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# Measurement error and Types of errors in measurement

I have already written articles about the basics of units and measurement, different conversion charts, dimensions and dimensional analysis etc. In this article, I focus on the error in measurement, different types of errors and the combination of errors which occurs during the measurement of a physical quantity.

So let’s start with the measurement error. While measuring any physical quantity, it is practically impossible to find its true value. The difference between the true value and the measured value of a physical quantity is called the error in its measurement. In other words, we can say, the result of every measurement by any measuring instrument contains some uncertainty and this uncertainty is called the error.

## Distinguish between Accuracy, Least Count and Precision.

Before going into the types of errors, let’s distinguish between three terms: accuracy, least count, and precision. The accuracy of a measurement is the relative exemption from errors. That is, accuracy is the measure of how close the measured value is to the actual value of the quantity.

For every instrument, there is a minimum value that can be measured accurately. This is called the least count of that instrument. It is 0.1 cm for an ordinary scale, 0.01 cm for an ordinary vernier calliper and 0.001 cm for an ordinary screw gauge.

Precision describes the limit or resolution of the quantity measured. For example, consider an iron rod of length 12 cm. The scale 1 measures it to be 11.9 cm and scale 2 measures it to be 12.426 cm. Here scale 1 is more accurate but scale 2 is more precise. Now another scale 3 measures it to be 12.0056 cm. We can say scale 3 is both accurate and precise.

Also, learn significant figures and the rules for rounding off the uncertain digits.

## What are the different types of errors in measurement?

The errors that may occur in the measurement of a physical quantity can be classified into six types: constant error, systematic error, random error, absolute error, relative error and percentage error. Each type of error in measurement are explained below.

• ### Constant error

Constant errors are those which affect the result by the same amount.
For eg: If the reading of a thermometer, when placed in melting ice at normal pressure, is 10 C, then the instrument has an error by 10 C.

• ### Systematic error

Systematic errors are due to some known causes according to a definite law and are tend to be in one direction, either positive or negative. We can minimize the systematic errors by selecting better instruments, by improving the experimental techniques or procedures and by removing personal errors as far as possible. For a given experimental set-up, these systematic errors may be calculated to a certain extent and the necessary corrections may be applied to the observed readings.

#### Types of systematic error

There are four sources or types of systematic error: Instrumental error, gross error, error due to external causes and the error due to imperfections.

1. ##### Instrumental error

Instrumental errors are errors due to the apparatus or measuring instruments used. It may be errors due to the imperfect design or calibration of the measuring instrument, zero error in the instrument etc. It depends on the limit or resolution of the measuring instrument.

For eg, using a metre scale with graduations at one mm interval, the accuracy of the reading is limited to one mm. The error in the reading of the metre scale is taken to be of the order of half of the smallest division on the scale; that is, of the order of 0.5 mm. When vernier calliper with least count 0.1 mm is used for the measurement, the error is about 0.05 mm. These errors are called instrumental errors.

2. ##### Gross error

The gross error is another type of systematic error which are committed due to the personal peculiarities of the experiment like carelessness in taking observations without observing necessary precautions or lack of proper setting of the measuring instruments, etc. The gross error is also called as the personal error or observation error.

For example, while taking the reading from the instrument meter observer may read 41 as 47.
Another example: Parallax error which arises due to the habit of taking measurements by always holding the observer’s head a bit too far to the right or left while reading the position of a needle on the scale.
We can reduce the gross error by increasing the number of observers who are taking the readings. Also, proper care should be taken while reading and recording the data.

3. ##### Error due to external causes

These errors arise due to change of external conditions like temperature, wind velocity, pressure, humidity, electric field or magnetic field etc.
For eg: During summer, the length of the iron metre scale becomes more than one metre.

4. ##### Error due to imperfection in experimental technique or procedure

Some errors occur due to imperfection in the experimental arrangement.
For example, while determining the human temperature, a thermometer placed under the armpit will always give a temperature lower than the actual value of the body temperature.
Another example: The loss of heat due to radiation in a calorimeter.

