To create distribution histograms

Distribution Histograms are used to ascertain the variation by displaying a standard distribution curve of measured values for an item.

To plot this chart you must select the combination of item or item/supplier, inspection order source, aspect/characteristic as well as the relevant time period. This chart is based only on actual inspection results.

The centre line of the distribution curve is the LN calculated mean (µ). The over/under tolerance limits of the process are the limits within which the process is capable of producing parts of an acceptable quality. These tolerance limits are generally expressed as the process mean plus or minus 3 standard deviations (σ) that can capture 95 percent of the normal variance spread.

To plot this type of chart, complete the following steps:

Calculate measured values for a range of periods.
Determine the spread R of the measured values: R = Xmax – Xmin
Determine the class width: W = R / SQRT (number of measurements)
Compose the classed: Class 1 Lower Tolerance (or Xmin in case Xmin < Lower Tolerance) then Class2 = Class1 + W and so on
Populate the classes based on the measured values. Determine the frequency within each class.
Calculate the arithmetic means of the measured values.
Calculate the standard deviation
Plot the histogram based on the classes calculated.

Example

Assume that 5 inspection orders are processed, each with 1 sample, resulting in one sample group for each order. All 5 inspection orders have a sample size of 10 pieces and a test quantity of 1 piece. The following results are displayed in the test data table:

Sample Group	Sample Number	Measured Value
1	1	1
1	2	1
1	3	1.002
1	4	0.997
1	5	1
1	6	1.001
1	7	1
1	8	1
1	9	1
1	10	0.999
2	1	1
2	2	0
2	3	0
2	4	0
2	5	0
2	6	0
2	7	0
2	8	0
2	9	0
2	10	0
3	1	1.001
3	2	1
3	3	0.9
3	4	0.988
3	5	1.001
3	6	1.004
3	7	0.999
3	8	0.989
3	9	1.012
3	10	1.03
4	1	1.001
4	2	1
4	3	0.9
4	4	0.988
4	5	1.001
4	6	1.004
4	7	0.999
4	8	0.989
4	9	1.012
4	10	1.03
5	1	1.001
5	2	1
5	3	0.9
5	4	0.988
5	5	1.001
5	6	1.004
5	7	0.999
5	8	0.989
5	9	1.012
5	10	1.03

Calculate the spread

Determine the spread of the measured values. The highest measured value is 1.03 (Sample Group 1, Sample Number 10). The lowest measured value is 0.9 (Sample Group 1, Sample Number 3).

Spread = 1.03 - 0.9 = 0.13

Calculate the class width:

The class width is 0.13 / √50 = 0.02055480479109446565799280803881. This value is rounded off to 0.02.

Compose the classes

The classes are composed as follows Class 1 Lower Tolerance (or Xmin in case Xmin < Lower Tolerance) then Class2 = Class1 + W and so on. The following classes are generated:

Class 1	0.900000
Class 2	0.920000
Class 3	0.940000
Class 4	0.960000
Class 5	0.980000
Class 6	1.000000
Class 7	1.020000

Populate the classes

The values of the different measurements can be grouped into a class, if the value is greater or equal to the class value, and less than the class value + the class width. The result is:

Class	Number of measurements
1	1
2	0
3	0
4	0
5	12
6	36
7	1

Calculate the mean

For every measurement, the difference to the mean is calculated and the square of the differences is added together. If the first sample number has a measurement value of 1:

(1 - 0.995850)² = (0.00415)² = 0.0000172225

The square of the differences are calculated and summed together to form a total square difference. For the above example the total is 1.311734.

Mean = Standard Deviation - √ 1.311734 /
		  50 = 0.160000

Plot the chart

The following figure displays the chart plotted with the above data:

On the X-Axis the characteristics unit is displayed. It is however possible that for a specific standard test procedure or for an inspection order line, the measurement value is expressed in a different unit that are later converted to the characteristic unit.