Error Types, Accuracy, Precision in Measurements – Absolute, Relative, Percentage Error

Errors in measurements are an unavoidable part of experimental physics and arise due to limitations in instruments, observer skills, and environmental conditions. Understanding different types of errors—absolute error, relative error, and percentage error—is essential for analyzing the accuracy and precision of measured values. These error concepts help in comparing results, improving measurement techniques, and ensuring reliability in scientific calculations, especially in Class 11 Physics and competitive exams.

THEORY OF ERRORS (PRECISION OF MEASUREMENT)

Foundation of a science, particularly of physics, is experiment. In physics, the discovery of a new law or principle is acceptable only, when the experiment approves it. In order to get as close to the truth as possible, physicists have not only been trying to design more and more perfect instruments but have also developed a theory of errors, which helps in eliminating possible errors in the observations.

The theory of error originates from the following two assumptions :

1. That every observer performing an experiment is careless to some extent and is thus liable to commit mistake

2. That every instrument used in an experiment is defective to some extent. Hence, the observation made with an instrument has got some inherent error.

In fact, errors and mistakes are two separate things. The term mistake is used to denote a gross error. It can be avoided by care on the part of the experimenter and should be avoided as the result cannot be corrected for them. On the other hand, the uncertainty in a measurement is called an error. It tells the limits, within which the true value may lie.

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TYPES OF ERRORS IN MEASUREMENTS

There is only a limit up to which measurements can be made with a given measuring instrument. This limit is called the least count of the instrument. For example, a metre rod can measure length accurately up to 0.1 cm, whereas a vernier callipers can measure length accurately up to 0.01 cm. However, when we make use of various measuring instruments, various types of errors creep into the observations.

There can be the following possible types of errors in measurements :

📚 1. Constant Error
When the result of a series of observations are in error by the same amount, the error is said to be a constant one. For example, in measuring the length of a cylinder by a vernier callipers whose graduations are faulty, say one centimetre on the scale is actually 0.9 centimetre, the measured length will always be greater than the true value by a constant amount. In order to avoid constant error, the measurements are made by as many different methods as possible.

📚 2. Systematic Error
A systematic error is one that always produces an error of the same sign. These are due to known causes. This type of error is eliminated by detecting the source of the error and the rule governing this error. Systematic error may be subdivided into the following types :

(i) Instrumental errors. These are inherent errors of the apparatus and the measuring instruments used. A simple example is the zero-error of a measuring instrument. All instrumental errors come under this category. The instrumental error, if any, can be detected by interchanging two similar instruments or by using different methods for measuring the same physical quantity.

(ii) Observational or personal errors. The errors due to the personal peculiarities of the experimenter are known as personal errors. For example, parallax error while reading the positions of uprights on the optical bench. The more experienced the experimenter is, the lesser it is.

These errors can be minimized by obtaining several readings carefully and then taking their arithmetical mean.

(iii) Errors due to external causes. These errors are caused by external conditions (pressure, temperature, wind, etc). For example, expansion of a scale due to increase in temperature. These errors can be taken care of by applying suitable corrections.

(iv) Errors due to imperfection. Sometimes, even when we know the nature of the error, it cannot be eliminated due to imperfection in experimental arrangement. For example, in calorimetry, loss of heat due to radiation, the effect on weighing due to buoyancy of air, etc. These errors will always exist but observations can be corrected for them.

📚 3. Random Errors
These errors are due to unknown causes and are sometimes termed as chance errors. In an experiment, even the same person repeating an observation may get different reading every time. For example, measuring diameter of a wire with a screw gauge, one may get different readings in different observations. It may happen due to many reasons. For example, due to non-uniform area of cross-section of the wire at different places, the screw might have been tightened unevenly in the different observations, etc. In such a case, it may not be possible to indicate, which observation is most accurate. However, if we repeat the observation a number of times, the arithmetical mean of all the readings is found to be most accurate or very close to the most accurate reading for that observation. That is why, in an experiment, it is recommended to repeat an observation a number of times and then to take their arithmetical mean.

If $a_1, a_2, a_3, \dots, a_n$ are the $n$ different readings in an experiment, their arithmetic mean is given by

$$
\bar{a} = \frac{a_1 + a_2 + a_3 + \dots + a_n}{n}
$$

or

$$
\bar{a} = \frac{1}{n} \sum_{i=1}^{n} a_i
$$

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GROSS ERRORS (OR MISTAKES)

These are the results of sheer carelessness on the part of experimenter. No correction can be applied for them and hence must be avoided by exercising due care.

