Errors during the measurement process

 

Distribution of manufacturing tolerances

Many aspects of manufacturing processes are subject to random variations caused by factors that are similar to those that cause random errors in measurements. In most cases, these random variations in manufacturing, which are known as tolerances, fit a 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] = P[z < 0.2] - P[z < -0.2] (for z = (X - µ)/α)

For X = 20.2; z = 0.1 and for X = 19.8; z = - 0.1

From tables, P[z < 0.1] = 0.5398 and thus P[z < - 0.1] = 1 - P[z < 0.1] = 1 - 0.5398 = 0.4602

Hence, P[z < 0.1] - P[z < -0.1] = 0.5398 - 0.4602 = 0.0796

Thus 0.0796 × 10⁵ = 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] = 1 - P[x < 17] = 1 - P[z < -1.5] = P[z < +1.5] = 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 therefore the line drawn will not be entirely straight even if the distribution is Gaussian. Considerable experience is needed to judge whether the line is straight enough to indicate a Gaussian distribution. This will be easier to understand if the data in measurement set C is used as an example. Using the same five ranges as used to draw the histogram, the following table is first drawn:

The normal probability plot drawn from the above table is shown in Figure 3.9. This is sufficiently straight to indicate that the data in measurement set C is Gaussian.

(c) Chi-squared test: A further test that can be applied is based on the chi-squared (x²) distribution. This is beyond the scope of this book but full details can be found in Caulcott (1973).