Measurement Uncertainties and Error Types

Uncertainties in Measurement

• Uncertainty, denoted as u(G), is always a positive value.

Random Errors

• Random errors can stem from several sources.
• Human error in reading instruments or results.
• Limitations of the equipment being used.
• External factors influencing measurements.
• Lack of reproducibility in the measurement, represented by T.
• Non-optimal measurement methods.
• Performing a series of measurements accompanied by statistical treatment reduces error.
• A reliable experiment has low random error.
• An accurate experiment has low systematic error.
• Systematic error results in values skewed in a particular direction relative to the true value, such as "zero error" on a scale.
• An exact measurement is both accurate and reliable.

Type A Uncertainty (Repeated Measurements)

• For a set of estimated values (x), ranging from x1 to xn.
• The result is written with a number of significant figures to match uncertainty.
• u(x) = σn-1 / (√n), where σ represents standard deviation.
• Round up to the nearest significant figure by excess.
• Calculations can be performed with a calculator using the stat D (CALC) function, which takes single-variable statistics.

Experimental Conditions

• State if the sample size affects the measurements
• State the type of uncertainty associated with a measurement.

Compatibility and Measurement

• Incompatibility exists when |measured value - reference value| / U(x) > 2.
• The measurement is not reliable with respect to a given reference value
• Compatibility exists when |sing| < 2.

Progressive Waves

• Progressive waves are disturbances propagating through a medium.
• Energy is transferred without matter displacement
• Displacement happens away from the source at a distance

Types of Waves

• Mechanical waves propagate through a material medium, like sound.
• Electromagnetic waves propagate through a vacuum and material mediums (e.g., satellite-T), including X-rays and infrared waves.

Wave Retardation

• ΔT represents the time interval separating two distinct points in a disturbance.

Wave Speed/Celerity

• Wave speed (v) relies on the material medium
• v = d/Δt, where d is distance, and Δt is time.
• Speed is measured in m/s.
• The speed of light (c) is 300,000 km/s in a vacuum.
• Mechanical progressive waves transfer energy and propagate without displacing matter.

Periodic Waves

• Periodic waves are disturbances that repeat identically over regular time intervals (period).
• Period (T) is the duration of one elementary repeating unit, measured in seconds (s).

Frequency

• Frequency (f) is the inverse of the period (f = 1/T).
• Hertz (Hz) is the unit of frequency (s⁻¹).
• Speed of sound in air (Vair) is 340 m/s, and speed of sound in water (Vwater) is 1500 m/s.

Measuring Duration in Two Different Mediums

• Time = distance * (1/Vair - 1/Vwater)

Vibratory State and Distance

• Small distance separates 2 points in the same vibratory state (in phase).

Speed of Ultrasound

• The speed of ultrasound is the same as that of sound waves.

Type B Uncertainty (Single Measurement)

• Expressed as U(L) = 1 graduation / 2
• If there is a double reading, U(L) is multiplied by 2.
• If the measurement on a scale is U(λ) = (λ) / 10

Measuring Wavelength

• λ = v/f and v = λ * f

Measuring the Epicenter of a Seism

• Method for estimating the epicenter of a seismic event.
• d = (Vs * Vp)(ts - tp) / (vp - vs)

Explore uncertainties in measurement, including random and systematic errors. Learn about Type A uncertainty calculations for repeated measurements and how statistical treatment reduces error. Understand the difference between reliable and accurate experiments.