They are of the following types :

📝 Neglect of the sources of error. This type of gross error results due to negligence towards sources of errors. For example, for plotting the field of a magnet, improper setting of the magnet along NS-line, presence of magnetic material in the vicinity of the magnet, etc.

📝 Reading the instrument incorrectly. Sometimes, over a metre scale, one cm is divided in 20 parts instead of 10 parts or in a voltmeter or ammeter, 1 volt or 1 ampere might have been divided in 20 or 5 parts instead of 10. The experimenter may read the instrument without paying due attention to the value of 1 division.

📝 Improper recording of the reading. This type of mistake is committed by the experimenter, when he records the reading wrongly. For example, he may record 21°3 in place of 23°1. It happens in undue haste or when the experimenter mentally carries the reading for a long time.

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CALCULATION OF ERRORS IN MEASUREMENTS (ABSOLUTE ERROR AND MEAN ABSOLUTE ERROR)

Define Absolute Error
The difference of the true value (standard value) and the experimental value (observed value) of a physical quantity is called absolute error.

It is expressed in the units of measured quantity. Therefore,

$$
\text{Absolute Error} = \text{true value} – \text{experimental value}
$$

Suppose in an experiment on determining the value of acceleration due to gravity, the value comes out to be 945 cm s⁻². We know that the value of the acceleration due to gravity is generally taken as 980 cm s⁻². Therefore,

Absolute error in the value of acceleration due to gravity is given by

$$
\text{Absolute Error} = 980 – 945 = 35 \text{ cm s}^{-2}
$$

📘 Calculation of Absolute Error in Measurement :
Absolute error refers to the magnitude of the difference between the true value and the measured value of a physical quantity.

Suppose a physical quantity is measured n times, and the measured values are: \begin{array}{l} a_1, a_2, a_3, …, a_n \end{array}

The arithmetic mean of these values is: \begin{array}{l} a_m = \dfrac{a_1 + a_2 + … + a_n}{n} \end{array}

This mean value is taken as the true value of the quantity if it is unknown otherwise.

The absolute error in each measurement is given by: \begin{array}{l} \Delta a_i = | a_i – a_m | \end{array} where Δa_i is the absolute error for each individual measurement.

Define Mean Absolute Error
Mean absolute error is the average of all absolute errors

Mean absolute error is given by:

\begin{array}{l} \bar{a} = \dfrac{\sum | a_i – a_m |}{n} \end{array}

This helps in determining the precision of the measurement. The final result of the measurement is written as:

\begin{array}{l} a = a_m \pm \bar{a} \end{array}

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RELATIVE ERROR (FRACTIONAL ERROR) AND PERCENTAGE ERROR

The knowledge of the relative error is more important than that of the absolute error in a measurement.

Define Relative or Fractional Error
The relative error is defined as the ratio of the absolute error to the true value (mean value) of the quantity measured.

$$
\text{Therefore, relative error} = \frac{\text{absolute error}}{\text{true value}}
$$

\begin{array}{l} \text{Relative Error} = \dfrac{\bar{a}}{a_m} \end{array}

It has no units. This gives an idea of how significant the error is compared to the measured value. Therefore, in the above example, the relative error in the value of acceleration due to gravity

$$
\text{Therefore, relative error} = \frac{35}{980} = 0.036\
$$

Define Percentage Error
The relative error expressed as percentage is known as percentage error.

$$
\text{Therefore, Percentage Error} = \frac{\text{absolute error}}{\text{true value}} \times 100
$$

\begin{array}{l} \text{Percentage Error} = \left( \dfrac{\bar{a}}{a_m} \right) \times 100 \% \end{array}

This helps in assessing the reliability of the measurement.

Therefore, in the above example, the percentage error in the value of acceleration due to g

$$
\text{Therefore, Percentage Error} = \frac{35}{980} \times 100 = 3.6\%
$$

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ACCURACY IN MEASUREMENTS

Accuracy and precision are fundamental concepts in measurements, crucial for fields like physics, engineering, and medicine. Understanding these concepts helps improve the reliability of measurements and ensures correctness in scientific and technical work.

