Question

Using same vernier calipers, the length of a cylinder in different measurements to be 2.36 cm, 2.27 cm, 2.26 cm, 2.28 cm, 2.31 cm, 2.28 cm and 2.29 cm. Calculate the percentage error.

Answer 1

Given :-

Seven values are given, measured by Vernier calipers to measure the length of a cylinder

these measurements are :

a₁ = 2.36 cm , a₂ = 2.27 cm , a₃ = 2.26 cm ,

a₄ = 2.28 cm , a₅ = 2.31 cm , a₆ = 2.28 cm ,

a₇ = 2.29 cm .

To find :-

→ Percentage error in the measurement

Formulae used :-

→  to calculate true value of quantity

$\boxed{\tt{a_{mean}=\frac{a_1+a_2+a_3+.....+a_n}{n}}}$

where , a_(mean) is the taken as the true value of quantity .

→to Calculate absolute error of the individual measurement

The magnitude of the difference b/w individual measurement and true value of quantity is known as absolute error of the measurement.

$\boxed{\tt{\triangle a_n= a_n - a_{mean}}}$

( where n denotes the nth measurement )

( and Δ aₙ represent the individual absolute error )

important to note here is that

Δ aₙ may be negative in some cases , but

| Δ aₙ |  is always positive .

→ to Calculate mean absolute error

The arithmetic mean of all absolute errors is called final or mean absolute error .

$\boxed{\tt{\frac{\triangle a_{mean}=(\mid \triangle a_1 \mid+\mid \triangle a_2\mid+...+\mid\triangle a_n\mid)}{n}}}$

where, Δ a_(mean) is the mean absolute error

→ to Calculate percentage error

$\boxed{\tt{percentage\:error , \triangle a =\frac{\triangle a_{mean}}{a_{mean}} \times 100 \%}}$

Solution :-

● Calculating true value of quantity

$\rm{a_{mean}=\frac{2.36+2.27+2.26+2.28+2.31+2.28+2.29}{7}}$

$\blue{\rm{a_{mean}=2.29 \:cm }}$

●Calculating individual absolute errors of measurements

$\rm{\triangle a_1 = a_1 - a_{mean} = 2.36 - 2.29 = 0.07\:cm }$

$\rm{\triangle a_2 = a_2 - a_{mean} = 2.27 - 2.29 = -0.02\:cm }$

$\rm{\triangle a_3 = -0.03\:cm}$

$\rm{\triangle a_4 = -0.01\:cm}$

$\rm{\triangle a_5 = 0.02\:cm}$

$\rm{\triangle a_6 = -0.01\:cm}$

$\rm{\triangle a_7 = 0 }$

●Calculating mean absolute error

$\rm{\triangle a_{mean}=\frac{\mid 0.07\mid+\mid-0.02\mid+\mid-0.03\mid+\mid-0.01\mid+\mid0.02\mid+\mid-0.01\mid+\mid0\mid}{7}}$

$\blue{\rm{\triangle a_{mean} = 0.02 \: cm}}$

●Calculating percentage error

$\rm{percentage \:error ,\:\triangle a = \frac{0.02}{2.29}\times100\%}$

$\boxed{\red{\bf{percentage\:error,\triangle a = 0.87 \:\%}}}$