Question

$\large{\underline{\mathrm{\red{Question-}}}}$

How to calculate the true value if it is not given?

Answer this question on the basis of systemic error - units and measurement.

Class 11th, physics​

Answer 1

SoluTion:

Firstly, what is a systematic error?

Answer - Errors that tend to be in one direction only are termed as systematic errors.

Sources :

• Instrumental errors
• Imperfection in experimental techniques
• Personal errors

Now, there is an another error, named absolute error.

Absolute error: Difference in magnitude of measured value and true value is called absolute error.

So, mathematically,

$\large{\boxed{\sf{\red{Absolute\:error\:=\: |measured \: value \: - \: true \: value|}}}}$

$\rule{200}2$

Let's come to the question now,

Here, we have to find the true value, if it is not given in the question or numerical.

Some steps are as follows :

$\purple{\sf{\underline{Step\:1)}}}$

Take larger number of readings.

$\sf{a_{1} , a_{2} , a_{3} ------ a_{n}}$ Where, $\sf{a_{1} , a_{2}\:etc}$ are measured values.

$\purple{\sf{\underline{Step\:2)}}}$

Mean of all measured values will be taken as true value.

$\large{\boxed{\sf{\green{a_{m} = \dfrac{a_{1} + a_{2} + a_{3} ----- a_{n}}{n}}}}}$

where, $\sf{a_m}$ is true value.

$\purple{\sf{\underline{Step\:3)}}}$

Calculate all absolute errors.

• $\sf{\Delta a_{1} = |a_{1} - a_{m}|}$

• $\sf{\Delta a_{2} = |a_{2} - a_{m}|}$

• $\sf{\Delta a_{3} = |a_{3} - a_{m}|}$

• $\sf{\Delta a_{n} = |a_{n} - a_{m}|}$

$\purple{\sf{\underline{Step\:4)}}}$

Mean of all absolute errors will be taken as the absolute errors of the readings.

$\large{\boxed{\sf{\blue{\overline{\Delta{a}} = \Delta a_{1} + \Delta a_{2} + \Delta a_{3} + -------- + \Delta a_{n}}}}}$

$\purple{\sf{\underline{Step\:5)}}}$

Final answer will be written as :

$\huge{\boxed{\sf{\pink{a_{m} \pm \bar{\Delta\:{a}}}}}}$

Hence, we get the answer.

Answer 2

Solution :

▪ While doing an experiment several error can enter into the results. Errors may be due to faulty equipment, carelessness of the experimenter or random causes.

▪ The first two types of errors can be removed after detecting their cause but the random errors still remains.

▪ No specific cause can be assigned to such errors.

↪ When an experiment is repeated many times, the random errors are sonetimes positive and sometimes negative.

↪ Thus, the average of a largr number of the results of repeated experiments is close to the true value.

↪ However, there is still some uncertainty about the truth of this average.

↪ The uncertainty is estimated by calculating the standard derivation described below....

Let X1, X2, X3, ... , Xn are the results of an experiment repeated n times. The standard derivation $\sigma$ is defined as

$\boxed{\pink{\bf\sigma=\lim\:(i=1\:to\:n)\sqrt{\dfrac{1}{n}\sum(X_i-\overline{X})^2}}}$

where $\sf\sigma=\dfrac{1}{n}\sum(X_i)$ is the average of all the values of X.

✴ Additional information :-

✏ The best value of X derived from these experiment is $\overline{X}$ and the uncertainty is of the order of $\pm{\sigma}$.

✏ In fact $\overline{X}\pm 1.96\sigma$ is quite often taken as the interval in which the true value should lie. It can be shown that there is a 95% chance that the true value lies within $\overline{X}\pm 1.96\sigma$.

▪ All this is true if the number of observations n is large. In practice if n is greater than 8, the results are reasonably correct.