IEMDaily - Video Lecture Notes

Definition

Define Percentage error

Definition:
The relative error expressed in percentage (i.e. multiplied by 100) is called percentage error.

∴ P e r c e n t a g e e r r o r = \frac{Δ a_{m}}{a_{m}} \times 100 %

Example

Problem on Mean Absolute error

Problem:
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 Mean absolute error of measurements.

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

\frac{| 1.51 - 1.54 | + | 1.51 - 1.53 | + | 1.51 - 1.44 | + | 1.51 - 1.54 | + | 1.51 - 1.56 | + | 1.51 - 1.45 |}{6}

= 0.04

Definition

Mean Absolute Error

Definition:The arithmetic mean of all the absolute errors is called

m e a n a b s o l u t e e r r o r

in the measurement of the physical quantity.

Suppose that 'n' readings taken for the measurement of a physical quantity are

a_{1}, a_{2}, . . . . . . . . . . a_{n}

then the mean value is

a_{m e a n} = \frac{a_{1} + a_{2} + . . . . . . . . a_{n}}{n}

Mean absolute error =

\frac{| a_{m e a n} - a_{1} | + | a_{m e a n} - a_{2} | + . . . . . . . . | a_{m e a n} - a_{n} |}{n}

Definition

Accuracy of Measured Quantities

Accuracy: The accuracy of a measurement is a measure of how close the measured value is to the true value of the quantity. The accuracy in measurement may depend on several factors, including the limit or the resolution of the measuring instrument.
For example, suppose the true value of a certain length is near

3.678 c m

. In one experiment, using a measuring instrument of resolution

0.1 c m

, the measured value is found to be

3.5 c m

, while in another experiment using a measuring device of greater resolution , say

0.01 c m

, the length is determined to be

3.38 c m

. The first measurement has more accuracy because it is closer to the true value.

Definition

Precision

Definition:

Precision is a measurement of the repeatability, or consistency, of a measurement. It is possible to have a very precise measurement without scatter (or noise) that is repeatable and would be considered precise (repeatable), however, it can be inaccurate because of an instrument error.

Example: Let us consider a weight measuring instrument which measures the weight of steel cube. Exact weight of the steel cube is 50 kg.

The instrument shows the weight of the cube as 47.819 kg, 47.823 kg, 47.835 kg and 47.847 kg.

From the above observations, we can conclude that the instrument has some precision since it almost reproduces same output value 47.8 kg. But the instrument does not have accuracy since the measured values are not close to the actual weight 50 kg.

Result

difference between accuracy and precision

The level of agreement between the actual measurement and the absolute measurement is called accuracy. The level of variation that lies in the values of several measurements of the same factor is called as precision.
Accuracy represents the nearness of the measurement with the actual measurement. On the other hand, precision shows the nearness of an individual measurement with those of the others.
Accuracy is the degree of conformity, i.e. the extent to which measurement is correct when compared to the absolute value. On the other hand, precision is the degree of reproducibility, which explains the consistency of the measurements.
Accuracy is based on a single factor, whereas precision is based on more than one factor.
Accuracy is a measure of statistical bias while precision is the measure of statistical variability.
Accuracy focuses on systematic errors, i.e. the errors caused by the problem in the instrument. As against this, precision is concerned with random error, which occurs periodically with no recognizable pattern.

Definition

Least Count

The least count of an instrument is the smallest measurement that can be taken accurately with it.

Definition

Least Count

Definition: The smallest value up to which an instrument can measure is called least count of the instrument.
Example:
least count of meter rule is up to 1 mm.
least count of Vernier caliper is 0.01 cm.

Definition

Least Count Error

The smallest value that can be measured by the measuring instrument is called its least count. Measured values are good only up to this value. The least count error is the error associated with the resolution of the instrument.

Example: During Searle's experiment, zero of the Vernier scale lies between

3.20 \times 10^{- 2} m

and

3.25 \times 10^{- 2}

m of the main scale. The 20

^{t h}

division of the Vernier scale exactly coincides with one of the main scale divisions. When an additional load of 2 kg is applied to the wire, the zero of the Vernier scale still lies between