In the previous section, we completed a discussion on significant figures. We saw some solved examples also. In this section, we will see errors in measurements.
1. While conducting experiments, we take various measurements. For example, diameter of a cylinder, mass of a box, temperature of a liquid etc.,
2. Consider the diameter of a cylinder
• The measurement that we take using a ruler or vernier calipers may not be the true diameter
• If the true diameter is 3.26 cm, we may be getting it as 3.24 or 3.27 cm
■ If it is not 3.26 cm, it is clear that some ‘error in measurement’ has occurred
3. Complete elimination of errors is not possible. But we can minimize the errors
• For that, first we have to see the ‘different types of errors’
4. Errors in measurement can be broadly classified into two:
(a) Systematic errors
(b) Random errors
This can be represented in a diagrammatic form as shown in fig.2.21 below:
• We will add more items to the diagram as and when we come across them in our discussion
• We will now see each of (a) and (b) in detail
5. Systematic errors
• These errors tend to be in any one direction only. Either positive or negative
• This can be explained using an example:
(i) Let the true length be 2.63 cm
(ii) The measured length is 2.65 cm
(iii) Error = Measured value - True value
= 2.65 - 2.63
= +0.02 cm
• Another length is measured under the same conditions
('same conditions' indicate same instruments, same surrounding temperature, same surrounding humidity, same voltage supply etc.,)
(i) Let the true length this time be 5.24 cm
(ii) The measured length is 5.26 cm
(iii) Error = Measured value - True value
= 5.26 - 5.24
= +0.02 cm
• We can guess what is happening:
The measurement is always greater than the true value by 0.02 cm
Another example:
(i) Let the true diameter be 1.52 cm
(ii) The measured diameter is 1.49 cm
(iii) Error = Measured value - True value
= 1.49 - 1.52
= -0.03 cm
• Another diameter is measured under the same conditions
(i) Let the true diameter this time be 3.54 cm
(ii) The measured diameter is 3.51 cm
(iii) Error = Measured value - True value
= 3.51 - 3.54
= -0.03 cm
• We can guess what is happening:
The measurement is always lower than the true value by 0.03 cm
6. The above given examples illustrate systematic errors
• It will be either positive or negative
• In systematic errors, following two items do not occur together under the same conditions:
♦ Positive error in some measurements
♦ Negative error in some other measurements
7. Systematic errors occur mainly under three conditions:
(i) Instrumental errors
(ii) Imperfections in experimental technique or procedure
(iii) Personal errors
• So the diagram that we saw in fig.2.21 can be modified as shown below:
• We will now see each of them in detail
8. Instrumental errors
• This occur due to imperfect design or imperfect calibration of the measuring instrument
9. Example for imperfect design:
(i) A measuring cylinder is not perfectly cylindrical
• It is wider at the base
(A cylinder must have the same diameter throughout it's height. But here, due to manufacturing defect, the diameter goes on increasing as we move from it's top to bottom)
(iii) Take some liquid. Let the actual volume of that liquid taken be 5 mL
(iv) Pour it into our cylinder. The reading will be less than 5 mL
• A person using this cylinder will always get a lesser measurement than the actual
10. Example for imperfect calibration:
• A measuring cylinder is perfect in design and manufacture. But some errors crept in at the time of calibration. Let us see how it happened:
(i) Using a standard cylinder, the technician took 10 mL of water and poured into the new cylinder which is to be calibrated
(ii) He carefully examined the height of the water column in the new cylinder
(iii) He made two marks:
♦ The lower end of the column was marked as 0 mL
♦ The upper end of the column was marked as 10 mL
(iv) The portion between the two marks was divided into 10 equal parts
(v) A mark was made corresponding to each of those equal parts
• The volume between any two consecutive marks will be equal to 1 mL
(vi) But the initial mark made at the top of the column was done with some error
• It was done at a point higher than where it should be
• Then each of the intermediate marks will be at higher levels
(vii) Take some liquid. Let the actual volume of that liquid taken be 5 mL
• Pour it into our cylinder. The reading will be less than 5 mL
(viii) Another possibility:
• The initial top most marking was done at a point lower than where it should be
• Then each of the intermediate marks will be at lower levels
(ix) Take some liquid. Let the actual volume of that liquid taken be 5 mL
• Pour it into our cylinder. The reading will be greater than 5 mL
11. Another example of imperfect calibration:
(i) A newly manufactured thermometer is being calibrated
• The technician places the bulb of the thermometer at a heat source
• The temperature of that heat source is known to be 100o c
(ii) The technician notes the level to which mercury rises
• He makes a mark at that level and writes 100o c there
(iii) But what if the temperature at the source was actually 104o c?
