Dealing with NumbersDealing with Numbers

A guide to Numerical & A guide to Numerical & Graphical MethodsGraphical Methods

1.0 The Importance of 1.0 The Importance of ExperimentsExperiments

Scientists and engineers spend a lot of time Scientists and engineers spend a lot of time performing experiments. Why ?performing experiments. Why ?

1.1. They form the basis for scientific and technical They form the basis for scientific and technical advances.advances.

2.2. They allow theory to be “put to the test”.They allow theory to be “put to the test”.3.3. They may reveal new, unexpected effects leading They may reveal new, unexpected effects leading

to new or modified theoretical models or to new or modified theoretical models or explanations.explanations.

In the case of students in a VCE class, experiments In the case of students in a VCE class, experiments are unlikely to break new ground, but they do are unlikely to break new ground, but they do provide you with the opportunity to acquire:provide you with the opportunity to acquire:KnowledgeKnowledgeSkillsSkillsUnderstanding Understanding

through investigating the “real world”through investigating the “real world”

1.1 Experimental Results – 1.1 Experimental Results – The DataThe Data

OK you’ve done an experiment and collected some OK you’ve done an experiment and collected some results.results.What are the important features of the data you have What are the important features of the data you have collected ?collected ?

Measurements made or taken during an experiment Measurements made or taken during an experiment generate “raw” data.generate “raw” data.This data must be recorded then presented and This data must be recorded then presented and analysed.analysed.

All data will have some uncertainty attached.All data will have some uncertainty attached.It doesn’t matter how good the experimenter, how well It doesn’t matter how good the experimenter, how well designed the experiment or how sophisticated the designed the experiment or how sophisticated the measuring device, ALL collected data has some measuring device, ALL collected data has some uncertainty.uncertainty.

(27.5 (27.5 ±± 0.5) 0.5)00CCThis statement of temperature indicates both its This statement of temperature indicates both its measured value and the uncertainty.measured value and the uncertainty.The temperature could be anywhere between The temperature could be anywhere between 27.5 – 0.5 = 27.027.5 – 0.5 = 27.000 and 27.5 + 0.5 = 28.0 and 27.5 + 0.5 = 28.000

1.2 Uncertainty 1.2 Uncertainty

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The uncertainty of the measurement The uncertainty of the measurement is determined by the scale of the is determined by the scale of the measuring device.measuring device.

The uncertainty quantifies (gives a number to) The uncertainty quantifies (gives a number to) the amount of variation that has been found in a the amount of variation that has been found in a measured value.measured value.An alternative term to that of uncertainty is to use An alternative term to that of uncertainty is to use the term EXPERIMENTAL ERROR.the term EXPERIMENTAL ERROR.This does NOT imply a mistake in your results, This does NOT imply a mistake in your results, but simply the natural spread in the values of a but simply the natural spread in the values of a repeatedly measured quantity.repeatedly measured quantity.

Uncertainty generally comes in three forms:Uncertainty generally comes in three forms:Resolution Uncertainty – how fine is the scale on the measuring Resolution Uncertainty – how fine is the scale on the measuring

device ?device ?Calibration Uncertainty – how well does the measuring device conform Calibration Uncertainty – how well does the measuring device conform

to the standard ?to the standard ?Reading Uncertainty - how well did the operator use the device ?Reading Uncertainty - how well did the operator use the device ?

1.3 Systematic and Random1.3 Systematic and Random UncertaintyUncertainty

Each form of uncertainty can have Each form of uncertainty can have 2 categories: 2 categories:

1.1. Systematic Uncertainty – can Systematic Uncertainty – can exist without the experimenters exist without the experimenters knowledge.knowledge.

Can skew all readings or values Can skew all readings or values one way. one way.

Mostly due to instruments Mostly due to instruments rather than humans.rather than humans.

2.2. Random Uncertainty – Random Uncertainty – produces scatter in produces scatter in measurements. measurements.

Environmental factors often Environmental factors often cause this type of error. cause this type of error.

Mostly due to humans rather Mostly due to humans rather than instruments.than instruments.

