Revision: Units and Measurements Physics HSC Science (General) 11th Standard Maharashtra State Board

Measurement is the process of determining the magnitude of a physical quantity by comparing it with a predefined standard unit of the same kind. 

The basic physical quantities that cannot be derived from other quantities and serve as the foundation for all measurements is called fundamental quantities.

Derived quantities are physical quantities that depend on and can be calculated using fundamental quantities.

One kilogram (kg) is the S.I. unit of mass. It is defined as the mass of the international prototype of the kilogram, which is a platinum-iridium alloy cylinder stored at the International Bureau of Weights and Measures in France.

Mass is the measure of the amount of matter in an object. It is a fundamental property of matter and does not change with location or the object’s state.

Accuracy is about how close your measured value is to the true, actual value of that quantity.

or

The quality or state cate of being accurate or the ability to work or perform without making mistakes.

Accuracy = Mean value - True Value

Precision is about getting reproducible results. If you measure the same thing multiple times and get nearly identical answers, your measurements are precise.

or

The quality, condition, or fact of being exact and accurate or the closeness of the set of values obtained from identical measurements of quantity.

Precision = Individual Value - Arithmetic Mean Value

In real experiments, it is very difficult to get exactly the same answer every single time. This difference or possibility of error is called uncertainty.

For a given set of measurements of the same quantity, the arithmetic mean of all the absolute errors is called mean absolute error in the measurement of that physical quantity.

`triangle "a"_"mean" = (triangle"a"_1 + triangle"a"_2 + ......+ triangle"a"_"n")/"n" = 1/"n"` \[\sum_{i=1}^n\triangle a_i\]

The ratio of the mean absolute error in the measurement of a physical quantity to its arithmetic mean value is called relative error.

Relative error = `(triangle "a"_"mean")/"a"_"mean"`

When we measure any physical quantity (length, mass, time, temperature, etc.), the value we obtain is usually not exactly equal to its true value. The difference between the measured value and the true value is called measurement error.

Random errors are unpredictable fluctuations in measurements that vary in both magnitude and direction.

OR

The error that occurs irregularly with respect to sign and size, being unpredictable and varying in magnitude and direction — which can be minimised by taking a large number of observations — is called random error.

When relative error is represented as percentage it is called the percentage error.

Percentage error = `(triangle"a"_"mean")/("a"_"mean") xx 100`

1. For a given set of measurements of a quantity, the magnitude of the difference between mean value (Most probable value) and each individual value is called absolute error (Δa) in the measurement of that quantity.
2. absolute error = |mean value - measured value|
   Δa₁ = |a_mean - a₁|
   Similarly,
   Δa₂ = |a_mean - a₂|,
             `\vdots           \vdots             \vdots`
   Δa_n = |a_mean - a_n|

The ratio of the mean absolute error to the mean value of the quantity measured is called relative error or fractional error.

When the relative/fractional error is expressed in percentage, it is called percentage error.

The magnitude of the difference between the true value and the measured value of a quantity is called absolute error.

The arithmetic mean of the magnitudes of absolute errors in all the measurements of a quantity is called the mean absolute error.

The measured value of a physical quantity denoting the number of digits in which we have confidence — where a larger number indicates greater accuracy of measurement — is called significant figures.

How far each reading is from the mean:

\[\Delta a_i=
\begin{vmatrix}
a-a_i
\end{vmatrix}\]

Average error over all readings:

\[\Delta a_{\mathrm{mean}}=\frac{\sum_{i=1}^n\Delta a_i}{n}\]

How big the error is, compared to the mean value (no units):

Relative Error: \[\frac {Δa_{mean}}{a}\]

The relative error as a percent: 

Percemtage Error: \[\frac{\Delta a_{\mathrm{mean}}}{a}\times100\%\]