ME - 495 Mechanical and Thermal Systems Lab

Fall 2011

Chapter 3: Assessing and Presenting

Experimental Data

Professor: Sam Kassegne, PhD, PE

Lecture Covers

Terminologies (error, precision, resolution, etc)

Statistical Theory Based on Population Statistical Theory Based on Sample

(student ‘t’, Chi squared, etc) Homework Problems

1) TERMINOLOGIES

ERRORS Error = ε = Xm – Xtrue:

Difference between measured and true values Can never be calculated exactly

Bias/Systematic errors: errors that occur the same way each time a measurement is taken Cannot be treated using statistics Commonly a zero offset or scale error

Precision/Random errors: different for each measurement but average to zero

COMBINED ERROR

Bias error > typical precision error typical precision error > bias error

CLASSIFICATION OF ERROR

Bias or systematic error Calibration errors Certain consistently recurring human errors Certain errors caused by defective equipment Loading errors Limitations of system resolution

Precision or random error Human error Errors caused by disturbance of the equipment Errors caused by fluctuating experimental

conditions Errors derived from insufficient measuring-

system sensitivity

OTHER ERRORS

Illegitimate error Blunders and mistakes during experimentation Computational errors in data reduction

Errors that are sometimes bias and sometimes precision error Errors from instrument backlash, friction,

hysteresis Errors from calibration drift, variation in test and

environmental conditions

Introduction to UNCERTAINTY

Uncertainty:

Where: Bx = bias uncertainty

Px = precision uncertainty

2x

2xx PBU

ESTIMATING PRECISION UNCERTAINTY

Distribution of error: probability that an error of given size will occur

Population: finite or infinite group from which samples are drawn

Sample: limited set of measurements Gaussian or Normal Distribution (bell curve) Probable difference/Confidence Interval:

Estimate of precision uncertainty of the measured sample

SAMPLE vs. POPULATION

A sample has a defined number of members (n)

A population may have: (n) members or may be infinite

A sample is randomly taken from a population of indefinite size

PROBABILITY DISTRIBUTIONSHistogram (Tossed Dice)Distribution Types:

Gaussian (standard)Students t-distributionχ2 distribution (lower right)

2) THEORY BASED ON POPULATION Probability Density Function (PDF)

Describes random precision error of sampling

Probability = area under curve

2

1

)(x

x

dxxf

GAUSSIAN DISTRIBUTION

For Gaussian Distribution:

x = magnitude of particular measurement μ = mean value of entire population σ = standard deviation of entire population

μ Xavg = (x1+x2+…+xn)/n for large nFor a measurement (x) the deviation (d)

d = x - μ

2

2

2

x

e2

1xf

GAUSSIAN DISTRIBUTION (ctd.)

σ = standard deviation Probability estimates

Standard curve

n

d...dd 2n

22

21

2

zexp

2

1)x(f

2

x

z

Probability

population entire the of deviation standardthe

population entire the of value mean the

tmeasuremen particular a of magnitude x

direction one iny probabilit z

xz

Question:

1) What is the area under the curve between z = -1.43 and z=1.43?

2) What range of z will contain 90% of the data?

3) THEORY BASED ON SAMPLE

Mean Standard Deviation

n

XX

ni

1ii

samples ofnumber n 1;-n freedom of Degree

)()(... 1

2222

2

2

1

n

ii

nxx

XnXXXXXXX

S

Useful Terms in Theory Based on Samples

Central Limit Theorem - For large number of n for a given number of samples, the distribution of the mean values is Gaussian. Eq 1.

Confidence Interval (c%) - used to indicate the reliability of an estimate. How likely the interval is to contain the parameter is determined by the confidence level.For a population with unknown mean  and known standard deviation , a confidence interval for the population mean, based on a simple random sample of size n is: Eq 2.

Remember that: Eq 3 & Eq 4.

nx /

nzx c

2/

n

xxz

x /

nzc

2/

A) Student’s t-statistic

Useful for estimating confidence interval for small samples (n << 30)

Assumes the underlying population satisfies the Gaussian distribution

Considers the distribution of a quantity ‘t’ for a specific confidence interval

samples ofnumber n

sample ofdeviation standard

average population true

average sample

x

x

S

x

nS

xt

Student’s t-statistic

Probability for the t-statistic approaches Gaussian as n infinity

t-statistic eliminates probable outliers and what remains must center around the true value = n-1

is called the degree of freedom)

Student’s t-statistic Example 3.6 on pg 74

samples ofnumber n

sample ofdeviation stan.

73) pgon 3.6 table(from

)( freedom of degree and )2(percent confidence afor on distributi- t

level confidencegiven afor avg x valuein they uncertaintprecision

,2

,2

x

x

xx

S

t

Pn

StP

designates the area (hence probability) outside a specified value of t

Conversely,the probability that a given value will lie inside of the limits t is 1-.

n = 14, = 1.009 oz. Sx = 0.04178, = 13.

From Table 3.6, t0.025,13 = 2.160

two sided interval = +/- t0.025,13 Sx/sqrt(n) = +/- 0.02412.

= 1.009 +/- 0.024 oz with CI of 95%

x

B) CHI SQUARED TEST

* The χ2 test measures ‘goodness of fit’

* Tests the probability that a sample of data is or is not normally distributed.

n

1i i

2ii2

E

EO

total area under curve = unity.

Curve starts at chi^2 = 0

The curve is not symmetrical. As number of data increases, it becomes similar to normal distribution.

CHI SQUARED TEST – measures the ‘goodness of fit’

The χ2 goodness of fit test Oi = the number of observed data in a bin

Ei = the number of expected data in a bin N = the number of data bins, or “class intervals” = degrees of freedom=1- number of bins

n

1i i

2ii2

E

EOLarge discrepancy between Oi and Ei

CHI SQUARED EXAMPLE

For a group of 50 trees on a set area of land, to determine if the trees prefer dry soil we examine the following:

If 50% of the land is dry, 30% is moist, and 20% is wet, then if there were no preference, we would expect:

25 trees on dry soil; 15 on moist; and 10 on wet If actually: 40 on dry; 8 on moist; 2 on wet 40-25=15; 152/25=9 | 8-15=7; 72/15=3.2 | 2-10=8; 82/10=6.4 χ2= (9+3.2+6.4)= 18.6

Examining the previous graph it is shown that it is extremely unlikely that there is no soil preference, as our Chi squared value falls far above the questionable probability (Note that bin-size, n = 3; nu = 2)