Elimination of these Elimination of these “experimental errors” is the “experimental errors” is the “holy grail” of experimental “holy grail” of experimental scientists and engineers.scientists and engineers.Systematic uncertainties can be Systematic uncertainties can be reduced or eliminated from the reduced or eliminated from the measuring device by measuring device by “calibrating” (comparing to a “calibrating” (comparing to a known standard) known to a high known standard) known to a high degree of both accuracy and degree of both accuracy and precision.precision.Random uncertainties can be Random uncertainties can be controlled (but not eliminated) by controlled (but not eliminated) by taking multiple readings and taking multiple readings and using statistical analysis on the using statistical analysis on the collected results.collected results.

1.4 Precision1.4 PrecisionPrecision is a measure of how closely a group of measurements agree Precision is a measure of how closely a group of measurements agree with one another.with one another.Close agreement translates to a small uncertainty.Close agreement translates to a small uncertainty.However, precision DOES NOT mean that the measurements are close to However, precision DOES NOT mean that the measurements are close to the “true value”. the “true value”.

An example here should explain:An example here should explain:The “true value” on a dart board is the bullseye.The “true value” on a dart board is the bullseye.

This player is precise - all This player is precise - all darts fall within a small area darts fall within a small area (small uncertainty) – but he is (small uncertainty) – but he is certainly not accuratecertainly not accurate

A player throws 5 dartsA player throws 5 darts

1.5 Accuracy1.5 AccuracyAccuracy is how closely the measurements agree with the true value.Accuracy is how closely the measurements agree with the true value.

Again using the darts analogy: Again using the darts analogy:

This player is BOTH accurate AND precise.This player is BOTH accurate AND precise.

What can you say about the following What can you say about the following measurements ? Each dot represents one measurements ? Each dot represents one person’s attempt to measure the length of person’s attempt to measure the length of a piece of stringa piece of string

True True ValueValue

InaccurateInaccurateImpreciseImprecise PrecisePrecise ImpreciseImprecise PrecisePrecise

InaccurateInaccurate AccurateAccurate AccurateAccurate

1.6 Significant Figures1.6 Significant FiguresSignificant Figures can be regarded as another method of indicating Significant Figures can be regarded as another method of indicating the uncertainty in a measured quantity.the uncertainty in a measured quantity.

Significant Figures – THE RULES:Significant Figures – THE RULES:1. All NON ZERO integers are significant.1. All NON ZERO integers are significant.

2. Zeros2. Zeros(a) Captive Zeros – (a) Captive Zeros – they fall between two non zero numbers they fall between two non zero numbers

they always count as significant they always count as significant figures.figures.

(b) Decimal Point Zeros – Zeros used to place a decimal (b) Decimal Point Zeros – Zeros used to place a decimal point are NOT significant.point are NOT significant.(c) Trailing Zeros – (c) Trailing Zeros – any zeros following a decimal point are any zeros following a decimal point are significant.significant.

Number 12.5 0.003002 49,000 0.000234 123.00

Significant Figures

3 4 2 3 5

1.7 Significant Figure 1.7 Significant Figure ManipulationManipulation

1.1. ADDITION & SUBTRACTION:ADDITION & SUBTRACTION:When adding and subtracting When adding and subtracting numbers, round the result of the numbers, round the result of the calculation to the same number of calculation to the same number of decimal places as the number with decimal places as the number with the fewest decimal places used in the fewest decimal places used in the calculation. the calculation.

16.5416.54 8.2698.269 0.470.47+21.1+21.1

46.37946.379

Rounding to the least number Rounding to the least number of decimal places of those of decimal places of those numbers added (21.1 with 1 numbers added (21.1 with 1 decimal place). decimal place).

2. 2. MULTIPLICATION AND MULTIPLICATION AND DIVISION:DIVISION: Identify the number in the Identify the number in the calculation with the least calculation with the least

number of significant figures. number of significant figures. Give your answer to the same Give your answer to the same number of sig figs. number of sig figs.

