Definitions [21]
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.
Define one kilogram, the S.I. unit of mass.
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.
Define mass.
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.
Think of dimensions as the "DNA" of physical quantities - they tell us the fundamental building blocks that make up any measurement.
Simple Definition: Dimensions are the powers to which the fundamental units (like length, mass, time) must be raised to express any physical quantity.
Understanding Dimensional Formulas
A dimensional formula shows how any physical quantity depends on the fundamental dimensions.
Format: [L0M0T¹I¹]
Where 0,0,1,1 are the powers (can be positive, negative, or zero).
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.
Define Mean absolute error.
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"` \[\sum_{i=1}^n\triangle a_i\]
Define relative error.
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.
OR
The type of error that consistently occurs in the same direction (either positive or negative), arising from imperfect design or calibration of measuring instruments, imperfection in experimental technique, or carelessness of an individual is called systematic 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 a physical quantity is measured incorrectly, it can result in an error.
Define percentage error.
When relative error is represented as percentage it is called the percentage error.
Percentage error = `(triangle"a"_"mean")/("a"_"mean") xx 100`
Define absolute error.
- 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.
- absolute error = |mean value - measured value|
Δa1 = |amean - a1|
Similarly,
Δa2 = |amean - a2|,
`\vdots \vdots \vdots`
Δan = |amean - an|
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 ratio of the mean absolute error to the mean value of the quantity measured is called relative error or fractional 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.
Formulae [5]
The best estimate (mean) of repeated readings:
\[\mathrm{mean~}a=\frac{a_1+a_2+\cdots+a_n}{n}\]
How far each reading is from the mean:
\[\Delta a_i=
\begin{vmatrix}
a-a_i
\end{vmatrix}\]
Average error over all readings:
\[\Delta a_{\mathrm{mean}}=\frac{\sum_{i=1}^n\Delta a_i}{n}\]
The relative error as a percent:
Percemtage Error: \[\frac{\Delta a_{\mathrm{mean}}}{a}\times100\%\]
How big the error is, compared to the mean value (no units):
Relative Error: \[\frac {Δa_{mean}}{a}\]
Concepts [15]
- Quantitative Science
- System of Units
- Derived Quantities and Units
- Rules and Conventions for Writing SI Units and Their Symbols
- Measurement of Length
- Measurement of Mass
- Measurement of Time
- Dimensions and Dimensional Analysis
- Accuracy, Precision and Uncertainty in Measurement
- Errors in Measurements>Systematic Errors
- Errors in Measurements>Random Errors
- Estimation of Errors
- Combination of Errors
- Significant Figures
- Definitions of SI Units and Constants
