Question

#Moderatorschallenge☑️✅

Can u pls tell which formula is this and tell me some little bit about it.

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Answer 1

✴ Errors in Measurement :

▪ 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.

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

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

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▶ True value of the quantity :

✒ If a1, a2, a3, ... , an are the readings of an experiment, then true value of the quantity is given by the arithmetic mean,

$\underline{\boxed{\bf{\pink{\overline{a}=\dfrac{a_1+a_2+a_3+...+a_n}{n}=\dfrac{1}{n}\sum{a_i} }}}}$

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⏭ Additional information :

✒ Absolute error = True value - Measured value

$\underline{\boxed{\bf{\red{\Delta{a_i}=\overline{a}-a_i}}}}$

✒ Final absolute error = Arithmetic mean of absolute errors

$\underline{\boxed{\bf{\green{\Delta\overline{a}=\dfrac{|\Delta{a_1}|+|\Delta{a_2}|+|\Delta{a_3}|+...+|\Delta{a_n}|}{n}}}}}$

✒ Relative error or fractional error = (Final absolute error) ÷ (True value)

$\underline{\boxed{\bf{\orange{\delta{a}=\dfrac{\Delta\overline{a}}{\overline{a}}}}}}$

✒ Percentage error = Relative error × 100

Answer 2

Answer:

ABSOLUTE, RELATIVE AND PERCENTAGE ERROR –

➸ Let us suppose different observational values : $\tt{a_1, a_2, a_3, a_4 ....a_n.}$

➸ Arithmetic mean = $\sf{a_{mean} = \dfrac{(a_1+a_2+a_3....a_n)}{n} = \dfrac{\displaystyle\sum\limits^{n}_{i=1} \: a_{i}}{n} = \dfrac{1}{n}\displaystyle\sum\limits^{n}_{i=1} \: a_{i}}.$

[This is the formula you wanted to know about.]

➸ The arithmetic mean of the observational values is said to be the true value of the quantity.

➸ The difference between measurement are the observational value and the true value of the quantity is referred to as the absolute error of the measurement.

➸It is denoted by |∆a|.

➸ $\sf{\triangle\:a_1 = a_1 - a_{mean}}$

$\sf{\triangle\:a_2 = a_2 - a_{mean}}$

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$\sf{\triangle\:a_n = a_n - a_{mean}}$

➸ ∆a can be positive/negative whereas |∆a| is always changed to positive.

➸ Now, mean of the Absolte error : $\sf{\Delta\:a_{mean}=\dfrac{|\Delta a_1|+|\Delta a_2|+|\Delta a_3|...+...|\Delta a_n|}{n}= \dfrac{\displaystyle\sum\limits^{n}_{i=1}\:a_i}{n}}.$

➸ The true value of the observation i.e a is between $\tt{a_{mean} \pm \Delta a_{mean}}$, hence, $\tt{a_{mean}-\Delta a_{mean} \leq a \leq a_{mean} + \Delta a_{mean}}$.

➸ Relative error is is the ratio of mean absolute error, $\tt{\Delta a_{mean}}$ and mean value of observation, $\tt{a_{mean}}$.

➸ $\sf{\Delta a_{mean} = a_{mean}}$.

➸ Relative error when expressed in the form of percentage is called percentage error.

➸ $\tt{\delta\:a = \dfrac{\Delta a_{mean}}{a_{mean}} \times 100\%}$.