65.64 (32.787 + 98.443)65.64 (32.787 + 98.443)

56.456.4

= 152.729383= 152.729383

With 56.4 having 3 sig figs, With 56.4 having 3 sig figs, the answer should have 3 sig the answer should have 3 sig figsfigs

Answer = 153Answer = 153

Answer = 46.4Answer = 46.4

1.8 Scientific Notation1.8 Scientific NotationIt is not always clear how many It is not always clear how many figures in a number are significant. figures in a number are significant. By changing the unit in which a By changing the unit in which a number is expressed it can appear number is expressed it can appear that the amount of significant that the amount of significant numbers changes.numbers changes.For example a time measurement For example a time measurement could be 125 sec. could be 125 sec. Writing this time in milliseconds Writing this time in milliseconds would give 125,000 ms.would give 125,000 ms.Both numbers have 3 significant Both numbers have 3 significant figures.figures.

However, say somebody asks for the However, say somebody asks for the time measurement in ms and assumes time measurement in ms and assumes (incorrectly) that our measuring (incorrectly) that our measuring device is accurate to within device is accurate to within ±1 ms, ±1 ms, then the time would be seen as a 6 then the time would be seen as a 6 significant number. significant number.

To get around this problem, Scientific Notation can be To get around this problem, Scientific Notation can be used. used. This has all numbers expressed as a “number between 1 This has all numbers expressed as a “number between 1 and 10, multiplied by a power of 10” and 10, multiplied by a power of 10” The time 125,000 ms becomes 1.25 x 10The time 125,000 ms becomes 1.25 x 1055 sec and now sec and now only the numbers to the left of the multiply (x) sign are only the numbers to the left of the multiply (x) sign are significant.significant.

1.9 Orders of Magnitude1.9 Orders of MagnitudeWhen performing experiments, When performing experiments, such as measuring the distance to such as measuring the distance to the stars, determining the strength the stars, determining the strength of gravity or measuring the speed of gravity or measuring the speed of cars passing the college, you of cars passing the college, you expect to gets answers within a expect to gets answers within a certain range.certain range.

If your measurements and If your measurements and subsequent calculations gave subsequent calculations gave answers for g of 99 Nkganswers for g of 99 Nkg-1-1 or or speeds of 400 kmhspeeds of 400 kmh-1-1 hopefully hopefully you would suspect your you would suspect your calculations or calculations or measurements.measurements.

The ability to make an estimate of an The ability to make an estimate of an expected answer at least to within a factor expected answer at least to within a factor of 10 can often save an embarrassing and of 10 can often save an embarrassing and career threatening mistake.career threatening mistake.This ability is called knowing an answer to This ability is called knowing an answer to within an “order of magnitude”.within an “order of magnitude”.

For gravity you would expect to get an answer in the For gravity you would expect to get an answer in the range 9.7 to 9.9 Nkgrange 9.7 to 9.9 Nkg-1-1

For the speeds of the cars maybe a range between 40 to For the speeds of the cars maybe a range between 40 to 80 kmh80 kmh-1-1..

Chapter 2Chapter 2

Mathematical ProcessesMathematical Processes

2.0 Rounding2.0 RoundingWhen the result of a calculation has too many figures, When the result of a calculation has too many figures, which normally happens when using a calculator, you may which normally happens when using a calculator, you may need to reduce the number of figures that appear in the need to reduce the number of figures that appear in the answer, so that it is becomes both meaningful and answer, so that it is becomes both meaningful and acceptable.acceptable.

For example, you are asked to measure the length of a thigh For example, you are asked to measure the length of a thigh bone (femur) from a skeleton and put that measurement into bone (femur) from a skeleton and put that measurement into a formula to calculate the height of the person before death.a formula to calculate the height of the person before death.

Since the original measurement had 2 Since the original measurement had 2 significant figures, the answer you quote significant figures, the answer you quote should be no more that 2 sig figs.should be no more that 2 sig figs.Thus the height of the person was 2.1 m. Thus the height of the person was 2.1 m.

The process of The process of reducing the number reducing the number of significant figures is of significant figures is called ROUNDING the called ROUNDING the number.number.

When a calculation has a number of steps don’t round When a calculation has a number of steps don’t round until you get to your final answer, as rounding during the until you get to your final answer, as rounding during the calculation could lead to large errors in the final answer.calculation could lead to large errors in the final answer.

