Analysis of Errors - physicslab.sc.chula.ac.th

Analysis of ErrorsError analysis may seem tedious; however, without proper error analysis, no valid scientific

conclusions can be drawn. In fact, as the picture below illustrates, bad things can happen

if error analysis is ignored. Since there is no way to avoid error analysis,

it is best to learn how to do it right!

The derailment at Gare Montparnasse, Paris, 1895.

Error in a scientific measurement usually does not mean

a mistake.

Instead, the terms "error = ความผดิพลาด" and

"uncertainty = ความไม่แน่นอน"

both refer to unavoidable imprecision in measurements.

You might wonder why physicists pay so much attention

to errors. The reason is simple: the failure to specify the error

for a given measurement can have serious consequences

in science and in real life.

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Slides prepared by Assoc. Prof. Dr. Satreerat Hodak(Presentation day: August 20 and 24, 2018)

Analysis of Errors

The concepts of precision and accuracy are demonstrated by the series of targets below. If the

center of the target is the "true value", then A is neither precise nor accurate. Target B is precise

(reproducible) but not accurate. The average of target C's marks give an accurate result but

precision is poor. Target D demonstrates both precision and accuracy - which is the goal in lab.

Precision and Accuracy:

Two terms are commonly associated with any discussion of error:

"precision=ความแม่นย า" and "accuracy = ความถูกตอ้ง".

Precision refers to the reproducibility (การท าซ ้ าได)้ of a measurement

while accuracy is a measure of the closeness to true value (การไดค่้าใกลเ้คียงกบัความเป็นจริง).

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Slides prepared by Assoc. Prof. Dr. Satreerat Hodak

Errors: Random errorsRandom (or indeterminate) errors are caused by uncontrollable fluctuations

in variables that affect experimental results. For example, air fluctuations

occurring as students open and close lab doors cause changes in pressure

readings. A sufficient number of measurements result in evenly distributed

data scattered around an average value or mean. This positive and negative

scattering of data is characteristic of random errors. The estimated standard

deviation (the error range for a data set) is often reported with

measurements because random errors are difficult to eliminate. Also, a

"best-fit line" is drawn through graphed data in order to "smooth out"

random error.

Example: You measure the mass of a ring three times using the same balance

and get slightly different values: 17.46 g, 17.42 g, 17.44 g

How to minimize it: Take more data. Random errors can be evaluated through

statistical analysis and can be reduced by averaging over a large number of observations.

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Errors: Systematic errors• Systematic (or determinate) errors are instrumental, methodological, or

personal mistakes causing "lopsided" data, which is consistently deviated in

one direction from the true value. Examples of systematic errors: an

instrumental error results when a spectrometer drifts away from calibrated

settings; a methodological error is created by using the wrong indicator for

an acid-base titration; and, a personal error occurs when an experimenter

records only even numbers for the last digit of buret volumes. Systematic

errors can be identified and eliminated after careful inspection of the

experimental methods, cross-calibration of instruments, and examination of

techniques.

More Examples: The cloth tape measure that you use to measure the length of an object

had been stretched out from years of use. (As a result, all of your length measurements were

too small.) The electronic scale you use reads 0.05 g too high for all your mass measurements

throughout your experiment.

How to minimize it: Systematic errors are difficult to detect and cannot be analyzed

statistically, because all of the data is off in the same direction (either to high or too low).

Spotting and correcting for systematic error takes a lot of care.4


Systematic vs. Random errors

All experimental uncertainty is due to either random errors or systematic errors.

Random errors are statistical fluctuations (in either direction) in the measured data

due to the precision limitations of the measurement device. Random errors usually

result from the experimenter's inability to take the same measurement in exactly

the same way to get exact the same number. Systematic errors, by contrast, are

reproducible inaccuracies that are consistently in the same direction. Systematic errors

are often due to a problem which persists throughout the entire experiment.5


Errors: Gross errors

• Gross errors are caused by experimenter carelessness or equipment failure.

These "outliers" are so far above or below the true value that they are

usually discarded when assessing data.

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Precision of a Set of Measurements

• A data set of repetitive measurements is often expressed as a single

representative number called the mean or average. The mean is the sum of

individual measurements divided by the number of measurements (N).

• Precision (reproducibility) is quantified by calculating the average deviation

or the standard deviation (ideally for data sets with 5 or more measurements).

Precision is the opposite of uncertainty. Widely scattered data results in a

large average or standard deviation indicating poor precision.

x

ix

N

xx

i

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Precision of a Set of Measurements• For data sets with 5 or more measurements, the estimated standard

deviation (s or σ), is used to express the precision of the measurements.

• The number of degrees of freedom (N−1) is the total number of

measurements minus one

A plot of normal distribution

bell-shaped curve)

where each band has a width of

1 standard deviation

In statistics, the standard deviation (SD, also represented

by the Greek letter sigma σ or the Latin letter s) is a

measure that is used to quantify the amount of variation

or dispersion of a set of data values. A low standard

deviation indicates that the data points tend to be close to

the mean (also called the expected value) of the set, while

a high standard deviation indicates that the data points are

spread out over a wider range of values.

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https://en.wikipedia.org/wiki/Sigma


https://en.wikipedia.org/wiki/S


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N

xx

If we make N measurements x1,..., xN of a quantity x (that is normally distributed),

the best estimate of the true value x is the mean of x1,..., xN . The uncertainty is this best

estimate is the standard deviation of the mean (SDOM) or standard error or standard

error of the mean. SDOM would slowly decrease as we increase N, the number of

experiments (but decrease rather slow e.g. improve the precision by a factor of 10 by

increasing N 100 times).

xx

The uncertainty in any one measurement of your experiment is therefore approximately

the standard deviation (SD) (s or σ), about 68% confidence that the second new experiment

lies in the range. On the other hand, SD represents the average uncertainty in the individual

measurements. SD will not change that much if using the same technique.



• Accuracy of a Result: The accuracy of a result can be quantified by

calculating the percent error. The percent error can only be found if the

true value is known. Although the percent error is usually written as an

absolute value, it can be expressed a negative or positive sign to indicate

the direction of error from true value.

10Slides prepared by Assoc. Prof. Dr. Satreerat

Hodak


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Propagation of errors

• The estimation of uncertainties involves > two

steps

• How these uncertainties “propogate” through

the calculations to produce an uncertainty in

the final answer

13Slides prepared by Assoc. Prof. Dr. Satreerat Hodak

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19Slides prepared by Assoc. Prof. Dr. Satreerat

Hodak

Proof in the class

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Slides prepared by Assoc. Prof. Dr. Satreerat Hodak 22

Chi squared

In all cases, chi squared is a sum of squares with the general form as above, and is an indicator

of the agreement between the observed and expected values of some variable. If the agreement

is good, chi squared will be of order n, if it is poor it will be much greater than n.



Linear regression or the Least-squares

fit for a straight line

Homework

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Slides prepared by Assoc. Prof. Dr. Satreerat Hodak27

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