Answer: The calibration would be wrong
(iv) Let a person use that wrongly calibrated thermometer at a later stage
• He places the thermometer at a point where the actual temperature is 100o c
• The reading will be 96o c
• The schematic diagram in the fig.2.23 below shows how the decrease to 96o occurs
12. Imperfections in experimental technique or procedure
• Every experiment has it's own specific procedure. We must strictly follow them. Other wise the results will not be accurate
13. An example for imperfect procedure is given below:
(i) A person measures the body temperature of a patient
(ii) He puts the thermometer under the armpit
(iii)This is wrong procedure. The thermometer placed under the armpit will always give a lower temperature than the actual temperature of the body
• So a patient having high fever may be reported as having a low fever
• Also, a patient having low fever may be reported as having no fever at all
■ In such conditions we must be very careful and seek expert advice
14. Another example for imperfect procedure:
(i) An experiment is to be conducted at room temperature
(ii) But in hot climatic conditions, the results obtained will all be incorrect
• We must wait till conditions become favorable
(iii) Like this, humidity, wind velocity etc., can also affect the results
15, Personal errors
• This arise due to individual’s bias
• 'Bias' means: ‘to be influenced’
• In our present discussion, bias means: Always conducting (unknowingly) an experiment in a particular wrong way
16. An example:
• If a person always look from a point far from the right or far from left, the reading obtained will be wrong
• This is an error due to parallax. It is shown in fig.2.24 below:
• In the fig., the tip of the yellow arrow indicates the end point of an object whose length is to be measured
• We can see that, the correct length is 7.30 cm
• But if the observer views from a point on the left, the length will appear to be greater than 7.30 cm
• Similarly, if the observer views from a point on the right, the length will appear to be less than 7.30 cm
17. Another example:
(i) Consider the experiment to ‘determine the size of soil particles’ in a sample of soil
(ii) The solution of the sample is first taken in a measuring cylinder or jar
(iii)The top of the cylinder is closed
(iv) The solution is shaken well by uniform vertical movements (alternate inverted and upright positions) of the cylinder
(v) In this way, distinct vertical layers of various soil particles will be obtained
(vi) The layers will be stacked one above the other. Some images can be seen here
(vii) A person is biased and prefers to shake the cylinder horizontally
• This will give all wrong results. Because, the particles will move towards one side of the cylinder. Vertically stacked layers will not be obtained
18. Now we can discuss about random errors
• These errors occur in an irregular manner
19. An example:
(i) An experiment is designed to be conducted at normal temperature
• If it is conducted in hot climatic conditions, we will get wrong results
• This comes under: Systematic error → Error due to imperfect procedure
♦ It is a climatic condition
♦ The temperature would be high for prolonged days or months
♦ That experiment should not be conducted in such a condition
(ii) But what if the surrounding temperature increase suddenly?
• That would be unexpected. There will be error in the reading obtained
■ This error is a random error
• There can be unexpected increase in humidity, unexpected vibrations in the room, unexpected fluctuations in voltage etc., All those will lead to random errors
19. Least count errors
• We have see that. the smallest value that can be measured by the measuring instrument is called it's least count
• All the readings or measured values are good only up to this value
• Least count error is the error associated with the resolution of the instrument
20. Let us see an example:
(i) In fig.2.11 that we saw in a previous section on significant figures, the least count of the ruler is 1 mm
• The fig. is shown again below:
(ii) The yellow arrow is closer to the 7.4 mm mark
• If the person taking measurement is not careful, a difference of 1 mm may occur. He may record it as 7.5 cm or 7.3 cm
• That is., a error of '± least count' occurs
• This type of error is called least count error
Another example:
• A measuring cylinder has the least count of 0.1 mL
• The actual reading when a liquid is taken in it is 3.7 mL
• But the observer may record the reading as 3.6 or 3.8 mL
21. We would not always pick a greater value on the right side of the yellow arrow
• Also we would not always pick a lesser value on the left side of the yellow arrow
• The error can occur on both sides
• So least count error can be assumed to come under the category of random errors
22. This error can be minimized by using instruments of greater precision
• For example, if the ruler in fig.2.11 above can measure upto the next lower level (one tenth of a mm), the least count error will come down by one level. It will become: ± 0.01 cm