You do this and the calculator gives you an answer You do this and the calculator gives you an answer of 2.064655089. Your original measurement for the of 2.064655089. Your original measurement for the femur was 0.33 mfemur was 0.33 m

2.1 The Mean2.1 The Mean

Speed of Sound (ms-1)

341.5 342.4 342.2 345.5 341.1 338.5 340.3 342.7

A team of students collected the following data in A team of students collected the following data in an experiment aimed at finding the Speed of an experiment aimed at finding the Speed of Sound. Sound.

x = x = 2734.22734.2 88 = 341.775 ms= 341.775 ms-1-1

To determine the average or MEAN (usually labelled as x) of these values:To determine the average or MEAN (usually labelled as x) of these values: add them and divide by the number of measurements: add them and divide by the number of measurements:

How many Significant Figures should the Mean be quoted to ?How many Significant Figures should the Mean be quoted to ?The data has 4, so the mean should also have 4, right ? The data has 4, so the mean should also have 4, right ?

So, in this case the Mean or So, in this case the Mean or Average speed for sound on Average speed for sound on this day was 341.8 msthis day was 341.8 ms-1-1

Is there an uncertainty in the Mean ? If so, how is it calculated ?Is there an uncertainty in the Mean ? If so, how is it calculated ?

2.2 Uncertainty in the Mean2.2 Uncertainty in the MeanWhat is the uncertainty What is the uncertainty associated with the calculated associated with the calculated speed of sound of 342.8 msspeed of sound of 342.8 ms-1-1 ? ?

To calculate the uncertainty in To calculate the uncertainty in the mean:the mean:

1.1. Calculate the range of values (largest – smallest)Calculate the range of values (largest – smallest)2.2. Divide the range by the number of termsDivide the range by the number of terms

Thus uncertainty = Thus uncertainty = 345.5 – 338.5345.5 – 338.588

= 0.875 ms= 0.875 ms-1-1

Since uncertainties are about Since uncertainties are about determining the probable range determining the probable range of a measured or calculated of a measured or calculated quantity, there is little use in quantity, there is little use in quoting them to any more than 1 quoting them to any more than 1 Significant Figure. Significant Figure. So the uncertainty here So the uncertainty here becomes becomes ± 0.9 ms± 0.9 ms-1-1 (NOTE: if the first number of the (NOTE: if the first number of the uncertainty is a 1, then quote to uncertainty is a 1, then quote to two sig figs., so an uncertainty two sig figs., so an uncertainty of of ±±1.425 becomes 1.425 becomes ± ± 1.4)1.4)

Thus, the speed of sound, and its Thus, the speed of sound, and its associated uncertainty, as associated uncertainty, as determined by the students is determined by the students is (342.8 (342.8 ± 0.9) ms± 0.9) ms-1-1

2.3 Fractional and Percentage 2.3 Fractional and Percentage UncertaintyUncertainty

The function of uncertainties is to quantify the The function of uncertainties is to quantify the probable probable range of the values of the measured quantity.range of the values of the measured quantity.

Thus it is usual to quote uncertainty to, at the most, 2 Thus it is usual to quote uncertainty to, at the most, 2 significant figures and often only 1 significant figure.significant figures and often only 1 significant figure.

For the speed of sound -For the speed of sound - (342.8 (342.8 ± 0.9) ms± 0.9) ms-1-1

FRACTIONAL UNCERTAINTY = FRACTIONAL UNCERTAINTY = Uncertainty in QuantityUncertainty in Quantity

Value of QuantityValue of Quantity = = 0.90.9

342.8342.8 = 0.0026= 0.0026

NOTE: Fractional Uncertainty has NO unitsNOTE: Fractional Uncertainty has NO units

PERCENTAGE UNCERTAINTY = Fractional Uncertainty x 100PERCENTAGE UNCERTAINTY = Fractional Uncertainty x 100 = 0.0026 x 100 = 0.0026 x 100 = 0.26%= 0.26%

2.4 Combining Uncertainties2.4 Combining UncertaintiesIn experiments often you will In experiments often you will collect two or more sets of collect two or more sets of data which need to be used in data which need to be used in an equation to calculate a final an equation to calculate a final result. result.

The uncertainties in each of the The uncertainties in each of the pieces of data will affect the final pieces of data will affect the final result in a process called result in a process called error error propagation.propagation.

