I Introductory MaterialA. Mathematical Concepts

Scientific Notation andSignificant Figures

• Scientific Notation – A method devised to express very small and very large numbers.

– Based on powers of 10 e.g 123 billion is 123,000,000,000 expressed in scientific notation = 1.23 x 10 11

– 1.23 is called the coefficient•It must be greater than 1 and less than 10

– 10 11 is called the base. It is a power of 10. In this example the power of 10 is the 11

• Significant Figures– Tells how well known a measurement is.

– Using Significant Figures is a way to express the error in a measurement.

– The more significant figures the more accurate the measurement.

– The last number listed is the value that is uncertain•e.g.

– 2.3 value known to units, estimated to tenths

– 2.34 value known to tenths, estimated to hundredths

• Determining the number of Significant Figures– All non - zero numbers are significant– Zeros within a number are significant

•Both 40.05 and 3201 have 4 significant figures

– Zeros used to set the decimal point are NOT significant.•0.00235

– 3 significant figures

•47,000– 2 significant figures

» If you want it to have more, use scientific notation: 4.7000 x 104 denotes 5 significant figures

» Other possibility: 47,000. is sometimes used to denote 5 significant figures.

• Types of Numbers– Exact

•Numbers from counting•Conversions in the same system of measurement

•Defined conversions between different systems of measurement

•Have an infinite number of significant figures

– Inexact•Numbers from measurements•Conversions between different systems of measurement

•Have a finite number of significant figures

• Working with Significant Figures– When doing calculations, you can not gain or lose precision. The answer can only be as good as the least precise number

– Multiplication and Division•The number of significant figures in the answer depends on the number with the least number of significant digits.

•Determine the number of significant figures in: (16.300)(0.00235)/1.6300 – can only have 3 digits in the answer.

Determine the correct answer for:

(4.10)(3.023 x 109)/(1.5)

1. 8.26 x 109

2. 8.3 x 109

3. 8.263 x 109

4. 8.2628 x 109

– Addition and subtraction•The number of significant figures in the answer depends on the number with the least number of significant decimal places.

•Determine the answer: 4.72 + 0.653 + 9.2– 9.2 is only known to the tenths place, so the answer can only be known to the tenths place

4.72 0.653 9.214.573

14.6

What is the sum of 9 x 103 + 7.2 x 102?

A) 1 x 104

B) 9.7 x 103

C) 1.0 x 104

D) 9.72 x 103

9 x 103 known only to the thousands place

7.2 x 102 known to the tens place, so answer can only be known to the thousands place.

9000 720 9720 rounds to 1000010 000How many significant digits do you

have?210. x 103 = 1.0 x 104

• Rounding– When the answer contains too many digits, (more than the number of significant digits) the answer must be rounded to the correct number of significant digits.

– General Rules: If the digit after the last significant digit is•0 - 4, the significant digit stays the same

– 15.635 to the tenths place is 15.6

•6 - 9, the significant digit goes up– 15.625 to the units place is 16

•5, the significant digit goes to an even number– 15.625 to the hundredths place is 15.62 – While 15.635 to the hundredths place is 15.64– The reason for this is that in a series of calculations any error will be averaged out.

Error Analysis• Types of Error

– Determinate Error (ED)• Source of the error is known and can be determined.

• e.g. a watch that is ten minutes fast• Determinate error will most often effect accuracy

• It is the difference between the true value and the measured value

– Indeterminate Error (EI)• Source of the error is unknown or not controllable

• e.g. the vibration in a balance table• Indeterminate error will most often effect precision

• Can be expressed as:– Standard deviation – Variance– Coefficient of Variation– Tolerances

– Total Error• Sum of all the determinate and indeterminate error

• If the true value is known, this can be expressed as the percent relative error: %ER

• Describing Indeterminate Error– Population

• Complete set of individual data points in a study– The female population of the United states

– Sample• A specific subset of the population that is assumed to be representative of the whole– The female population of Newark, De.

