Measurement and Uncertainty

Uncertainties and Errors

Calculating uncertainties and errors is an important part of scientific inquiry and experimental design.

Random vs Systematic Uncertainties

Random uncertainties are due to the inherent limitations of your measuring device or the variations in the quantity being measured. Systematic uncertainties, on the other hand, arise from consistent errors in the experimental setup or measuring instruments.

Random Uncertainties in Analog and Digital devices

Uncertainty in a Scale Measuring Device is equal to the smallest increment divided by 2.
Uncertainty in a Digital Measuring Device is equal to the smallest increment.

Propagating uncertainties

For the product rule (when $A \times B = C$), the formula is: $∆C/C = ∆A/A + ∆B/B$, where $C$ is the final result, $A$ and $B$ are the input values, and $∆A$ and $∆B$ are their respective uncertainties.

For the sum rule (when $A + B = C$), the formula is: $∆C = ∆A + ∆B$, where $C$ is the final result, $A$ and $B$ are the input values, and $∆A$ and $∆B$ are their respective uncertainties.

These formulas work for both random and systematic uncertainties.

Percentage and relative uncertainties

The absolute uncertainty is the actual amount by which the value is uncertain, e.g. 6.0 ± 0.1 cm

Relative uncertainty is the ratio of the uncertainty to the measured quantity, expressed as a percentage.

To change between them use the formula:
$R = \frac{∆A}{A} \times 100\%$, where $R$ is the relative uncertainty, $A$ is the value, and $∆A$ its absolute uncertainty.

SI Units

The seven base units

The SI system defines seven base units that are used to express all physical quantities. These are:

• Length: meter (m)
• Mass: kilogram (kg)
• Time: second (s)
• Electric current: ampere (A)
• Temperature: Kelvin (K)
• Amount of substance: mole (mol)
• Luminous intensity: candela (cd)

Prefixes

The SI system uses prefixes to denote multiples or submultiples of the base units. For example, the prefix “kilo” (k) means 1000 times the base unit, so one kilogram (kg) is equal to 1000 grams (g). Similarly, the prefix “milli” (m) means 1/1000th of the base unit, so one millisecond (ms) is equal to 1/1000th of a second (s).

Derived units

In addition to the base units, the SI system defines derived units, which are combinations of the base units. For example, the unit for velocity is meters per second (m/s), which is derived from the base units of length and time. Other derived units include units for force (Newton), energy (Joule), and power (Watt).

The SI system is designed so that all derived units are expressed in terms of the base units in a coherent way. This means that the units are related to each other by a set of defined conversion factors, and that there are no numerical constants involved in the conversions. For example, the unit of force (Newton) is defined as 1 kilogram-meter per second squared (kg m/s^2), which is a coherent combination of the base units of mass, length, and time.

Scalars vs Vectors

Definitions:

Scalar

• A scalar quantity is a quantity that is completely described by its magnitude. Examples of scalar quantities include length, mass, time, temperature, speed, and energy
• Scalar quantities are represented using a single numerical value with a unit. For example, the length of a pencil is 15 cm, or the temperature of a room is 25°C.
• Scalar quantities can be added, subtracted, multiplied, and divided using regular arithmetic operations. For example, the sum of two masses, 2 kg and 3 kg, is 5 kg.

Vector

• A vector quantity is a quantity that is described by both its magnitude and its direction. Examples of vector quantities include displacement, velocity, acceleration, force, and momentum.
• Vector quantities are represented using both a numerical value and a direction. They can be represented graphically as an arrow, where the length of the arrow represents the magnitude of the vector, and the direction of the arrow represents the direction of the vector.

Mathematical operations

• Scalar quantities can be added, subtracted, multiplied, and divided using regular arithmetic operations. For example, the sum of two masses, 2 kg and 3 kg, is 5 kg.
• Vector quantities cannot be added, subtracted, multiplied, and divided using regular arithmetic operations. Vector addition and subtraction require the use of vector components, and vector multiplication requires the use of vector dot product or vector cross product.

Units

• Scalar quantities have only one type of unit associated with them. For example, the unit for length is meters, the unit for time is seconds, and the unit for mass is kilograms.
• Vector quantities have both a magnitude and a direction, so they require two types of units. The magnitude of a vector is expressed in terms of a scalar unit, while the direction of a vector is expressed using a direction unit. For example, the unit for velocity is meters per second (m/s), where meters represent the scalar unit and seconds represent the direction unit.