1. 1 ENGINEERING MEASUREMENTS Prof. Emin Korkut

2. 2 Statistical Methods in Measurements

3. 3 Average of the deviations of all readings is zero The average of absolute values of the deviations is given by:

4. 4 The standard deviation or root-mean-square deviation is given by: and the square of the standard deviation  2 is called variance. The above definition is valid if there is a large number of samples. For small sets of data sample standard deviation is given by:

5. 5 For small sets of data sample standard deviation is given by:

6. 6 Simple Probability Concepts Consider a coin is flipped a large number of times. The frequency of occurence is the same for both heads and tail. (½, ½) Probability is a mathematical quantity that is linked to the frequency with which a certain phenomenon occurs after a large number of tries.

7. 7 Probability Distributions Consider we toss a horseshoe some distance x. Although we try to throw the horseshoe the same time ach time, we wouldn’t always meet with success. On the first throw the horseshoe might travel a distance x 1, 2nd throw x 2, etc.

8. 8 For a large number of tosses (throws) the probability that it will travel a distance is obtained by dividing the number of traveling this distance by the the total number of tosses. The distribution of throws will be obtained as and this curve is called probability distribution. Where x m denotes the position of target.

9. 9 Histograms Probability distribution is obtained when we observe frequency of occurence over a large number of observations. When a limited number of observations is made and the raw data is plotted, we call this plot a histogram.

10. 10 Distance from target Number of Throws Frequency = Number of throw / Total number of throws Cumulati ve frequency 1050.05 20150.150.20 30130.130.33 40110.110.44 5090.090.54 6080.080.62 70100.100.72 8060.060.78 9070.070.85 10050.050.90 11050.050.95 12030.030.98 13020.021.00 Total99

11. 11 Gaussian or Normal Distribution Consider an experimental observation is made and some particular result is recorded. We know that the observation has been subjected to many random errors. These random errors may make the final reading either too large or too small. If the measurement is designated by x the gaussian distribution gives the probability that the measurement will lie between x and x+dx and is given by

12. 12 The Gaussian or normal distribution for two values of the standard deviation

13. 13 Confidence Interval and Level of Significance