Data Analysis Techniques David Butler School of MAE Nanyang Technological University Singapore

2

Hmmmm 3

Error Every measurement has a degree of uncertainty associated with the measurement Error is the difference between the measured value and the true value of the object being measured. Uncertainty is a quantification of the doubt about the measurement result. 4

How many apples are there in this picture? Type of answers How many apples are there in this picture? How many particles are there in this picture? 5

Data & Data Analysis Data refers to the collection of organized information. It can be numbers, words, images of measurements or observations. Data Analysis The process of looking at and summarising data to extract useful information and develop conclusions. Exploratory Data Analysis discovering new features in the data Confirmatory Data analysis- confirming or falsifying existing hypotheses 6

Examples of data QUALITATIVE QUANTITATIVE Scores in a test Measurement of wind speed Number of mosquitos collected in a trap The speed of light in a vacuum A robot Graph of intensity vs angle High speed video of a bullet Videos of interviews 7

Numerical Data Analysis There is no general method for treating qualitative data For quantitative data, there are a number of standard techniques Not all standard numerical techniques are applicable to all types of research Be careful when trying to quantify qualitative data 8

Precision is ability to tell the same story over again Accuracy & Precision Accuracy is the proximity of answer to the true value and the absence of systematic errors Precision is ability to tell the same story over again The influence of random error can be minimized by averaging a large data set 9

Types of error Systematic errors – typically can be attributed to the measurement instrument used. Two examples are calibration error (failure to calibrate) and zero error Random errors arise due to accidental errors in measurement such as estimating the reading, random fluctuations, incorrectly copying down the results 10

Systematic errors Can not be estimated by repeating the equipment with the same equipment Can be minimized Careful calibration Best possible techniques 11

These can be reliably estimated by repeating measurement Random Errors These can be reliably estimated by repeating measurement Can typically be quantified using the standard deviation formula This is the most common error 12

How to determine the measurement error numerically Determination of Boiling Point of a liquid You measure 32oC. You then repeat the experiment many times and collect a set of results Trial Number 1 2 3 4 5 Measured value 31.9 32.1 31.8 32.2 The best estimate is the average of the 5 measured results 13

How to determine the measurement error numerically Take the difference between the highest measured value and the average will give the maximum deviation 32.2-32.02 = 0.18 The best estimate is to get the average deviation known as standard deviation. The boiling point of the liquid is 32.0 oC ± 0.2 oC 14

Histograms Take a large number of measurements and count the number of occurrence of each value 31.9, 32.1, 31.8, 32.2, 32.1, 31.8, 32.1, 32.4, 32.1, 32.2, 32.3 15

Normal Distribution The uncertainty of an average decreases as more data points are averaged. The plot will typically follow a normal distribution curve It is a property of a normal distribution curve that 68% of the measurements lie within one standard deviation on either side of the mean 16

17

So a typical Singaporean male is taller than an Indonesian Male! 158 170 18

So a typical Singaporean male is taller than an Indonesian Male! 158 170 19

So a typical Singaporean male is taller than an Indonesian Male! 158 170 20

So a typical Singaporean male is taller than an Indonesian Male! Be careful of making statements without a full data analysis!ca 158 170 21

f (V0 – σ) < f (V0)< f (V0 + σ) Error Propagation Consider the ideal gas equation. Aim: to determine the number of moles of gas in a sample We need to measure Pressure, Volume and Temperature to get the value of n. Each measured quantity has an error value To determine the error in n we need to propagate the error in the individual measurement We want to measure a quantity V The result is f (V) (V0 – σ) < V0< (V0 + σ) Type equation here. The desired property f s then within the range : f (V0 – σ) < f (V0)< f (V0 + σ) 22

Error Propagation If a function only contains additions and subtractions the propagated error can be calculated as: If a function only contains multiplications and division operations the propagated error can be calculated as: 23

Example Lets measure the molar heat of solvation in LiCl in water. Weighing an mount of LiCl Measuring the volume of water Measuring the temperature change when the material dissolve The heat of solvation (H) is expressed in terms of the three measurements as: The function H contains multiplication and division operations. 24

Lets measure the molar heat of solvation of LiCl in water Example Lets measure the molar heat of solvation of LiCl in water If two separate masses were weighed and added to the solvent, the equation is given as: Step 1: to calculate the uncertainty in m = m1+m2 using the error propagation rule Step 2: Then use the total mass and its calculated error and determine H and σH 25

Determine the diameter of the bottle using a set of digital calipers Example Determine the diameter of the bottle using a set of digital calipers 26

Possible Error Sources Man Machine Varying force Eyesight Stiffness No calibration alignment resolution Different grips Reading of scale Calculation errors Measurement variation exact reference? Temperature stability calibration Appropriate equipment Lighting Unclear instructions hardness datum Alignment of jaws shape Ergonomics Method Measurand Environment 27

Watch your number of decimal places Calculated Values This looks impressive Watch your number of decimal places Calculated Average 24.498 mm Std Dev 0.635625676 Used 24.50 0.64 Realistically this is what you can measure 28

Calculated Values mm1 29

Summary Error does not mean a mistake or blunder but refers to imprecision in measurement Random or indeterminant errors are inherent in all measurements The source of systematic errors are Instrumental errors, method errors and personal errors Numerical results without error analysis are as good as useless 30

Thank You Acknowledgement to Dr G. Rooshan Deen, NTU http://www.sigmaxi.org/programs/ethics 31

32