Uncertainties in measured or Uncertainties in measured or calculated quantities are quoted calculated quantities are quoted in a number of ways:in a number of ways:If the quantity measured is a If the quantity measured is a Volume (V), its uncertainty could Volume (V), its uncertainty could be quoted as be quoted as ∂V or ∆V or ∂V or ∆V or σσVV

Mathematical OperationsMathematical Operations::1.1. Sums and DifferencesSums and Differences

If V = a + bIf V = a + b or V = a - bor V = a - bthen ∂V = ∂a + ∂bthen ∂V = ∂a + ∂b

In words, the uncertainty In words, the uncertainty in V is the sum of the in V is the sum of the uncertainties of a and b uncertainties of a and b

Mathematical Operations:Mathematical Operations:Products and QuotientsProducts and Quotients

If V = a x bIf V = a x b or V = a / bor V = a / b

In words, the fractional In words, the fractional uncertainty in V equals the uncertainty in V equals the sum of the fractional sum of the fractional uncertainties in a and buncertainties in a and b

then then ∂V∂V = = ∂a∂a + + ∂b∂b V a bV a b

2.5 Data Selection2.5 Data SelectionA vital question for all experimental A vital question for all experimental scientists and engineers is:scientists and engineers is:Are ALL my data equal ?Are ALL my data equal ?

For many investigators ALL data For many investigators ALL data is valid and NONE can ever be is valid and NONE can ever be rejected.rejected.While others can simply look at a While others can simply look at a set of data and label it as set of data and label it as spurious and reject the lot.spurious and reject the lot.And there are yet others who can And there are yet others who can look at individual data points and look at individual data points and reject them whilst keeping the reject them whilst keeping the rest.rest.

Confidence in the “correctness” of Confidence in the “correctness” of experimental data really comes when you experimental data really comes when you are satisfied that the experiment is are satisfied that the experiment is repeatable. repeatable. If you do have a suspect data point the If you do have a suspect data point the best thing to do is to repeat the best thing to do is to repeat the experiment.experiment.Of course this is not always possible, Of course this is not always possible, especially when testing to destruction, especially when testing to destruction, as in breaking a wire or bursting a as in breaking a wire or bursting a balloon.balloon.

Statistical tests which help eliminate “spurious” data do exist, but Statistical tests which help eliminate “spurious” data do exist, but their rigid and unquestioning application to all data may mask a trend their rigid and unquestioning application to all data may mask a trend that you should know about.that you should know about.

There are situations where a data point There are situations where a data point may be neglected or rejected. For may be neglected or rejected. For example, during a series of events being example, during a series of events being hand timed, the operator lost hand timed, the operator lost concentration during one of the events. concentration during one of the events.

Chapter 3Chapter 3

Graphical MethodsGraphical Methods

3.0 Why Graphs ?3.0 Why Graphs ?A picture is worth a thousand words.A picture is worth a thousand words.Humans generally find it easier to Humans generally find it easier to understand information when presented as understand information when presented as a picture rather than as a table of figures.a picture rather than as a table of figures.

A graph will indicate:A graph will indicate:(a)(a) The range of the measurements takenThe range of the measurements taken

(d) The existence of “outlying data” (d) The existence of “outlying data” (c) The existence or otherwise of trends(c) The existence or otherwise of trends

(b) The uncertainty in each measurement(b) The uncertainty in each measurement

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Data point showing Data point showing error bars for both error bars for both Temperature (vertical) Temperature (vertical) and Time (horizontal)and Time (horizontal)

outlying data pointoutlying data point

X axisX axisAbscissaAbscissa

3.1 Graphs – The Basics3.1 Graphs – The BasicsThe most used graph in The most used graph in science is the Cartesian science is the Cartesian Coordinate Graph, better Coordinate Graph, better known as the x – y graph.known as the x – y graph.The y axis is known as the The y axis is known as the ordinate and the x axis as the ordinate and the x axis as the abscissa.abscissa.

