Experiment 3: experimental errors and uncertainty

 

EXPERIMENT 3:

 

Experimental Errors and Uncertainty

 

Read the entire experiment and organize time, materials, and work space before beginning.

Remember to review the safety sections and wear goggles when appropriate.

 

Objective: To gain an understanding of experimental errors and uncertainty.

Materials: Student Provides: Pen and pencils

Paper, plain and graph

Computer and spreadsheet program

From LabPaq: No supplies are required for this experiment.

 

Discussion and Review: No physical quantity can be measured with perfect certainty;

there are always errors in any measurement. This means that if we measure some

quantity and then repeat the measurement we will almost certainly measure a different

value the second time. How then can we know the gtrueh value of a physical quantity?

The short answer is that we cannot. However, as we take greater care in our

measurements and apply ever more refined experimental methods we can reduce the

errors and thereby gain greater confidence that our measurements approximate ever

more closely the true value.

gError analysish is the study of uncertainties in physical measurements. A complete

description of error analysis would require much more time and space than we have in

this course. However, by taking the time to learn some basic principles of error analysis

we can:

.. Understand how to measure experimental error;

.. Understand the types and sources of experimental errors;

.. Clearly and correctly report measurements and the uncertainties in

measurements; and

.. Design experimental methods and techniques plus improve our measurement

skills to reduce experimental errors.

Two excellent references on error analysis are:

.. John R. Taylor, An Introduction to Error Analysis: The Study of Uncertainties in

Physical Measurements, 2d Edition, University Science Books, 1997; and

.. Philip R. Bevington and D. Keith Robinson, Data Reduction and Error Analysis

for the Physical Sciences, 2d Edition, WCB/McGraw-Hill, 1992.

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Accuracy and Precision

Experimental error is the difference between a measurement and the true value or

between two measured values. Experimental error itself is measured by its accuracy

and precision.

Accuracy measures how close a measured value is to the true value or accepted

value. Since a true or accepted value for a physical quantity may be unknown, it is

sometimes not possible to determine the accuracy of a measurement.

Precision measures how closely two or more measurements agree with each other.

Precision is sometimes referred to as grepeatabilityh or greproducibilityh. A measurement

that is highly reproducible tends to give values which are very close to each other.

Figure 1 defines accuracy and precision with an analogy of the grouping of arrows in a

target.

Figure 1: Accuracy vs. Precision

Types and Sources of Experimental Errors

When scientists refer to experimental errors they are not referring to what are commonly

called mistakes, blunders, or miscalculations or sometimes illegitimate, human or

personal errors. Personal errors can result from measuring a width when the length

should have been measured, or measuring the voltage across the wrong portion of an

electrical circuit, or misreading the scale on an instrument, or forgetting to divide the

diameter by 2 before calculating the area of a circle with the formula A = ƒÎ r2. Such

errors are certainly significant but they can be eliminated by performing the experiment

again correctly the next time.

On the other hand, experimental errors are inherent in the measurement process. They

cannot be eliminated simply by repeating the experiment, no matter how carefully.

There are two types of experimental errors: systematic errors and random errors.

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Systematic Errors: Systematic errors are errors that affect the accuracy of a

measurement. Systematic errors are gone-sidedh errors because, in the absence of

other types of errors, repeated measurements yield results that differ from the true or

accepted value by the same amount. The accuracy of measurements subject to

systematic errors cannot be improved by repeating those measurements. Systematic

errors cannot easily be analyzed by statistical analysis. Systematic errors can be

difficult to detect, and once detected they can only be reduced by refining the

measurement method or technique.

Common sources of systematic errors are faulty calibration of measuring instruments,

poorly maintained instruments, or faulty reading of instruments by the user. A common

form of this last source of systematic error is called gparallax errorh, which results from

the user reading an instrument at an angle resulting in a reading which is consistently

high or consistently low.

Random Errors: Random errors are errors that affect the precision of a measurement.

Random errors are gtwo-sidedh errors because, in the absence of other types of errors,

repeated measurements yield results that fluctuate above and below the true or

accepted value. Measurements subject to random errors differ from each other due to

random, unpredictable variations in the measurement process. The precision of

measurements subject to random errors can be improved by repeating those

measurements. Random errors are easily analyzed by statistical analysis. Random

errors can be detected and reduced by repeating the measurement or by refining the

measurement method or technique.

Common sources of random errors are problems estimating a quantity that lies between

the graduations (the measurement lines) on an instrument and the inability to read an

instrument because the reading fluctuates during the measurement.

Calculating Experimental Error

When a scientist reports the results of an experiment the report must describe the

accuracy and precision of the experimental measurements. Some common ways to

describe accuracy and precision are described below.

Significant Figures: The least significant digit in a measurement depends on the

smallest unit that can be measured using the measuring instrument. The precision of a

measurement can then be estimated by the number of significant digits with which the

measurement is reported. In general, any measurement is reported to a precision equal

to 1/10 of the smallest graduation on the measuring instrument, and the precision of the

measurement is said to be 1/10 of the smallest graduation.

