IEMDaily - Video Lecture Notes

Definition

Backlash Error

Sometimes, due to wear and tear of threads of screw in instruments such as micrometer screw gauge, it is observed that on reversing the direction of rotation of the thimble, the tip of the screw does not start moving in the opposite direction at once due to slipping, but it remains stationary for a part of rotation. This causes error in observation which is called the backlash error. To avoid this, we should rotate the screw only in one direction.

Example

Absolute error

Definition:
The magnitude of the difference between mean value and each individual value is called

a b s o l u t e e r r o r .

Example:The diameter of a wire as measured by a screw gauge was found to be 1.002 cm, 1.004 cm and 1.006 cm. The absolute error in the readings will be:Solution:

M e a n = \frac{1.002 + 1.004 + 1.006}{3}

= 1.004 c m

Hence, absolute errors

Δ a_{1} = 1.002 - 1.004

= 0.002 c m

Δ a_{3} = 1.004 - 1.004

= 0.000 c m

Δ a_{3} = 1.006 - 1.004

= 0.002 c m

Definition

Mean Absolute Error

The magnitude of the difference between the individual measurement and the true value of the quantity is called the absolute error of the measurement. The arithmetic mean of all the absolute error is taken as the mean absolute error of the value of the physical quantity.

Example

Calculate Relative error

Definition:The ratio of the mean absolute error in the measurement of a physical quantity to its most probable value is called

r e l a t i v e e r r o r .

R e l a t i v e e r r o r = \frac{| Δ a_{m} |}{a_{m}}

Example:
In an experiment, the values of refractive indices of glass were found to be 1.54, 1.53, 1.44, 1.54,1.56 and 1.45 in successive measurements. Find the Relative error.
Solution:

M e a n = \frac{1.54 + 1.53 + 1.44 + 1.54 + 1.56 + 1.45}{6}

= 1.51

| M e a n a b s o l u t e e r r o r | = \sqrt{\frac{1}{6} \sum (\bar{x_{1}} - \bar{x})^{2}} = 0.04

R e l a t i v e e r r o r = \frac{A b s o l u t e e r r o r}{M e a n} = \frac{0.04}{1.51} = 0.026 \approx 0.03

Definition

Relative Error

The relative error is the ratio of the mean absolute error (

δ a_{m e a n}

) to the mean value (

a_{m e a n}

) of the quantity measured.
Relative error

= \frac{δ a_{m e a n}}{a_{m e a n}}

Definition

Accuracy, precision and error

Accuracy: Measure of closeness of the measured value to its actual value.
Precision: Measure of repeatability of a set of values of measurement.
Error: Difference between the true value and measured value.
To minimize error, instruments need to calibrated well, noise from the surroundings need to be reduced, multiple measurements must be recorded, etc.

Definition

Absolute error

Definition:
The magnitude of the difference between mean value and each individual value is called

a b s o l u t e e r r o r .

For the measurement

a_{1}

the absolute error is

| Δ a_{1} | = | a_{m} - a_{1} |

Similarly in the measurment

a_{2}

it is

| Δ a_{2} | = | a_{m} - a_{2} |

Example:
The length of metal plate was measured using a vernier callipers of least count 0.01 cm. The measurements made were 3.11 cm, 3.13 cm, 3.14 cm and 3.10 cm. Find absolute error for measurements.
Solution:
Mean length

a_{m}

\frac{3.11 + 3.13 + 3.14 + 3.10}{4} = 3.12 c m

Absolute error for measurements

| Δ a_{1} | = | a_{m} - a_{1} | = | 3.12 - 3.11 | = 0.01 c m

| Δ a_{2} | = | a_{m} - a_{2} | = | 3.12 - 3.13 | = 0.01 c m

| Δ a_{3} | = | a_{m} - a_{3} | = | 3.12 - 3.14 | = 0.02 c m

| Δ a_{4} | = | a_{m} - a_{4} | = | 3.12 - 3.10 | = 0.02 c m

Example

Example on Relative error

Example:
John measures the size of the metal ball as 3.97 cm but the actual size of it is 4 cm. Calculate the relative error.
Solution:The measured value of metal ball