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. 

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.

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

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.

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

When a physical quantity is measured incorrectly, it can result in an error.

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

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"`

Systematic errors are consistent deviations from the true value caused by flaws in the measurement system.

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|

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