# PHY 1321 Lecture Notes - Lecture 1: Observational Error, Error Bar, Measuring Instrument

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27 Jan 2016

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Tutorial – Experimental errors

1

Tutorial – Experimental errors

Content

Tutorial – Experimental errors ....................................................................................................................................... 1

Content ...................................................................................................................................................................... 1

Precision vs. Accuracy ................................................................................................................................................ 1

Types of Errors ........................................................................................................................................................... 2

Statement of a measurement .................................................................................................................................... 3

Quantitative comparisons ......................................................................................................................................... 4

Significant figures ...................................................................................................................................................... 5

Precision of errors ..................................................................................................................................................... 6

Precision vs. Accuracy

In ordinary conversation we tend to use the terms accuracy and precision interchangeably, but in the context of

scientific measurement, they give very different meanings. They are actually two different ways of expressing the

uncertainty of experimental data.

Accuracy: Refers how closely a measured value of a quantity corresponds to its true value.

Precision: Expresses the degree of reproducibility of a result when the experiment is repeated under the same

conditions. In other words precision refers to how closely individual measurements agree with each

other.

A result can be measured precisely yet still be inaccurate.

An imprecise result can be accurate.

In order to better understand the difference between accuracy and precision, let us take the example of an archer

shooting a total of thirteen arrows at a target. From the archer’s first results (shown in Figure 1a), we can conclude

that on average the archer is accurate but no precise since all the arrows have a large deviation with respect to

each other.

Several months before the next competition the archer has went through rigorous training and have increased his

precision considerably. During a practice session on the eve of her second competition, the archer calibrates the

sights on her bow and hits the bull’s-eye of the target 10 of 13 times. The morning of the competition the wind

conditions have changed; due to her lack of experience the archer fails to readjust (or recalibrate) the sights on her

bow to compensate for the wind factor. Her arrows end up hitting a localized region in the top left-hand corner of

the target (see Figure 1b). The results show an improvement in precision relative to her first competition, but

unfortunately her shots are all inaccurate.

In her next competition the more experienced archer made sure to properly calibrate the sights of her bow

moments before her competition in order to compensate for the current weather conditions. As a result all of her

arrows hit the target near the bull’s-eye (as shown in Figure 1c) demonstrating high precision as well as high

accuracy.

Tutorial – Experimental errors

2

Trying to hit the bull’s eye of a target with the use of a bow and an arrow is analogous to making a measurement in

the goal of obtaining to the true value of a quantity within an acceptable range of uncertainty. A quantity’s true

value is the value that would be obtained in the absence of errors. An archer’s precision can be improved through

training, while a scientist can improve the precision of his/her measurements with the use of a better experimental

technique and/or the use of a measuring instrument having a greater precision. However, each instrument has a

precision limit that cannot be overcome, i.e., independent of the amount of training spent by the archer she will

never be able to hit the target at precisely the same spot thirteen times in a row due to random wind fluctuations.

Similarly, a scientist will always encounter random fluctuations during experiments that cannot be eliminated. As

in the case of the archer who readjusted the sights on her bow, the inaccuracy of measurements can be improved

with the proper calibration of measuring instruments.

Figure 1 - Archer shooting 13 times at a target. (a) Accurate (the average is accurate) but not

precise. (b) Precise not accurate. (c) Accurate and precise.

Types of Errors

It is impossible to obtain an exact measurement due to the lack of precision of instruments and the experimental

techniques.

Here are the two types of errors an experimentalist can encounter:

Random errors

Random errors are those which come up differently each time a reading is taken. They are statistical in origin and

can be treated using statistical methods. Repeated readings of the same quantity will give a statistical sample and

this serves both to provide a better answer and to estimate the random error. Random errors are seen as

deviations between the measured values and the mean value (see Figure 2).

Random errors affect the precision of measurements not its accuracy.

Systematic errors

These are deviations between the mean of a large number of measured values and the true value. This type of

error is due to limitations of the measurement equipment or improper calibration. These types of errors will shift

all the measurements relative to the true value (see Figure 2). Examples of such errors are the displacement of the

zero point on a micrometer, the unaccounted loss of heat during a calorimeter experiment or a meter scale drawn

with slightly wrong spacing.

Systematic errors affect the accuracy of measurements not its precision.