– Members of the Data Set• Individual items (xi) within the sample of a known number (N) of items

% ER X

XtrueXtrue

x 100

• Measuring the Central Tendency of a Population or Sample– Mean

•Average value of all the members in the set•A population mean is the true or expected value; it is represented by the symbol µ

•A sample mean is the calculated average of the sample population; it is represented by the symbol x

•When a sample is used, we assume that x is a good approximation of µ

– Median•The middle value of all the members in the set

– Mode•The value that appears with the greatest frequency in the set

• Measuring the Spread of Data– Range

• Difference between the highest (largest) value and the lowest (smallest) value in the set

• Range = xHi - xLo

– Standard deviation( for population; s for sample)• Measure of how closely the individual values are clustered around the mean

– Variance• Square of the standard deviation• Variance = s2

s

X

Xi

2

i 1

n

n 1

– Accuracy•Closeness of population mean and sample mean

– Precision•Closeness of individual data points to each other

•Measured by range or standard deviation

• Determining and Describing Indeterminate Error (Uncertainty Statistics)– Standard deviationPopulation Sample

– Where xi is the data point, x is the sample mean, n is the number of data points

– Variance• s2

X

Xi

2

i 1

n

n

s

X

Xi

2

i 1

n

n 1

– Relative standard deviation(RSD) and Coefficient of Variation (CV)•Coefficient of Variation is a measure of the dispersion of a distribution

•It allows us to compare populations or samples that have different mean values

– Standard deviation of the mean,sm

RSD s

X

CV s

X x 100

sm s

n

• Ways of Representing DataData Table

• Ways of Representing DataData Table

DATA118.2

120.7

119.4

121.3

118.6

119.7

120.2

114.8

121.2

119.3

Collection of 10 data points

Histograma tabulation of the frequency of each measurement

• How do you decide if all you data should be used or is some can be ignored?

• Sort the data from low to highDATA

114.8

118.2

118.6

119.3

119.4

119.7

120.2

120.7

121.2

121.3

Look at the Range of the data:114.4 - 121.3Difference = 6.5

Apply Q TestLook at the gap between the smallest two numbers (Low Gap) and the largest two numbers (High Gap).Low Gap = 118.2 - 114.8 = 3.4High Gap = 121.3 - 121.2 = 0.1

1. Calculate the Gap to Range Ratio

Qcalc GapRange

Qcalc Low3.4

6.50.523

Qcalc High 0.1

6.50.015

2. Compare Qcalc to Q from the table (90% confidence)

Q Number of data points

0.76 4

0.64 5

0.56 6

0.51 7

0.47 8

0.44 9

0.41 10

If Qcalc < Qtable data CAN NOT be rejected

Qlo = 0.523Qhi = 0.016Qtable = 0.41 for 10 data points

So: Qlo > Qtable so can reject 114.8

Always make sure you mention the method used to reject data

Calculate mean, deviation and RSD

x = 119.8; s = 1.1; RSD = 0.009

118.2

118.6

119.3

119.4

119.7

120.2

120.7

121.2

121.3

Data now consists of 9 points.

x = 119.8

s

X

Xi

2

i 1

n

n 1

1.6

1.2

0.5

0.4

0.1

-0.4

-0.9

-1.4

-1.5

X - Xi(X - Xi)2

2.56

1.44

0.25

0.16

0.01

0.16

0.81

1.96

2.25

s

9.6

8ÊÊÊÊ1.095ÊÊ1.1

RSD s

X

Ê1.1

119.8ÊÊ0.009

x = 119.8

s = 1.1

RSD = s/ x = 0.009

119.8 ± 1.1119.8 ± 1 68.3 % of data in this range

119.8 ± 2

119.8 ± 3

95.8 % of data in this range99.7 % of data in this range

Confidence Limit• How certain you are that the true value [population mean] (µ) lies within a range around the sample mean(x)

• Confidence Limits are usually calculated at 90%, 95% or 99%

• Confidence Limits are calculated using the Student t table

DF Confidence

Limit

90% 95% 99%

1 6.31 12.7 63.7

2 2.92 4.3 9.92

3 2.35 3.18 5.84

4 2.13 2.78 4.60

5 2.02 2.57 4.03

6 1.94 2.45 3.71

7 1.90 2.36 3.50

8 1.86 2.31 3.36

10 1.81 2.23 3.17

15 1.75 2.13 2.95

20 1.73 2.09 2.85

40 1.68 2.02 2.70

inf 1.65 1.96 2.58

DF = degrees of freedomIt is equal to n - 1Where n is the number of data points

To calculate the Confidence Limit

CL() X t s

n

From our data, for the 9 data pointsx = 119.8; s = 1.1; DF = 8, t = 2.31

CL() 119.8 (2.31)(1.1)

9CL() 119.8 0.8

at 95% confidence:

119.0 120.6