LECTURER PROF.Dr. DEMIR BAYKA AUTOMOTIVE ENGINEERING LABORATORY I

STATISTICAL TREATMENT OF EXPERIMENTAL DATA DISCRETE FREQUENCY DISTRIBUTIONS

FREQUENCY GRAPH

MEASURES OF CENTRAL TENDENCY ARITHMETIC MEAN IT PROVIDES THE BEST ESTIMATE OF AN UNBIASED DISTRIBUTION OF DATA

MEASURES OF CENTRAL TENDENCY MEDIAN IT IS THE VALUE AT THE MIDDLE POSITION OF A DISTRIBUTION OF DATA IT IS USUALLY USED WHEN THE DISTRIBUTION IS BIASED

MEASURES OF CENTRAL TENDENCY MODE IT IS THE VALUE HAVING THE HIGHEST FREQUENCY IN THE SAMPLE DISTRIBUTION

MEASURES OF DISPERSION OF DATA VARIANCE (MEAN SQUARE DEVIATION )

MEASURES OF DISPERSION OF DATA STANDARD DEVIATION

UNBIASED ESTIMATES A) THE BEST AVAILABLE ESTIMATE OF THE UNKNOWN STANDARD DEVIATION OF THE UNIVERSE IS GIVEN BY 

THE USE OF THIS EXPRESSION BECOMES IMPORTANT ESPECIALLY WHEN n IS SMALL FOR LARGE VALUES OF n HOWEVER, S >  sample ALWAYS

CONTINUOUS DISTRIBUTIONS IF WE HAD A SET OF 100 DATA VALUES SUCH AS 23.26, , etc THEN THE FREQUENCY GRAPH WOULD PROBABLY HAVE VERY FEW VALUES THAT WERE THE SAME

CONTINUOUS DISTRIBUTIONS

THE ONLY APPARENT MEANINGFUL QUANTITY APPEARS TO BE THE DENSITY OF THE “DOTS”

CONTINUOUS DISTRIBUTIONS 16 LET US DIVIDE THE DATA BY INCREMENTS

CONTINUOUS DISTRIBUTIONS NOW LET US COUNT HOW MANY DATA POINTS ARE BETWEEN AND

IF MORE MEASUREMENTS WITH A MORE ACCURATE DEVICE WERE TAKEN

AND IF THE DATA WERE INCREASED

THE INTERVAL MUST BE CHOSEN *LARGE ENOUGH TO BE MEANINGFUL *SMALL ENOUGH TO GIVE DETAIL

TYPICAL FREQUENCY CURVES 1. BINOMIAL DISTRIBUTION IF AN EVENT CAN OCCUR IN 2 WAYS (A) OR (B) AND IF (p) IS THE PROBABILITY OF OCCURRENCE OF (A) AND (q) IS THE PROBABILITY OF OCCURRENCE OF (B)

THE PROBABILITY P OF (A) OCCURRING n TIMES OUT OF N TRIALS IS : 1. BINOMIAL DISTRIBUTION TYPICAL FREQUENCY CURVES

IF EVENTS ARE OCCURRING RANDOMLY INDEPENDENT OF EACH OTHER THEN THEY WILL MAKE UP A POISSON DISTRIBUTION TYPICAL FREQUENCY CURVES 2. POISSON DISTRIBUTION

P(n) IS THE PROBABILITY OF A DISCRETE EVENT OCCURRING (n) TIMES DURING A SAMPLING INTERVAL (T) WHERE (a) IS THE AVERAGE OF THESE EVENTS DURING (T) AND

TYPICAL FREQUENCY CURVES 3. GAUSSIAN DISTRIBUTION WHEN THE PROBABILITY OF (A) OR (B) OCCURRING IS THE SAME AND IF THE NUMBER OF TRIALS ARE LARGE THEN THIS DISTRIBUTION IS CALLED THE NORMAL DISTRIBUTION

TYPICAL FREQUENCY CURVES 3. GAUSSIAN DISTRIBUTION THIS IS TYPICAL OR UNBIASED RANDOM ERROR DISTRIBUTION

3. GAUSSIAN DISTRIBUTION /   x m x 1 x /  P x

3. GAUSSIAN DISTRIBUTION  (NORMALIZED) P

3. GAUSSIAN DISTRIBUTION (NORMALIZED) P  -2 +2

3. GAUSSIAN DISTRIBUTION (NORMALIZED)  P

3. GAUSSIAN DISTRIBUTION (NORMALIZED)    …...

PROPERTIES OF GAUSSIAN DISTRIBUTION 1. P(x) >0 FOR ALL FINITE VALUES OF x AND P(x) -->0 AS X-->oo 2. P(x) IS SYMMETRICAL ABOUT ITS MEAN

3. MEAN ( x ) DETERMINES ITS LOCATION AND (  DETERMINES ITS AMOUNT OF DISPERSION

4. 5. WHEREIS THE PROBABILITY OF HAVING  BETWEEN   AND  

INTERPRETATION OF FREQUENCY CURVES 1. EXPECTED VALUE BECAUSE OF THE UNCERTAINTY INVOLVED IN MEASUREMENT THE TRUE VALUE WILL NEVER BE KNOWN

INTERPRETATION OF FREQUENCY CURVES 1. EXPECTED VALUE THE VALUE OF THE QUANTITY TO BE MEASURED CAN ONLY BE ESTIMATED

INTERPRETATION OF FREQUENCY CURVES 1. EXPECTED VALUE THE PROBLEM IS TO BE AS SURE AS POSSIBLE ABOUT THE ESTIMATION

INTERPRETATION OF FREQUENCY CURVES 1. EXPECTED VALUE IF THE SAME VALUE IS MEASURED AGAIN AND AGAIN WITH THE SAME INSTRUMENT THEN THE EXPECTED VALUE WILL BE THE MEAN OF ALL THE MEASUREMENTS

