54 Errors during the measurement process Gaussian distribution, and the previous analysis of random measurement errors can be applied to analyse the distribution of these variations in manufacturing parameters. Example 3.5 An integrated circuit chip contains 105 transistors. The transistors have a mean current gain of 20 and a standard deviation of 2. Calculate the following: (a) the number of transistors with a current gain between 19.8 and 20.2 (b) the number of transistors with a current gain greater than 17. Solution (a) The proportion of transistors where 19.8 < gain < 20.2 is: P[X < 20] P[X < 19.8] D P[z < 0.2] P[z < 0.2] for z D X / For X D 20.2; z D 0.1 and for X D 19.8; z D 0.1 From tables, P[z < 0.1] D 0.5398 and thus P[z < 0.1] D 1 P[z < 0.1] D 1 0.5398 D 0.4602 Hence, P[z < 0.1] P[z < 0.1] D 0.5398 0.4602 D 0.0796 Thus 0.0796 ð 105 D 7960 transistors have a current gain in the range from 19.8 to 20.2. (b) The number of transistors with gain >17 is given by: P[x > 17] D 1 P[x < 17] D 1 P[z < 1.5] D P[z < C1.5] D 0.9332 Thus, 93.32%, i.e. 93 320 transistors have a gain >17. Goodness of fit to a Gaussian distribution All of the analysis of random deviations presented so far only applies when the data being analysed belongs to a Gaussian distribution. Hence, the degree to which a set of data fits a Gaussian distribution should always be tested before any analysis is carried out. This test can be carried out in one of three ways: (a) Simple test: The simplest way to test for Gaussian distribution of data is to plot a histogram and look for a ‘Bell-shape’ of the form shown earlier in Figure 3.5. Deciding whether or not the histogram confirms a Gaussian distribution is a matter of judgement. For a Gaussian distribution, there must always be approximate symmetry about the line through the centre of the histogram, the highest point of the histogram must always coincide with this line of symmetry, and the histogram must get progressively smaller either side of this point. However, because the histogram can only be drawn with a finite set of measurements, some deviation from the perfect shape of the histogram as described above is to be expected even if the data really is Gaussian. (b) Using a normal probability plot: A normal probability plot involves dividing the data values into a number of ranges and plotting the cumulative probability of summed data frequencies against the data values on special graph paper.Ł This line should be a straight line if the data distribution is Gaussian. However, careful judgement is required since only a finite number of data values can be used and Ł This is available from specialist stationery suppliers.

MEASUREMENT & AUTOMATION PRINCIPLES 3RD EDITION by GLOBALAUTOMATION