25.4, 25.4, 25.4, 25.4, 25.5, 25.5, 25.6, 25.6, 25.7, 25.8, 26.5 gap Qcalculated =

(26.5-25.8) (26.5-24.7)

24.0 24.1 24.2 24.3 24.4 24.5 24.6 24.7 24.8 24.9 25.0 25.1 25.2 25.3 25.4 25.5 25.6 25.7 25.8 25.9 26.0 26.1 26.2 26.3 26.4 26.5 26.6 26.7 26.8 26.9 27.0

Number

Example: You calibrate a 25-mL Class A pipet by dispensing 25 mL of water into a pre-weighed volumetric flask, and “weigh by difference” to measure the mass of water and subsequently determine the actual volume of water. You do this thirty (30) times, and obtain the data shown graphically below.

= 0.7/1.8 = 0.4 0.4 > 0.298

Discard?

n 3 4 5 6 7 8 9 10 15 20 25 30

Qtable (95% confidence)

0.970 0.829 0.710 0.625 0.568 0.526 0.493 0.466 0.384 0.342 0.317 0.298

7

We can assign a confidence level to our measurements….. If we can accept a 5% error level, we can say that these values are reported with a 95% confidence limit. For a small number of measurements, we must consult a t value table () • choose confidence level • determine number of degrees of freedom (n-1) • plug t value into the following equation:

Rectangular Distribution

Frequency

Frequency

Bimodal Distribution

49

50

51

52

53

54

55

56

57

58

59

60 49

Measurements

50

51

52

53

54

55

56

57

Measurements

Systematic Error

Frequency

Frequency

Skewed Distribution

46

48

50

52

Measurements

54

56

-4

-2

0

2

4

6

8

Measurements

8