In previous lessons, we have covered physical quantities including classification of physical quantities, scientific notation and prefixes.
In this lesson, we are going to move on to measurement.
In this lesson, we will discuss what is error in measurement, sensitivity of measuring instrument, accuracy of a measurement and consistency of measurement.
Under error, where we will discuss what is error. What is systematic error, including the sources of systematic error, how to reduce it and zero error as a systematic error. We will also discuss random error, sources of random error, how to reduce it and parallax error as a random error.
Under accuracy, we will discuss what is relative error and how to improve the accuracy of a measurement. Under consistency, we will also discuss what is relative deviation and how to improve the consistency of readings.
Error in Measurement
The figure above shows a straight line of length 8.2cm. When a student made a measurement, due to a mistake, he recorded his reading as 8.4cm. The different between the actual length and the measured length is called the error of measurement. In this case, the error is 0.2cm.
Error is the difference between the actual value of a quantity and the value obtained in measurement.
Types of Error
There are 2 main types of error: the systematic error and the random error.
Examples of systematic errors are zero error and the error due to inaccurately calibrated measuring instrument. At this stage, you don’t need to know what they really means. We will come to that later.
Examples of random errors are parallax error, careless in making reading or record, use of wrong techniques in making measurements and etc.
Let’s see this example. I have a bathroom scale. This scale always give readings 0.5kg higher than its actual reading. The table shows the actual weight and the reading taking from the bathroom scale of my family members.
My actual weight is 72kg, but the scale showed a reading of 72.5kg. Therefore, the reading has an error of 0.5kg. This is also happen to my family members. We can see that, in every measurement, the bathroom scale give a reading 0.5kg higher than the actual weight. The scale shift all measurements in a systematic way, causing the readings 0.5kg higher than their actual value.
The 0.5kg difference are called the error of measurement. This kind of errors are called the systemic error or systematic error.
Systematic errors are errors which tend to shift all measurements in a systematic way so that the readings are always differ from the true value by a fixed amount.
The diagram shows an analog stopwatch. It shows a reading of 12.8s. When the reset button is pressed, the pointer will kick back to the zero position. If the pointer points exactly at zero, we say this stopwatch has no zero error.
However, due to certain reason, the pointer doesn’t point exactly at zero, even though the stopwatch is reset. If the pointer is a little bit in front of zero, we call this the positive zero error. In this case, the zero error is +0.4s.
If it is a little bit behind the zero, we call it the negative zero error. In this case, the zero error is -0.2s.
Zero error is a systematic error. A zero error arises when the measuring instrument does not start from exactly zero.
Zero error must be identified when we take any readings from an instrument. The zero error can be eliminated by deducting it from the actual reading.
For example, the zero error of this stopwatch has been identified as –0.02s. The stopwatch shows a reading of 12.8s. Therefore, the correct reading for this measurement should be the actual reading minus the zero error, which is equal to 12.8s – (-0.2s), which is then equal to 13.0s.
Sources of Systematic Error
Let’s see the picture above. There are 2 rulers putting side by side. Compare the scale of the rulers. You will find that the scale is not exactly the same. If you try to compare your ruler with your friends’ ruler, you will probably find out the same thing: The rulers are not properly calibrated.
An incorrectly calibrated instrument can cause systematic error. In this case, the plastic ruler is not calibrated correctly. It shows an error of +0.2cm in a measurement of 15.0 cm. The percentage of the error is +0.2/15.0 x 100%, which is equal to +1.33%.
Whenever you use this ruler to measure length, the readings will be 1.33% higher than the actual readings. The error is categorised as systematic error because it systematically shift the readings 1.33% higher than what it should be.
The diagram above shows a reading on a Vernier caliper. A Vernier caliper has 2 scale: the main scale and the Vernier scale. The reading on the main scale is determined with reference to the “0” mark on the Vernier scale. In this case, the reading on the main scale is 1.2cm.
However, some students mistakenly take the reading with reference from the front end of the Vernier scale rather than the “0” mark, and this mistake is done consistently. As a consequence, all readings taken are systematical shifted 0.2cm lower than what they should be. This consistently improper use of equipment can be a source of systematic error.
From previous slides, we can conclude that there are 3 possible sources of systematic error when making a measurement. The zero error, consistently improper use of equipment and incorrectly calibrated instrument.
Systematic error can be reduced by
- Conducting the experiment with care.
- Repeat the experiment by using different instruments.
- Proper calibration or calibrate the instrument frequently.
We should take notes that systematic error cannot be reduced by repeating the same measurement using the same instrument and then calculate the average of the readings. Finding the average of the readings does not reduce systematic error.
Let’s see this case. A pendulum was oscillating at a frequency of 0.5Hz. It take 20.0s to make 10 complete oscillation.
A student was asked to find the time taken for the pendulum to make 10 oscillations. He repeated the experiment for 5 times and the reading that he obtained was recorded in a table.
We can see that the readings vary in each measurement due to his responding time. We can also see that the readings distributed randomly: sometime it is higher than 20.0s, sometime it is lower. The readings are not systematically shifted to one direction. Instead, they distributed randomly. The errors are sometime positive, sometime negative.
This type of error is called the random error. Random errors arise from unknown and unpredictable variations in condition. It varies from one measurement to the next.
Random error can cause by
- wrong technique of measurement such as parallax error.
- interference of the environment with the measurement such as wind blow or temperature change.
- the instrument fail to respond to small changes.
A parallax error is an error in reading an instrument due to the eye of the observer and pointer are not in a line perpendicular to the plane of the scale.
As shown in the diagram, if the position of the eye is in a line perpendicular the plane of the scale of the measuring cylinder, the reading taken will have no parallax error. If the eye is not in a line perpendicular the plane of the scale, the readings are subject to parallax error.
Parallax error is a random error.
Random error can be reduced by finding the average value of the readings. Since random errors distributed randomly, the positive error may be cancelled off by the negative error. Therefore, the average value of the readings will be closer to the actual reading.
Let’s see the example of the pendulum above. The readings taken consist of positive and negative error. However, if we calculate the average value of the reading, we will find that the error cancel off and the reading become accurate.
Random error can also be reduced by repeating the experiment with different manipulated variables and then find the relevant value by using graphical method or statistics. You will learn this in your add maths in the chapter called Linear Law.
Sensitivity of a measuring tool is its ability to detect small changes in the quantity that is being measured. A measuring tool that has scale with smaller division is more sensitive.
For example, the diagram shows a section on a ruler and a section on a measuring tape. The ruler can measure a minimum length up to 0.1cm. Therefore we say the sensitivity of the ruler is 0.1cm. The measuring tape can measure a minimum length up to 0.5cm. Therefore its sensitivity is 0.5cm.
The accuracy of a measurement is how close the measurement made is to the actual value of the quantity of physics.
Let’s see this example. The length of this line is 12.231567 cm. 3 students, John, Jack and June used different instrument to measure the length of this line. Here is the reading that they obtained. Among the 3 readings, which one is the most accurate?
Yes! Jack’s reading is more accurate than John. June’s reading is the most accurate among the 3.