• ### Random error

Random error is the error caused by the individual who measures the quantity. The random error depends on the qualities of the measuring person and the care taken in the measuring process. It is also called as the chance error. In order to minimise random errors, the measurements are repeated several times and the average (arithmetic mean) value is taken as the correct value of the measured quantity. The mean value would be very close to the most accurate reading. When the number of observation is made ‘n’ times, the random error reduces to 1/n times.
If a1, a2, a3 ….. an are the n different readings of a physical quantity when it is measured, the most accurate value is its arithmetic mean value which is given by large begin{aligned} a_{mean}=frac{a_{1}+a_{2}+.....+a_{n}}{n}=frac{1}{n};sum_{i=1}^{n}a_{i} end{aligned}

• ### Absolute error

The magnitude of the difference between the true value of the quantity and the measured value is called the absolute error in the measurement. Since the true value of the quantity is not known, the arithmetic mean of the measured values may be taken as the true value.
If a1, a2, ….. are the measured values of a certain quantity, the errors in ∆a1, ∆a2, ……… in the measurements are
∆a1 = amean – a1
∆a2 = amean – a2

…………………………………
The arithematic mean of all the absolute errors is taken as the final absolute error in the measurement and is known as mean absolute error. large begin{aligned} Delta a_{mean}=frac{left | Delta a_{1} right |+left | Delta a_{2} right |+....+left | Delta a_{n} right |}{n} = frac{1}{n}sum_{i=1}^{n}left | Delta a_{i} right |end{aligned}
The value obtained in a single measurement may be in the range
amean  ±  ∆ amean

• ### Relative error

The ratio of the absolute error to the true value of the measured quantity is called the relative error or fractional error. Since the arithmetic mean value is taken as the true value, the relative error is given by, large begin{aligned} Relative;error,delta a = frac{Delta a_{mean}}{a_{mean}}end{aligned}

• ### Percentage error

It is the relative error exprressed in percentage. large begin{aligned}Percentage;error = frac{Delta a_{mean}}{a_{mean}}times 100%end{aligned}

Example:

When the diameter of a wire is measured using a screw gauge, the successive readings are found to be 1.10 mm, 1.12 mm, 1.14 mm, 1.08 mm, 1.16mm and 1.17mm. Calculate the absolute errors and the relative error in the measurement.

Ans: Arithmetic mean value of the mesurement is largebegin{aligned}a_{mean}=frac{1.10+1.12+1.14+1.08+1.16+1.17}{6} = 1.128 mmend{aligned}

 Difference between amean and measured value Magnitude of errors 1.128 – 1.10 =    0.028 mm 0.028 mm 1.128 – 1.12 =    0.008 mm 0.008 mm 1.128 – 1.14 = –0.012 mm 0.012 mm 1.128 – 1.08 =   0.048 mm 0.048 mm 1.128 – 1.16 = –0.032 mm 0.032 mm 1.128 – 1.17 = –0.042 mm 0.042 mm

The arithmetic mean of the absolute errors (mean of the magnitudes of the errors), large begin{aligned}Delta a_{mean}=frac{0.028+0.008+0.012+0.048+.032+.042}{6}=0.028 mmend{aligned} large begin{aligned} Relative;error,delta a = frac{Delta a_{mean}}{a_{mean}}=frac{0.028}{1.128}=0.0248end{aligned}

Percentage error = largebegin{aligned} = frac{Delta a_{mean}}{a_{mean}}times 100%=frac{0.028times 100}{1.128}=pm 2.48%end{aligned}

More solved problems for the calculation of errors are given in the last section of this article.

## Combination of errors

When a quantity is determined by combining several measurements, the errors in the different measurements will combine in some way or other.

### Error in the sum of the quantities

Let two quantities, A and B have measured values A ± ∆ A and B ± ∆ B respectively.
∆A and  ∆B are the absolute errors in their measurements.

To find the error ∆Z that may occur in the sum Z = A + B, consider
Z ± ∆Z = (A ± ∆A) + (B ± ∆B)
= A + B ± ∆A ± ∆B

The maximum possible error in the value of Z is given by ∆Z = ∆A + ∆B.

Thus, when two quantities are added, the absolute error in the result is the sum of the absolute errors in the measured quantities.

### Error in the difference of the quantities

Let two quantities, A and B have measured values A ± ∆ A and B ± ∆ B respectively.
∆A and  ∆B are the absolute errors in their measurements.

To find the error ∆Z that may occur in the difference Z = A – B, consider
Z ± ∆Z = (A ± ∆A) – (B ± ∆B)
= A – B ± ∆A ± ∆B

Similarly, the maximum possible error in the value of Z is given by ∆Z = ∆A + ∆B.