Y axisY axisOrdinateOrdinate

The quantity that is controlled or The quantity that is controlled or deliberately varied throughout the deliberately varied throughout the experiment is the INDEPENDENT experiment is the INDEPENDENT Variable and is plotted on the x Variable and is plotted on the x axisaxis

Independent VariableIndependent Variable

The quantity that varies in The quantity that varies in response to changes in the response to changes in the independent variable is called the independent variable is called the DEPENDENT Variable and is DEPENDENT Variable and is plotted on the y axisplotted on the y axis

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Temperature versus TimeTemperature versus Time

ALL graphs require a TITLE, ALL graphs require a TITLE, and AXIS labels and UNITS and AXIS labels and UNITS

Temp (Temp (00C)C)

Time (Sec)Time (Sec)

3.2 Graphs – Origins, Scales & 3.2 Graphs – Origins, Scales & SymbolsSymbols

On most graphs the numbering On most graphs the numbering of both the axes begins at zero, of both the axes begins at zero, so the bottom left hand corner of so the bottom left hand corner of the graph is the point (0,0) and is the graph is the point (0,0) and is called the ORIGIN.called the ORIGIN.

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ORIGINORIGIN

The scale should be The scale should be chosen so as to allow the chosen so as to allow the graph to fill the whole graph to fill the whole page, while leaving enough page, while leaving enough space for labels units and space for labels units and a titlea title

However there is no law that states However there is no law that states that an origin must be included in a that an origin must be included in a graph.graph.Sometimes including an origin will Sometimes including an origin will produce too coarse a scale which produce too coarse a scale which may hide important information.may hide important information.

Data points (with or Data points (with or without error bars) should without error bars) should be too big rather than too be too big rather than too small so as they cannot be small so as they cannot be mistaken for a smudge on mistaken for a smudge on the pagethe page

Good data pointsGood data points

..

Bad data pointBad data point

3.3 Error Bars & Line Drawing3.3 Error Bars & Line DrawingUncertainties in the quantities Uncertainties in the quantities being graphed are indicated being graphed are indicated by attaching “error bars” to by attaching “error bars” to each of the data points. They each of the data points. They can be vertical, horizontal or can be vertical, horizontal or both. both.

Error bars may:Error bars may:•remain the same size remain the same size for all data points or,for all data points or, •vary in size from data vary in size from data point to data point.point to data point.

Where error bars are very small, Where error bars are very small, due to the scales used, it is due to the scales used, it is advisable to omit them from the advisable to omit them from the graph.graph.TemperatureTemperature

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Their length indicates the size Their length indicates the size of the uncertainty associated of the uncertainty associated with that data point. with that data point.

When connecting data points it is When connecting data points it is difficult to draw freehand “smooth difficult to draw freehand “smooth curve” curve” A rubberised flexible ruler called a A rubberised flexible ruler called a “flexi - curve” is probably the best “flexi - curve” is probably the best way to draw curves through data.way to draw curves through data.

As long as the curve fits within the As long as the curve fits within the error bars, the data has been joined error bars, the data has been joined together in a valid way.together in a valid way.

3.4 Linear Graphs3.4 Linear GraphsIt is hard to determine exact It is hard to determine exact mathematical relationships mathematical relationships from curved graphs. from curved graphs.

Converting the graph to a linear Converting the graph to a linear or “straight line” graph allows or “straight line” graph allows quantitative relationships to be quantitative relationships to be determined.determined.

Linear graphs are important in Linear graphs are important in the analysis of experimental the analysis of experimental data because:data because:

(a)(a) The slope or gradient and y The slope or gradient and y intercept can be calculatedintercept can be calculated

(b)(b) Departure from linearity can Departure from linearity can be easily seenbe easily seen

(c)(c) Outliers are easily identifiedOutliers are easily identified(d)(d) A mathematical relationship A mathematical relationship

between “y” and “x” is easily between “y” and “x” is easily determineddetermined

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Recognising that the Recognising that the temperature – time graph temperature – time graph shown previously indicates an shown previously indicates an inverse relationship (Temp inverse relationship (Temp αα 1/Time) and manipulating the 1/Time) and manipulating the data will give:data will give:

3.5 Line of Best Fit3.5 Line of Best FitTemperatureTemperature

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Is the red line the only line Is the red line the only line that can be drawn to join the that can be drawn to join the data points ?data points ?Obviously not, other lines Obviously not, other lines can be drawn.can be drawn.