• Also, if we have better measuring techniques like optical magnifiers, we will get greater precision
• So least count error can be assumed to come under the category of systematic errors also
1. While conducting experiments, we take various measurements. For example, diameter of a cylinder, mass of a box, temperature of a liquid etc.,
2. Consider the diameter of a cylinder
• The measurement that we take using a ruler or vernier calipers may not be the true diameter
• If the true diameter is 3.26 cm, we may be getting it as 3.24 or 3.27 cm
■ If it is not 3.26 cm, it is clear that some ‘error in measurement’ has occurred
3. Complete elimination of errors is not possible. But we can minimize the errors
• For that, first we have to see the ‘different types of errors’
4. Errors in measurement can be broadly classified into two:
(a) Systematic errors
(b) Random errors
This can be represented in a diagrammatic form as shown in fig.2.21 below:
 |
| Fig.2.21 |
• We will now see each of (a) and (b) in detail
5. Systematic errors
• These errors tend to be in any one direction only. Either positive or negative
• This can be explained using an example:
(i) Let the true length be 2.63 cm
(ii) The measured length is 2.65 cm
(iii) Error = Measured value - True value
= 2.65 - 2.63
= +0.02 cm
• Another length is measured under the same conditions
('same conditions' indicate same instruments, same surrounding temperature, same surrounding humidity, same voltage supply etc.,)
(i) Let the true length this time be 5.24 cm
(ii) The measured length is 5.26 cm
(iii) Error = Measured value - True value
= 5.26 - 5.24
= +0.02 cm
• We can guess what is happening:
The measurement is always greater than the true value by 0.02 cm
Another example:
(i) Let the true diameter be 1.52 cm
(ii) The measured diameter is 1.49 cm
(iii) Error = Measured value - True value
= 1.49 - 1.52
= -0.03 cm
• Another diameter is measured under the same conditions
(i) Let the true diameter this time be 3.54 cm
(ii) The measured diameter is 3.51 cm
(iii) Error = Measured value - True value
= 3.51 - 3.54
= -0.03 cm
• We can guess what is happening:
The measurement is always lower than the true value by 0.03 cm
6. The above given examples illustrate systematic errors
• It will be either positive or negative
• In systematic errors, following two items do not occur together under the same conditions:
♦ Positive error in some measurements
♦ Negative error in some other measurements
7. Systematic errors occur mainly under three conditions:
(i) Instrumental errors
(ii) Imperfections in experimental technique or procedure
(iii) Personal errors
• So the diagram that we saw in fig.2.21 can be modified as shown below:
 |
| Fig.2.22 |
8. Instrumental errors
• This occur due to imperfect design or imperfect calibration of the measuring instrument
9. Example for imperfect design:
(i) A measuring cylinder is not perfectly cylindrical
• It is wider at the base
(A cylinder must have the same diameter throughout it's height. But here, due to manufacturing defect, the diameter goes on increasing as we move from it's top to bottom)
(iii) Take some liquid. Let the actual volume of that liquid taken be 5 mL
(iv) Pour it into our cylinder. The reading will be less than 5 mL
• A person using this cylinder will always get a lesser measurement than the actual
10. Example for imperfect calibration:
• A measuring cylinder is perfect in design and manufacture. But some errors crept in at the time of calibration. Let us see how it happened:
(i) Using a standard cylinder, the technician took 10 mL of water and poured into the new cylinder which is to be calibrated
(ii) He carefully examined the height of the water column in the new cylinder
(iii) He made two marks:
♦ The lower end of the column was marked as 0 mL
♦ The upper end of the column was marked as 10 mL
(iv) The portion between the two marks was divided into 10 equal parts
(v) A mark was made corresponding to each of those equal parts
• The volume between any two consecutive marks will be equal to 1 mL
(vi) But the initial mark made at the top of the column was done with some error
• It was done at a point higher than where it should be
• Then each of the intermediate marks will be at higher levels
(vii) Take some liquid. Let the actual volume of that liquid taken be 5 mL
• Pour it into our cylinder. The reading will be less than 5 mL
(viii) Another possibility:
• The initial top most marking was done at a point lower than where it should be
• Then each of the intermediate marks will be at lower levels
(ix) Take some liquid. Let the actual volume of that liquid taken be 5 mL
• Pour it into our cylinder. The reading will be greater than 5 mL
11. Another example of imperfect calibration:
(i) A newly manufactured thermometer is being calibrated
• The technician places the bulb of the thermometer at a heat source
• The temperature of that heat source is known to be 100o c
(ii) The technician notes the level to which mercury rises
• He makes a mark at that level and writes 100o c there
(iii) But what if the temperature at the source was actually 104o c?
