Friday, May 04, 2007

Using statistics to design a holder

I recently designed a bracket to hold some scintillation detectors. The detectors were not all uniformly sized. To determine the opening size to make for the detectors I followed the procedure outlined below.

I measured the detector widths from a sample and collected the data listed in the table 1. The data has a mean of 1.192 inches (3.0277 cm) and a standard deviation of 0.008071 inches (0.0205 cm).

Table 1. Detector sizes.
data (in) data (cm)
1.18 2.9972
1.184 3.0074
1.185 3.0099
1.193 3.0302
1.195 3.0353
1.197 3.0404
1.200 3.048
1.202 3.054

At this point I assumed the data to follow a normal distribution. I determined to make the opening fit the 99th percentile detector. From a z-score table I determined that the 99th percentile is at 2.4 standard deviations above the mean (2.4 std devs above mean is actually 99.2 percentile). The z-score is a dimensionless number obtained by subtracting the population mean from an individual (raw) score and then dividing the difference by the population standard deviation. The z score reveals how many units of the standard deviation a case is above or below the mean. Figure 1 shows a normal distribution with cumulative percentiles and z-scores.

Figure 1. Z-Scores and normal distribution (source: Wikipedia)
This gives the size for the 99th percentile detector of

1.192”+2.4X.008071”=1.211” (3.076 cm)

The number above is the width of a 99th percentile detector. To this number I added 0.005”. This addition was to allow for any inaccuracies in locating the holes for the dowel pins. This addition would allow the dowel pin on each side of a detector to move in by 0.0025”.

The mount was designed with nominal opening for a detector of 1.216” (3.0886 cm).

Unfortunately, designing for 99th percentile detector to ensure that almost all detectors will fit inside the provided opening means that the average detector with a width of 1.192” (3.028 cm) will have 0.024” of side to side travel if the pins are in their correct nominal position.

It turns out that due to the excessive play in the small detectors I will need to redesign the holder.

At this point I created a normal plot to see if the assumption that the data was normally distributed. The normal plot is shown in figure 2.The straight line on the plot indicates normally distributed data. With the limited sample size of the data deviation from a straight line is to be expected. The data is approximately normal. Given the small sample size there is no point to conducting a more formal test for normality.

Figure 2. Normal plot.

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