Question

Define percentage error.

Answer 1

Error:

1. Sometimes, the accuracy of the result is limited by the least count of the instrument.
2. Sometimes, environmental changes in temperature and pressure affect both the experiments as well as the measuring instruments.
3. Thus, all measurements involve errors.

Absolute error:

1. It is the difference between the true value and the measured value of the quantity.
2. If a_1, a_{2, …….}a_n are the measured values when the quantity is measured n times.
3. The arithmetic mean of all quantities is given by,

$$\begin{gathered} a_m=\frac{a_1+a_2+........+a_n}{n} \\ where,a_m=truevalue \\ n=numberoftimesthequantityhasbeenmeasured. \end{gathered}$$

4. Absolute errors for individual measurements can be written as:

$$\begin{gathered} △a_1=a_m-a_1 \\ △a_2=a_m-a_2 \\ . \\ . \\ . \\ △a_n=a_m-a_n \end{gathered}$$

Mean absolute error:

1. It is the arithmetic average of the magnitudes of absolute errors in all the measurements of a quantity.
2. It is given by formula,

$\overline{△a}=\frac{△a_1+△a_2+△a_3+.....△a_n}{n}$

Relative error:

1. It is also called as a fractional error.
2. It is defined as the ratio of mean absolute error to the mean value of the quantity measured.
3. It is given by the formula,

$$\begin{gathered} Relativeerror=\frac{Meanabsoluteerror}{Meanvalue}=\frac{\overline{△a}}{a_m} \\ where,\overline{△a}=\frac{△a_1+△a_2+.........△a_n}{n} \\ {a}_m=absoluteerror \end{gathered}$$

Percentage error:

1. When the relative error is expressed in percentage, it is called percentage error.
2. Percentage error is given by,

$Percentageerror=\frac{\overline{△a}}{a_m}\times 100$

3. Alternately, the percentage error can be calculated using the formula:

$Percentageerror=\frac{Leastcountoftheinstrument}{Measuredvalue}\times 100$