Thus, when two quantities are subtracted, the absolute error in the result is the sum of the absolute errors in the measured quantities.

### Error in the product of the quantities

Let two quantities, A and B have measured values A ± ∆ A and B ± ∆ B respectively.
∆A and  ∆B are the absolute errors in their measurements.

To find the error ∆Z that may occur in the product Z = AB, consider
Z ± ∆Z = (A ± ∆A)(B ± ∆B)
= AB  ± A∆B ± B∆A ± ∆A∆B

Dividing L.H.S by Z and R.H.S by AB we get, large begin{aligned}1pm frac{Delta Z}{Z}=1pm frac{Delta B}{B}pm frac{Delta A}{A}pm frac{Delta ADelta B}{AB}end{aligned}

Since ∆A and ∆B are small, their products $large frac{Delta ADelta B}{AB}$ is very small and can be neglected. Hence, the maximum fractional error in Z is given by large begin{aligned}frac{Delta Z}{Z}=frac{Delta A}{A}pm frac{Delta B}{B}end{aligned}

Thus, when two quantities are multiplied, the fractional error in the result is the sum of the fractional errors in the measured quantities.

### Error in the quotient of the quantities

Let two quantities, A and B have measured values A ± ∆ A and B ± ∆ B respectively.
∆A and  ∆B are the absolute errors in their measurements.

To find the error ∆Z that may occur in the quotient $large Z=frac{A}{B}$, consider $large Zpm Delta Z=frac{Apm Delta A}{Bpm Delta B}$

Similarly, on solving, we get the maximum fractional error in Z as $large frac{Delta Z}{Z}=frac{Delta A}{A}pm frac{Delta B}{B}$

Thus, when two quantities are divided, the fractional error in the result is the sum of the fractional errors in the measured quantities.

### Error when a quantity is raised to a power

The error ∆Z that may occur when a quantity is raised to its nth power is n times the fractional error in the quantity itself ie., if Z = An , $large frac{Delta Z}{Z}=n;frac{Delta A}{A}$   ——————- eqn (1)

By this equation $large frac{Delta Z}{Z}=frac{Delta A}{A}pm frac{Delta B}{B}$ it is clear that the maximum percentage error in Z is the sum of the maximum percentage error in A and maximum percentage error in B. $large i.e,;;; frac{Delta Z}{Z}times 100=frac{Delta A}{A}times 100+frac{Delta B}{B}times 100$

Similarly from eqn (1), the maximum percentage error in Z is given by $large frac{Delta Z}{Z}times 100=ntimes frac{Delta A}{A}times 100$

If $large Z = frac{A^{textsc{p}}.B^{textsc{q}}}{C^{textsc{r}}}$, then the maximum % error in Z is given by $large frac{Delta Z}{Z}times 100=textsc{p}frac{Delta A}{A}times 100+textsc{q}frac{Delta B}{B}times 100+textsc{r}frac{Delta C}{C}times 100$

For example:- The volume V of a cube of side L is given by
V = L3 = L.L.L
Thus as before, $large frac{Delta V}{V} = frac{Delta L}{L}+frac{Delta L}{L}+frac{Delta L}{L}$
or, the maximum fractional error $large frac{Delta V}{V} = 3;frac{Delta L}{L}$

## Error calculation solved problems

1. A physical quantity p is related to four observations a, b, c and d as follows $large p=frac{a^{2}b^{3}}{csqrt{d}}$ . The percentage error in the measurements in a, b, c and d are 2 %, 3 %, 1 % and 4 % respectively. Calculate the percentage error in p?

Ans: $large p=frac{a^{2}b^{3}}{csqrt{d}}$ $large therefore frac{Delta p}{p}times 100=2frac{Delta a}{a}times 100+3frac{Delta b}{b}times 100+frac{Delta c}{c}times 100+frac{1}{2}frac{Delta d}{d}times 100$

ie, % error in p = 2×(% error in a)+3×(% error in b)+(% error in c)+ ½ ×(% error in d).
= 2 x 2 % + 3 x 3 % + 1 % + ½ x 4 %
= 16 %

2. Period of oscillations of a simple pendulum is measured in an experiment to be 2.01 s, 2.03 s, 1.99 s, 1.98 s, 2.05 s and 2.04 s. Calculate (1) Mean time period of the pendulum, (2) Absolute error in each measurements (3) Average absolute error (4) Relative error and (5) Percentage error.

