1. Measurement errors and uncertainties
Northrop : pgs 7 – 11
Hughes : pgs
Measurement errors and uncertainties

2. Errors in Measurement Types of errors in measurement :-
1. Gross Errors – due to human errors
2. System Errors – due to technical errors

3. Gross Errors
1. Reading the instrument before it has reached its steady state. 2. Parallax error when reading an analog meter scale. 3. Mistakes in recording measured data and in calculating a derived measurand. 4. Misuse of the instrument.

4. System Errors
1. The instrument is not calibrated, and has an offset - Loss of calibration and zero error can occur because of long term component value changes due to ageing, or changes associated with temperature rise. 2. Reading uncertainty due to the presence of random noise. 3. Reading uncertainty due to slow, or long term drift in the system. Drifts can cause slow changes in system sensitivity and/or zero. Drift may arise as the result of a slow temperature change as a system warms up. Drift or system offset can also arise from dc static charges.

5. Definition of terms Error in nth measurement 𝜀 𝑛 = 𝑋 𝑛 − 𝑌 𝑛
𝜀 𝑛 = 𝑋 𝑛 − 𝑌 𝑛
% 𝜀= 𝜀 𝑛 / 𝑌 𝑛 ×100%
𝑋 𝑛 =𝑡ℎ𝑒 𝑛 𝑡ℎ 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑣𝑎𝑙𝑢𝑒
𝑌 𝑛 =𝑡ℎ𝑒 𝑎𝑐𝑡𝑢𝑎𝑙, 𝑡𝑟𝑢𝑒, 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑜𝑟 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒
Error: deviation of the measured value from the true value.
True value: standard or reference of known value or a theoretical value

6. Accuracy, An of the nth measurement is defined as :-
𝐴 𝑛 =1− 𝑌 𝑛 − 𝑋 𝑛 𝑌 𝑛
% 𝑎𝑐𝑐𝑢𝑟𝑎𝑐𝑦=% 𝐴 𝑛 =100−%𝜀= 𝐴 𝑛 ×100%
Accuracy = closeness of a measurement to the true value
Accurate measurements require the use of a precision measurement system which is calibrated against a certified, accurate standard.

7. The precision of the nth measurement is defined as:
𝑃 𝑛 =1− 𝑋 𝑛 − 𝑋 𝑋
𝑋 =𝑠𝑎𝑚𝑝𝑙𝑒 𝑚𝑒𝑎𝑛 𝑜𝑓 𝑁 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡𝑠
𝑋 = 1 𝑁 𝑛=1 𝑁 𝑋 𝑛
Precision: a measure of the reproducibility of the measurements; given a fixed value of a variable, precision is a measure of the degree to which successive measurements differ from one another i.e., a measure of reproducibility or agreement with each other for multiple trials.

8. 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛, 𝑑 𝑛 = 𝑋 𝑛 − 𝑋
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛, 𝐷 𝑁 = 1 𝑁 𝑛=1 𝑁 𝑑 𝑛
𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛, 𝑆 𝑁 = 1 𝑁 𝑛=1 𝑁 𝑑 𝑛 2 = 𝜎 𝑥
𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒, 𝑆 𝑁 2 = 1 𝑁 𝑛=1 𝑁 𝑋 𝑛 2 − 𝑋 2 = 𝜎 𝑥 2
Tolerance: maximum deviation allowed from the conventional true value.

9. Resolution is related to the precision of the measurement
and is basically the smallest unit of the measured quantity which can be reliably be detected.
Oscilloscopes generally give poor resolution.
Limiting Error (LE) is an important parameter used in specifying instrument accuracy. The limiting error, or guarantee error, is given by manufacturers to define the outer bounds or the expected worst case error.
Example, a voltmeter is specified as having an accuracy of 2% of its full scale reading. If on the 100V scale, the meter reads 75V,
The LE in this reading, V max = | V /V |
V = 0.02 x 100 = 2 V
V max = (2/75) =
The limiting error in this reading = 2.67%
The reading is expressed as 75 V  2.67%

10. Uncertainty analysis
Any system that relies on a measurement system will involve some amount of uncertainty (doubt). The uncertainty may be caused by individual inaccuracy of sensors, limitations of the display devices, random variations in measurands, or environmental conditions. The accuracy of the total system depends on the interaction of components and their individual accuracies.
Two methods are commonly used for determining the uncertainty.
Using limiting error
Square roof of sum of squares

11. Limiting Error in the calculation of the dc power in a resistor
Power dissipated in the resistor with uncertainty = 6400 W  2.6% or  W

12. Using Square Root of some of square

13. Analysing resistors in series
Rs = R1 + R2, R1 = 10   0.5 , R2 = 10   0.5 

14. Analysing resistors in parallel

15. Try exercise : 13, 14, 19, 20, 21, 24 - hughes