Is the red line the “best” line Is the red line the “best” line to join the data ?to join the data ?

Yes, because it meets the Yes, because it meets the criteria for a “line of best fit”.criteria for a “line of best fit”.It passes through all the error It passes through all the error bars.bars.It has as many data points It has as many data points above the line as below and above the line as below and the distances above and below the distances above and below total about the same.total about the same.

Rules for drawing a Line of Best Fit:Rules for drawing a Line of Best Fit:1. Place a clear plastic ruler over the data 1. Place a clear plastic ruler over the data points.points.2. Move the ruler until the data points are 2. Move the ruler until the data points are equally placed above and below the straight equally placed above and below the straight edge.edge.3. Generally the origin is not a special point, 3. Generally the origin is not a special point, don’t force the line through it.don’t force the line through it.4. Use a pencil to draw a fine line along the 4. Use a pencil to draw a fine line along the straight edge.straight edge.

3.6 Determining Relationships3.6 Determining RelationshipsLinearising the relationship between variables allows you to use the Linearising the relationship between variables allows you to use the general equation for a straight line (y = mx + c) to determine the general equation for a straight line (y = mx + c) to determine the mathematical law which relates the variables.mathematical law which relates the variables.

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In this case:In this case:y = Temperature (y = Temperature (ooC)C)m = Slope of Graphm = Slope of Graphx = 1/Time (sec)x = 1/Time (sec)c = Temperature axis interceptc = Temperature axis intercept

Slope = Rise/RunSlope = Rise/Run = (75 – 5)/(5 x 10= (75 – 5)/(5 x 10-2 -2 - 0) - 0) = 1400 = 1400 = 1.4 x 10= 1.4 x 1033

RunRun

RiseRise

c = +5c = +5 Thus: Temp = 1.4 x 10Thus: Temp = 1.4 x 1033 (1/Time) + 5 (1/Time) + 5

3.7 Interpolation & Extrapolation3.7 Interpolation & Extrapolation

When the “y” or “x” value falls When the “y” or “x” value falls within the range of known data within the range of known data points INTERPOLATION is points INTERPOLATION is occurring.occurring.

Determining a value of a Determining a value of a variable (y and/or x) outside the variable (y and/or x) outside the range of those already known, range of those already known, EXTRAPOLATION is occurring.EXTRAPOLATION is occurring.

Once a line of best fit has been drawn for the available data, Once a line of best fit has been drawn for the available data,

Interpolation RegionInterpolation Region

ExtrapolationExtrapolation RegionsRegions

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it becomes quite easy to determine a “y” value from a given “x” value it becomes quite easy to determine a “y” value from a given “x” value or visa versa.or visa versa.

Of the two processes, Of the two processes, interpolation is inherently more interpolation is inherently more reliable than extrapolation. reliable than extrapolation.

4.0 In Summary4.0 In Summary

It is very easy to enter data incorrectly It is very easy to enter data incorrectly into a calculator or computer which will into a calculator or computer which will ultimately lead to ridiculous values for ultimately lead to ridiculous values for gradients and intercepts. gradients and intercepts. This can go unnoticed unless you have This can go unnoticed unless you have an approximate value obtained from a an approximate value obtained from a hand drawn graph for comparison.hand drawn graph for comparison.

Computers and calculators are excellent Computers and calculators are excellent for fast and repetitive calculations. for fast and repetitive calculations. But they cannot match the eye/brain But they cannot match the eye/brain combination when it comes to spotting combination when it comes to spotting patterns or anomalies.patterns or anomalies.

Information Sources:Information Sources:1. Experimental Methods – An Introduction to the Analysis and Presentation of Data1. Experimental Methods – An Introduction to the Analysis and Presentation of DataLes Kirkup – Jacaranda Wiley Ltd. ISBN 0 471 33579 7Les Kirkup – Jacaranda Wiley Ltd. ISBN 0 471 33579 72. Dr. Fred Omega Garces - Chemistry 100 – Powerpoint - 2. Dr. Fred Omega Garces - Chemistry 100 – Powerpoint - Miramar CollegeMiramar College