Ans:

Let T1 = 2.01 s, T2 = 2.03 s, T3 = 1.99 s, T4 = 1.98 s, T5 = 2.05 s, T6 = 2.04 s

(1) Mean time period of the pendulum is $T_{mean}= frac{2.01+2.03+1.99+1.98+2.05+2.04}{6}=2.01667 s$
Since there are only three significant figures, it is proper to have only 3 significant figures in the mean also
ie, Tmean = 2.02 s

(2) Absolute errors in each measurement.

 Absolute errors = Difference between Tmean and measured value Magnitude of errors ∆T1 = Tmean – T1 = 2.02 – 2.01 =   0.01 s 0.01 s ∆T2 = Tmean – T2 = 2.02 – 2.03 =  -0.01 s 0.01 s ∆T3 = Tmean – T3 = 2.02 – 1.99 =   0.03 s 0.03 s ∆T4 = Tmean – T4 = 2.02 – 1.98 =   0.04 s 0.04 s ∆T5 = Tmean – T5 = 2.02 – 2.05 =  -0.03 s 0.03 s ∆T6 = Tmean – T6 = 2.02 – 2.04 =  -0.02 s 0.02 s

(3) Average absolute error.

Average absolute error = arithmetic mean of the magnitude of the errors $large (Delta T)_{mean}=frac{left | Delta T_{1} right |+left | Delta T_{2} right |+left | Delta T_{3} right |+left | Delta T_{4} right |+left | Delta T_{5} right |+left | Delta T_{6} right |}{6}$ $large ie, (Delta T)_{mean}= frac{0.01+0.01+0.03+0.04+0.03+0.02}{6}=0.02;s$

(4) Relative error. $large Relative;error,delta T = frac{Delta T_{mean}}{T_{mean}}=frac{0.02}{2.02}=0.01$

(5) Percentage error. $large Percentage;error = frac{Delta T_{mean}}{T_{mean}}times 100%=frac{0.02times 100}{2.02} approx 1%$

3. Calculate the maximum percentage error in P if $large p=pi r^{2}frac{X}{L}$ . Given that r = (0.32 ± 0.03) cm; X = (19 ± 1); L = (72 ± 0.2 )cm; π is a constant.

Ans:

Given $large p=pi r^{2}frac{X}{L}$ $large frac{Delta p}{p}=frac{Delta pi }{pi }+frac{2Delta r}{r}+frac{Delta X}{X}+frac{Delta L}{L} = 0+2times frac{0.03}{0.32}+frac{1}{19}+frac{0.2}{72}=0.244$
Maximum percentage of error in p = 0.244 × 100 = 24.4 %

4. Two resistances (80 ± 3)Ω and  (130 ± 4)Ω are connected in series. Calculate the effective resistance with error limit and percentage error?

Ans:

Let R1 = 80 Ω, ∆R1 = 3 Ω, R2 = 130 Ω, ∆R2 = 4 Ω

Effective resistance of this series connection

R = R1 + R2 = 80 + 130 = 210 Ω
∆R = ∆R1 + ∆R2 = 3 + 4 = 7 Ω

∴ Effective resistances with error limit = (210 ± 7)Ω $large Percentage;error = frac{Delta R}{R}times 100 % = frac{7}{210}times 100 = 3.33%$

5. The time period of oscillation of a simple pendulum is $large T = 2pi sqrt{frac{l}{g}}$. The length of the pendulum is measured with a scale of least count 1 mm is 60 cm. If the time for 20 oscillations is measured with a stop watch of resolution 0.1 s is 50 s, what is the percentage error in the determination of g?

Ans: $large \* Given;T = 2pi sqrt{frac{l}{g}}; newline ie,;T^{2}=4pi ^{2}frac{l}{g};newline or;g=4pi ^{2}frac{l}{T^{2}}$
Given ∆l = 1 mm = 0.1 cm
l = 60 cm
Time for 20 oscillations = 50 s
∆T = 0.1 s $large \* Percentage;error;in;g=frac{Delta l}{l}times 100% + 2times frac{Delta T}{T}times 100% \ linebreak = frac{0.1}{60}times 100% + 2times frac{0.1}{50}times 100% \ linebreak = 0.167+0.4 \ linebreak =0.567%$

I hope the information in this article helps you to get a brief idea about the errors and types of errors in measurement, and if you believe I missed something or if you have any suggestions, do let me know via comments.