2. 14.1 ARITHMETIC MEAN Experimental Readings are scattered around amean value Fig. 14.1 Scatter of the readings around the mean value

3. The most probable value of a measured is the arithmetic mean. Eq. 14.1 Where is the arithmetic mean, is the reading No. i, and n is the No. of readings The arithmetic mean does not define the precision of the results (how the readings scattered around the mean). A measure of the scatter of the readings must be established to completely define the experimental results

4. 14.2 DEVIATION (d) Deviation is the departure of a given reading from the arithmetic mean. Fig. 14.2 Deviations

5. Eq. 14.2 where d 1 is the deviation of reading No. 1 Eq. 14.3 Where d i is the deviation of reading No. I Note :- Eq. 14.4

6. 14.3 AVERAGE DEVIATION (D) Eq. 14.5 14.4 VARIANCE (V) Eq. 14.7

7. 14.5 STANDARD DEVIATION (σ) Eq. 14.6

8. Note: Average deviation, standard deviation, variance are measure of precision of readings (scatter).Highly precise measurements will yield low values of these parameters. Standard deviation is the most common used parameter to describe the reading precession.. Note: Eq. 14.7

9. 14.6 PROBABILITY OF ERRORS (ERROR DISTRIBUTION) Suppose that a number of readings are taken to a measured. The mean and deviation are computed for the readings. If the deviations are plotted against its No. of occurrence (frequency) the shown bell shape curve would be resulted.

10. Fig. 14.3. Error distribution.

11. This type of curve is known as Gaussian or Normal Curve. It is clear that:- a) Small errors are more probable than large errors. b)Large errors are very improbable. c)There is an equal probability of plus and minus errors so that the probability of a given error will be symmetrical about the mean value.

12. 14.7 AREA UNDER THE ERROR DISTRIBUTION CURVE Fig. 14.4 Areas under the error distribution curve

13. 14.9 COMBINATION OF ERRORS 14.9.1 Sum of Two Quantities: Eq. 14.8

14. Where,, are the uncertainties of quantities, and is the uncertainty of sum. 14.9.2 Difference of Two Quantities: Eq. 14.9

15. 14.9.3 Product of two quantities: Eq. 14.10 14.9.4 Division of two quantities: Eq. 14.11