# Error Analysis

Error(Error Analysis) :- The result of every measurement by any measuring instrument contains some uncertainty. This uncertainty is called error.

True Value and Measured Value :-  To understand the concept of errors in measurement, one should know the two terms, “true value” and “measured value”. The true value is impossible to find by experimental means. It may be defined as the average value of an infinite number of measured values. The measured value is a single measuring of the object with the aim of being as accurate as possible.

So “error may be defined as the difference between the measured value and the true value”.

Accuracy and Precision

(Error Analysis)

Accuracy refers to the “closeness of a measured value to a standard or true value” while precision refers to the “closeness of a series of measurements of a same quantity under similar conditions”. Precision tells us to what resolution or limit the quantity is measured. Precision is related to the least count of the measuring instrument.

For example, suppose the true value of a certain length is 3.678 cm and two instruments with different resolutions, up to 1 (less precise) and 2 (more precise) decimal places respectively, are used. If first measures the length as 3.5 cm and the second as 3.38 cm, then the first has more accuracy but less precision while the second has less accuracy and more precision.

Types of errors(Error Analysis) :-

Errors in measurement can be broadly classified as…

1. Systematic errors and
2. Random errors

Systematic errors :- Systematic errors are usually caused by measuring instruments that are incorrectly calibrated or are used incorrectly. These  errors tend to be in one direction, either positive or negative. Systematic errors can further be classified into three categories :-

1. Instrumental errors:- These errors are due to imperfect design or wrong calibration of the measuring instrument. For example inadequately calibrated thermometer, zero mark of vernier scale not coinciding with the zero mark of main scale, worn out meter scale at one end etc. These errors can be reduced by using more accurate instruments.
2. Errors due to imperfection in experimental technique :- If the technique of measurement is not accurate (for example measuring temperature of human body by placing thermometer under armpit resulting in lower temperature than actual) then the result will be erroneous. Other factors such as temperature variations, wind velocity, humidity etc. also contribute to the error.
3. Personal errors :- Personal error refers to faults introduced as a result of the observer’s fault, such as carelessness in taking observations, lack of proper setting of apparatus or individual bias.

To remove systematic errors, one must :-

• Improve experimental techniques
• select better instruments
• remove personal bias as far as possible.

Random Errors :- The errors which are random with respect to sign and size and occur irregularly are called random errors.

Causes of random errors :- Unpredictable fluctuations in experimental conditions conditions like unpredictable fluctuations in temperature, supply voltage, mechanical vibrations of experimental set-ups, etc.

Least Count :- The smallest value that can be measured by the measuring instrument is called its least count. All Measured values are good only up to the least count of the measuring instrument.

Least Count Error :- Consider the meter scale that we generally use. What is its least count? Its smallest division is in millimeter (mm). Hence, its least count is 1 mm i.e. 10-3 m i.e. 0.001 m. Clearly, this meter scale can be used to measure length from 10-3 m to 1 m.

Let let us suppose that a piece of rod is measured using the meter rod and result of measurement comes out to be 0.587 m (= 58.7 cm = 587 mm). Let us write out result of measurement as (580 mm + 7 mm). Now here the first part of the reading, i.e., 580 mm is undoubtable, because we are 100% sure that length is more than 580 mm, but the last part of our reading, i.e., 7 mm is suspicious because our scale has least count of 1 mm, so it may be 8 mm or 6 mm. Hence the correct way of reporting the length should be x = (0.587 ± 0.001) m.

Hence we can define least count error as “the error which arise due to least count or resolution of the measuring instrument”.

### (Error Analysis)

Absolute Error (|Δa|):- The difference between the true value and the individual measured value of the quantity is called the absolute error of the measurement. Suppose a physical quantity is measured n times and the measured values are a1, a2, a3……….an. The arithmetic mean (am) of these values is

$\displaystyle {{a}_{m}}=\frac{{{a}_{1}}+{{a}_{2}}+{{a}_{3}}+.............{{a}_{n}}}{n}=\frac{1}{n}\sum\limits_{i=1}^{n}{{{a}_{i}}}$

If the true value of the quantity is not given then mean value (am) can be taken as the true value.

Then the absolute errors in the in individual measured values are-

|Δa1| = |am – a1|

|Δa2| = |am – a2|

……………

……………

|Δan| = |am – an|

The arithmetic mean of all the absolute errors is defined as the final or mean absolute error (Δa)m  of the value of the physical quantity a

$\displaystyle {{(\Delta a)}_{m}}=\frac{|\Delta {{a}_{1}}|+|\Delta {{a}_{2}}|+.........+|\Delta {{a}_{n}}|}{n}=\frac{1}{n}\sum\limits_{i=1}^{n}{|\Delta {{a}_{i}}|}$

So if the measured value of a quantity be ‘a’, true value of the quantity is am( at = am) and the error in measurement be Δa, then the measured value (a) can be written as

a = am ± Δa

Or in other words if we do a single measurement, the value we get may be in the range….

(am – Δam)  ≤   a  ≤  (am + Δam)

Relative or Fractional Error :– It is defined as the ratio of the mean absolute error (Δam) to the true value or the mean value (am) of the quantity measured.

$\displaystyle \mathbf{Relative}\text{ }\mathbf{or}\text{ }\mathbf{Fractional}\text{ }\mathbf{Error}=\frac{Mean\,\,absolute\,\,error}{Mean\,\,value}=\frac{{{(\Delta a)}_{m}}}{{{a}_{m}}}$

Percentage Error :- When the relative error is expressed in percentage, it is known as percentage error.

Percentage error = relative error × 100

Or

$\displaystyle \mathbf{Percentage}\text{ }\mathbf{error}\text{ }=\frac{mean\,\,absolute\,\,error{{(\Delta a)}_{m}}}{true\,\,value({{a}_{m}})}\times 100%$

## By Manoj Kumar Verma

Hi, I'm Manoj Kumar Verma, a physics faculty having 7 years of teaching experience. I have done B.Tech (E.E.). I am also a YouTuber and Blogger. This blog is dedicated to help students learn the physics concepts easily.