Answer: The calibration would be wrong
(iv) Let a person use that wrongly calibrated thermometer at a later stage
• He places the thermometer at a point where the actual temperature is 100o c
• The reading will be 96o c
• The schematic diagram in the fig.2.23 below shows how the decrease to 96o occurs
 |
| Fig.2.23 |
• Every experiment has it's own specific procedure. We must strictly follow them. Other wise the results will not be accurate
13. An example for imperfect procedure is given below:
(i) A person measures the body temperature of a patient
(ii) He puts the thermometer under the armpit
(iii)This is wrong procedure. The thermometer placed under the armpit will always give a lower temperature than the actual temperature of the body
• So a patient having high fever may be reported as having a low fever
• Also, a patient having low fever may be reported as having no fever at all
■ In such conditions we must be very careful and seek expert advice
14. Another example for imperfect procedure:
(i) An experiment is to be conducted at room temperature
(ii) But in hot climatic conditions, the results obtained will all be incorrect
• We must wait till conditions become favorable
(iii) Like this, humidity, wind velocity etc., can also affect the results
15, Personal errors
• This arise due to individual’s bias
• 'Bias' means: ‘to be influenced’
• In our present discussion, bias means: Always conducting (unknowingly) an experiment in a particular wrong way
16. An example:
• If a person always look from a point far from the right or far from left, the reading obtained will be wrong
• This is an error due to parallax. It is shown in fig.2.24 below:
 |
| Fig.2.24 |
• We can see that, the correct length is 7.30 cm
• But if the observer views from a point on the left, the length will appear to be greater than 7.30 cm
• Similarly, if the observer views from a point on the right, the length will appear to be less than 7.30 cm
17. Another example:
(i) Consider the experiment to ‘determine the size of soil particles’ in a sample of soil
(ii) The solution of the sample is first taken in a measuring cylinder or jar
(iii)The top of the cylinder is closed
(iv) The solution is shaken well by uniform vertical movements (alternate inverted and upright positions) of the cylinder
(v) In this way, distinct vertical layers of various soil particles will be obtained
(vi) The layers will be stacked one above the other. Some images can be seen here
(vii) A person is biased and prefers to shake the cylinder horizontally
• This will give all wrong results. Because, the particles will move towards one side of the cylinder. Vertically stacked layers will not be obtained
18. Now we can discuss about random errors
• These errors occur in an irregular manner
19. An example:
(i) An experiment is designed to be conducted at normal temperature
• If it is conducted in hot climatic conditions, we will get wrong results
• This comes under: Systematic error → Error due to imperfect procedure
♦ It is a climatic condition
♦ The temperature would be high for prolonged days or months
♦ That experiment should not be conducted in such a condition
(ii) But what if the surrounding temperature increase suddenly?
• That would be unexpected. There will be error in the reading obtained
■ This error is a random error
• There can be unexpected increase in humidity, unexpected vibrations in the room, unexpected fluctuations in voltage etc., All those will lead to random errors
19. Least count errors
• We have see that. the smallest value that can be measured by the measuring instrument is called it's least count
• All the readings or measured values are good only up to this value
• Least count error is the error associated with the resolution of the instrument
20. Let us see an example:
(i) In fig.2.11 that we saw in a previous section on significant figures, the least count of the ruler is 1 mm
• The fig. is shown again below:
 |
| Fig.2.11 |
• If the person taking measurement is not careful, a difference of 1 mm may occur. He may record it as 7.5 cm or 7.3 cm
• That is., a error of '± least count' occurs
• This type of error is called least count error
Another example:
• A measuring cylinder has the least count of 0.1 mL
• The actual reading when a liquid is taken in it is 3.7 mL
• But the observer may record the reading as 3.6 or 3.8 mL
21. We would not always pick a greater value on the right side of the yellow arrow
• Also we would not always pick a lesser value on the left side of the yellow arrow
• The error can occur on both sides
• So least count error can be assumed to come under the category of random errors
22. This error can be minimized by using instruments of greater precision
• For example, if the ruler in fig.2.11 above can measure upto the next lower level (one tenth of a mm), the least count error will come down by one level. It will become: ± 0.01 cm
• Also, if we have better measuring techniques like optical magnifiers, we will get greater precision
• So least count error can be assumed to come under the category of systematic errors also
In the next section, we will see methods to obtain the 'quantity of error that has been made' when an experiment was done
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