Definition
Accuracy refers to how close a measured value is to the true or accepted value of a physical quantity. It indicates correctness.

📘 Examples of Accuracy

• If a clock shows the time as 3:00 PM and it is actually 3:00 PM, the clock is accurate.
• If a thermometer reads 100°C while the actual temperature is 99.9°C, it is considered accurate.

📘 Accuracy Formula

Accuracy is determined using the percent error formula:

Percent Error = {(Measured Value – True Value)/True Value} × 100

This formula gives us the accuracy as a percentage. The less the percent error the more accurate the value is.

📘 Types of Accuracy

1. Point Accuracy: Accuracy at a specific point on the instrument’s scale.
2. Accuracy as Percentage of Scale Range: Measured as a fraction of the instrument’s range.
3. Accuracy as Percentage of True Value: How close the measured value is to the true value.

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PRECISION IN MEASUREMENTS

Definition
Precision refers to how consistently a measurement produces the same result under the same conditions. It measures repeatability and reliability.

📘 Examples of Precision

• If you weigh an object five times and get 4.6 kg each time, your measurements are precise.
• A machine cutting metal pieces to exactly 5.00 cm repeatedly exhibits precision.

📘 Types of Precision

1. Repeatability: Consistency of measurements taken by the same instrument under the same conditions.
2. Reproducibility: Consistency when different people use different instruments over extended periods.

📘 Precision Formula

Precision is measured using the standard deviation formula: Standard Deviation (Precision) = √(∑(Measurement −Mean)²​​/ Number of measurements)

A lower standard deviation indicates higher precision.

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DIFFERENCE BETWEEN ACCURACY AND PRECISION IN MEASUREMENTS

Aspect	Accuracy	Precision
Definition	Closeness to the true value	Consistency of repeated measurements
Dependence	Depends on the true value	Independent of accuracy and true value
Example	A scale showing your true weight	A scale showing the same weight every time
Illustration	Darts hitting the bullseye	Darts landing close to each other but not necessarily on the bullseye
Measurement Error	Error is the difference from the true value	Error is the spread of repeated values

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PERMISSIBLE ERROR IN A RESULT

Even when an experimenter has managed to avoid gross error by exercising due care and the instrumental errors have been avoided by selecting a perfect apparatus for the experiment, still another type of error may creep in the result of an experiment. It is due to the limits put on the measuring abilities of various kinds of instruments used, while performing the experiment owing to their least counts. Such an error is called the permissible error.

For example, suppose that the temperature of a liquid is read at 25.4°C on a thermometer calibrated in degrees. If the reading is estimated to the nearest 0.1°C, the temperature should be recorded as 25.4 ± 0.1°C. This is the scientific way of recording a reading with the limits of error. In the present case, the temperature of liquid will be in the range 25.3°C to 25.5°C. Similarly, if the length of an object is read as 34.7 cm with a metre rod calibrated in cm and correct to the nearest 0.1 cm, then the length of the object should be recorded as 34.7 ± 0.1 cm.

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Solved Numerical Problem

In an experiment, the value of refractive index of glass was found to be $1.54$, $1.53$, $1.44$, $1.54$, $1.56$ and $1.45$ in successive measurements. Calculate (i) mean value of refractive index of glass (ii) absolute error in each measurement (iii) mean absolute error (iv) relative error and (v) percentage error. Also express the result in terms of absolute error and percentage error.

Ans. (i) Mean value of refractive index,

$$\mu = \frac{1.54 + 1.53 + 1.44 + 1.54 + 1.56 + 1.45}{6} = 1.51$$

(ii) Taking $\mu$ as the correct value, the absolute error in the six measurements are :

$1.51 – 1.54 = -0.03$;

$1.51 – 1.53 = -0.02$;

$1.51 – 1.44 = +0.07$;

$1.51 – 1.54 = -0.03$;

$1.51 – 1.56 = -0.05$ and

$1.51 – 1.45 = +0.06$

(iii) Mean absolute error in the value of $\mu$,

$$\Delta \mu = \frac{0.03 + 0.02 + 0.07 + 0.03 + 0.05 + 0.06}{6} = \frac{0.26}{6} = 0.04$$

(iv) Relative error in the value of $\mu$,

$$\delta \mu = \frac{\Delta \mu}{\mu} = \frac{0.04}{1.51} = 0.02649 = 0.03$$

(v) Percentage error in the value of $\mu = 0.03 \times 100 = 3$%

Value of $\mu$ expressed in terms of absolute error,

$$\mu = 1.51 \pm 0.04$$

Value of $\mu$ expressed in terms of percentage error,

$$\mu = 1.51 \pm 3\%$$

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Very Short Frequently Asked Questions (FAQs) and Answers

🎯 What are the three main types of errors in measurement ?