For example, a measurement of length using a meter tape with 1-mm graduations will

be reported with a precision of }0.1 mm. A measurement of volume using a graduated

cylinder with 1 mL graduations will be reported with a precision of }0.1 mL.

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Digital instruments are treated differently. Unless the instrument manufacturer indicates

otherwise, the precision of measurement made with digital instruments are reported with

a precision of }. of the smallest unit of the instrument. For example, a digital voltmeter

reads 1.493 volts; the precision of the voltage measurement is }. of 0.001 volts or

}0.0005 volt.

Percent Error: Percent error measures the accuracy of a measurement by the

difference between a measured or experimental value E and a true or accepted value A.

The percent error is calculated from the following equation:

Equation 1 % Error = | E . A| x 100%

A

Percent Difference: Percent difference measures precision of two measurements by

the difference between the measured or experimental values E1 and E2 expressed as a

fraction of the average of the two values. The equation used to calculate the percent

difference is:

Equation 2

Mean and Standard Deviation: When a measurement is repeated several times we

see the measured values are grouped around some central value. This grouping or

distribution can be described with two numbers: the mean, which measures the central

value and the standard deviation, which describes the spread or deviation of the

measured values about the mean. For a set of N measured values for some quantity x,

the mean of x is represented by the symbol <x> and is calculated by the following

formula:

Equation 3

Where xi is the i-th measured value of x. The mean is simply the sum of the measured

values divided by the number of measured values. The standard deviation of the

measured values is represented by the symbol ƒÐx and is given by the formula:

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Equation 4

The standard deviation is sometimes referred to as the gmean square deviation.h It

measures how widely spread the measured values are on either side of the mean. The

meaning of the standard

deviation can be seen

from the figure on the

right. This is a plot of

data with a mean of 0.5.

As shown in this graph,

the larger the standard

deviation, the more

widely spread the data is

about the mean. For

measurements that have

only random errors the

standard deviation

shows that 68% of the

measured values are within ƒÐx from the mean, 95% are within 2ƒÐx from the mean, and

99% are within 3ƒÐx from the mean.

Reporting the Results of an Experimental

Measurement

When a scientist reports the result of an experimental measurement of a quantity x, that

result is reported with two parts. First, the best estimate of the measurement is reported.

The best estimate of a set of measurement is usually reported as the mean <x> of the

measurements. Second, the variation of the measurements is reported. The variation in

the measurements is usually reported by the standard deviation ƒÐx of the

measurements.

The measured quantity is then known to have a best estimate equal to the average, but

it may also vary from <x>+ ƒÐx to <x> – ƒÐx. Any experimental measurement should then

be reported in the following form:

x = <x> } ƒÐx

Example: Consider Table 1 below that lists 30 measurements of the mass m of a

sample of some unknown material.

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Table 1: Measured Mass (kg) of Unknown

We can represent this data on a type of bar chart called a histogram (Figure 3), which

shows the number of measured values which lie in a range of mass values with the

given midpoint.

Figure 3: Mass of Unknown Sample

For the 30 mass measurements the mean mass is given by:

<m> = 1/30 (33.04 kg) = 1.10 kg

We see from the histogram that the data does appear to be centered on a mass value

of 1.10 kg. The standard deviation is given by:

We also see from the histogram that the data does, indeed, appear to be spread about

the mean of 1.10 kg so that approximately 70% (= 20/30×100) of the values are within

ƒÐm from the mean.

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49

The measured mass of the unknown sample is then reported as:

m = 1.10} 0.05 kg

PROCEDURES: The data table that follows shows data taken in a free-fall

experiment. Measurements were made of the distance of fall (Y) at each of the four

precisely measured times. From this data perform the following:

1. Complete the table.

2. Plot a graph <y> versus t (plot t on the abscissa, i.e., x-axis).

3. Plot a graph <y> versus t2 (plot t2 on the abscissa, i.e., x-axis). The equation of

motion for an object in free fall starting from rest is y = . gt2, where g is the

acceleration due to gravity. This is the equation of a parabola, which has the general

form y = ax2.

4. Determine the slope of the line and compute an experimental value of g from the

slope value. Remember, the slope of this graph represents . g.

5. Compute the percent error of the experimental value of g determined from the graph

in part d. (Accepted value of g = 9.8 m/s2)

6. Use a spreadsheet to perform the calculations and plot the graphs indicated.

Time, t

(s)

Dist.

y1 (m)

Dist.

y2 (m)

Dist.

y3 (m)

Dist.

y4 (m)

Dist.

y5 (m) <y> ƒÐ t2

0 0 0 0 0 0

0.5 1.0 1.4 1.1 1.4 1.5

0.75 2.6 3.2 2.8 2.5 3.1

1.0 4.8 4.4 5.1 4.7 4.8

1.25 8.2 7.9 7.5 8.1 7.4

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