INTERPRETATION OF FREQUENCY CURVES 1. EXPECTED VALUE

A REASON FOR CHOOSING X AS THE EXPECTED VALUE IS THAT IT COULD NOT VERY WELL BE ANY OTHER VALUE

1. EXPECTED VALUE FOR EXAMPLE, IF X 1 WERE CHOSEN THEN BASED SOLELY ON THE EVIDENCE OF THE FREQUENCY CURVE X 2 COULD ALSO BE CHOSEN

1. EXPECTED VALUE A. IT IS THE MEAN OF ALL THE MEASUREMENTS ONLY X HAS UNIQUE PROPERTIES

1. EXPECTED VALUE B. IT IS THE MODE (THE VALUE WITH THE GREATEST FREQUENCY) ONLY X HAS UNIQUE PROPERTIES

1. EXPECTED VALUE C.IT IS THE MEDIAN MEASUREMENTS ABOVE AND BELOW X OCCUR EQUALLY FREQUENTLY ONLY X HAS UNIQUE PROPERTIES

THE MEAN IS CALCULATED AS FOLLOWS : A. DISCRETE VALUE DISTRIBUTION

THE MEAN IS CALCULATED AS FOLLOWS : B. HISTOGRAM MULTIPLY THE MID-VALUE OF THE MEASUREMENT IN EACH INTERVAL BY THE RELATIVE FREQUENCY FOR THAT INTERVAL AND SUM OVER ALL POSSIBLE VALUES

THE MEAN IS CALCULATED AS FOLLOWS : B. HISTOGRAM

THE MEAN IS CALCULATED AS FOLLOWS : C. CONTINUOUS DISTRIBUTIONS

INTERPRETATION OF FREQUENCY CURVES 2. STANDARD DEVIATION THE PROBABILITY THAT IS 68.3 %

-- ++ x P xmxm

FOR FINITE NUMBER OF MEASUREMENTS

THIS IS THE ACTUAL STANDARD DEVIATION FOR THE FINITE NUMBER OF MEASUREMENTS

TERMINOLOGY x m =THE BEST ESTIMATE OF THE TRUE VALUE FOR INFINITE NUMBER OF MEASUREMENTS x =THE BEST ESTIMATE OF THE TRUE VALUE FOR FINITE NUMBER OF MEASUREMENTS

 =THE STANDARD DEVIATION FOR INFINITE NUMBER OF MEASUREMENTS TERMINOLOGY  a =THE STANDARD DEVIATION FOR FINITE NUMBER OF MEASUREMENTS

IF WE WANT TO ESTIMATE THE TRUE VALUE BY AS A RESULT OF A FINITE NUMBER OF MEASUREMENTS WE WILL ASSUME THAT

IF THEN

IN ORDER TO DETERMINE HOW ACCURATELY x IS ESTIMATING x m STANDARD DEVIATION OF x FROM x m

THEREFORE THIS CORRESPONDS TO A 68.3 % CONFIDENCE LEVEL FOR THE INTERVAL TO CONTAIN THE TRUE VALUE.

IF WE WANT TO EXPRESS x m WITH A CONFIDENCE LEVEL OF 95 % THEN AND FOR 99% CONFIDENCE LEVEL :

P 



ACCURACY OF THE STANDARD DEVIATION : NUMBER OF SIGNIFICANT FIGURES THIS IMPLIES THAT HAS A ROUGHLY 2:1 CHANCE OF LYING WITHIN THE INTERVAL

CHAUVENET’S CRITERION IF A MEASUREMENT IS THOUGHT TO BE WRONG (NOT IMPRECISE OR INACCURATE) PERHAPS DUE TO A FAULTY READING OR SO, THEN THE CHAUVENET’S CRITERION IS USED TO CONFIRM SUCH A SUSPICION.

CHAUVENET’S CRITERION CHAUVENET’S CRITERION IS USED FOR REJECTING OBSERVATIONS WHOSE ERRORS ARE “TOO GREAT”

CHAUVENET’S CRITERION THE LIMITING VALUE BEYOND WHICH ERRORS ARE CONSIDERED TO BE “TOO GREAT”

OK Not OK Not OK

IN GENERAL THIS TEST CAN BE APPLIED TO REJECT A HYPOTHESIS FOR ERROR ANALYSIS IT IS USED TO TEST IF THE ERROR IS RANDOM OR NOT THAT IS, CAN THE DATA BE ASSUMED TO HAVE A NORMAL (UNBIASED) DISTRIBUTION

n o = NUMBER OF OBSERVED OCCURRENCES n e = NUMBER OF EXPECTED OCCURRENCES

F=N-k F= DEGREES OF FREEDOM k= NUMBER OF CONSTRAINTS 5%

TESTING IF DISTRIBUTION OF DATA IS NORMAL TOTAL DATA POINTS : N TOTAL NUMBER OF INTERVALS : N-2 No. OF CONSTRAINTS F = N-2-3 = N-5 1) FIXED NUMBER OF DATA POINTS 2) n e IS CALCULATED BY FIXING x m 3) n e IS CALCULATED BY FIXING 

TESTING IF DISTRIBUTION OF DATA IS NORMAL CALCULATION OF 1. FOR THE MAXIMUM DATA VALUES AND HIGHER VALUES a) n o = 0 (THERE ARE OBVIOUSLY NO DATA POINTS WITH VALUES HIGHER THAN THE MAXIMUM)

b) CALCULATE

2. REPEAT THIS CALCULATION FOR OTHER FREQUENCY INTERVALS FOR RELIABLE RESULTS EACH INTERVAL SHOULD HAVE AT LEAST 5 DATA VALUES