📝 Gross Errors
📝 Random Errors
📝 Systematic Errors

🎯 How Gross Errors occur in Measurements ?

Gross errors occur due to human oversight and mistakes in reading, recording, or interpreting data. These are the most common types of errors and can be minimized through careful measurement and proper techniques.

🎯 Write the examples of Gross Errors

🔖 Misreading the scale of an instrument (e.g., reading 23 as 28)
🔖 Recording incorrect values
🔖 Calculation mistakes

🎯 How can we reduce gross errors in measurement?

Gross errors can be minimized by careful reading and recording of data, and by increasing the number of experimenters taking independent measurements at different points and use the average value.

🎯 How Random Errors occur in Measurements ?

Random errors occur unpredictably due to fluctuations in experimental conditions such as temperature changes, voltage variations, or mechanical vibrations. These errors cause variations in repeated measurements.

🎯 How to Reduce Random Errors ?

🔖 Take multiple readings and use statistical methods to determine the average and standard deviation.
🔖 Use precision instruments with fine calibration.

🎯 How Systematic Errors occur in Measurements ?

Systematic errors are consistent and repeatable inaccuracies that arise from known sources. They can be classified into three types:

a) Environmental Errors : These occur due to external conditions such as temperature, pressure, humidity, and electromagnetic fields.

Example: Measuring body temperature in a room where the air conditioner stops working, leading to increased temperature readings.

b) Observational Errors : These arise due to human bias, incorrect apparatus setup, or carelessness in observation. Parallax errors are a common type of observational error.

Example: Taking a reading from an angle rather than directly in front of the scale.

c) Instrumental Errors : These errors occur due to faulty instrument construction, calibration errors, or wear and tear of instruments. Zero errors in Vernier calipers and screw gauges are common examples.

Example: A weighing scale that does not reset to zero before measurement.

Instrumental Errors Arise Due To: ⭐ Inherent constraints of devices ⭐ Misuse of apparatus ⭐ Loading effects

🎯 Give an example of an environmental error.

Measuring body temperature in a room where the air conditioning fails, affecting the accuracy of the reading.

🎯 What is a parallax error?

A parallax error occurs when the observer views a scale from an angle rather than directly, leading to incorrect readings.

🎯 What are zero errors?

Zero errors occur when an instrument does not start from zero, leading to consistent measurement inaccuracies.

🎯 What is an error in measurement?

Error in measurement is the difference between the true value and the measured value of a physical quantity.

🎯 How is absolute error different from relative error?

Absolute error is the direct difference between measured and true value, whereas relative error is the ratio of absolute error to the true value.

🎯 Why is percentage error useful?

Percentage error helps compare the reliability of different measurements by giving a standardized percentage value.

🎯 Can a measurement be precise but not accurate?

Yes, if measurements are consistently incorrect but close to each other.

🎯 Can a measurement be accurate but not precise?

Yes, if measurements are close to the true value but vary widely.

🎯 How can accuracy and precision be improved?

By using high-quality instruments, minimizing environmental interference, and calibrating tools regularly.

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Units and Measurements – Chapter Summary

The chapter Units and Measurements builds the foundation of physics by introducing how physical quantities are measured and expressed using standard systems of units. It begins with the concept of physical quantities, followed by the classification into fundamental and derived quantities, and the use of SI units for universal consistency. You will also learn about different systems of units such as CGS, MKS, and FPS, along with important rules for writing units correctly.

The chapter further explains measurement techniques, accuracy, precision, and types of errors including absolute error, relative error, and percentage error, which are crucial for analyzing experimental data. Significant figures and rounding off rules help in maintaining proper precision in calculations. Additionally, dimensional analysis is introduced as a powerful tool to check the correctness of equations, derive relations, and convert units from one system to another.

This chapter is essential for solving numerical problems and forms the base for all future topics in physics. Make sure to explore detailed explanations, formulas, and JEE PYQs for each topic.

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