The sampling area of an inventory sampling usually contains a large number of stock management units with very different values. The larger the value range between the smallest value and the largest value within the sampling area, the more stock management units must be counted in order to obtain statistically correct results.

To reduce the number of elements to be counted, the random selection and extrapolation are carried out for individual strata. The stratification is performed on the basis of the classification. Individual consecutive classes are grouped together to form various strata.

During stratification, the system issues a warning message (M7 641) if the last stratum of the sampling area is going to be a complete-count stratum and subsequent random selection would thereby not be possible. The message long text explains what you have to do to prevent this.

Two parameters are predefined (on the parameter screen) for stratification:

• A variation interval

• The minimum sample size

During the stratification process, the system determines the following:

• The stratification for each number of strata within the variation interval

• The number of elements to be counted per stratum

• The optimum stratification of the generated stratifications

If the optimum stratification lies on a limit of the variation interval, you should move the variation interval beyond this limit. This is explained in the following example:

The optimum stratification of an inventory sampling is expected to contain between 5 and 10 strata. However, the actual number of strata of the optimum stratification would be 12. 5 and 10 strata are entered as the variation limits. When generating the stratification, the system determines an optimum stratification of 10 strata. Only after you have changed the variation limits (for example, to an interval between 10 and 15), can the system set (in a new stratification process) the optimum stratification to 12.

The system uses the Dalenius-Hodges procedure to calculate the stratification. For this purpose, the following information is important:

• How many elements are contained in a class?

• How many strata are to be generated?

For this calculation, the system proceeds as follows:

It determines the number of elements per class, and then calculates the square root of each number determined. Afterwards, it adds the square roots together. The quotient obtained by dividing this total by the number of strata results in the target figure for the stratum limit.

The square roots of the individual classes are now added together in succession until the total equals or exceeds the target figure. All classes including the class that reached the target figure are combined to form the first stratum. From the next class onwards, the square roots are again added together until the target figure is reached. This is how the second stratum is obtained. This process is repeated until all strata have been generated. The following example shows these calculations.

The system now calculates the individual stratifications, using various numbers of strata within the variation interval as a basis.

The target figure of the three strata is 300/3 = 100. The totals of the square roots are 60, 110, 153, and so on. As 110 is the first total to exceed the target figure, the classes 1 and 2 are grouped in the first stratum. The square roots of the third class are then totaled. The totals are 43, 74, 102, 127, and so on. As 102 is the first total to reach the target figure, classes 3,4, and 5 form the second stratum. Classes 6 to 10 form the third stratum.

The following table shows how the stratification is calculated for 3 to 6 strata.

For example, using three strata as a basis, classes 1 and 2 would form the first stratum, classes 3, 4, 5 the second stratum and classes 6 to 10 the third stratum.

It is possible that fewer strata are actually generated than the number of strata used as the basis would suggest. This case is shown in the above table with 5 strata used as the basis.

The following table shows which classes form a stratum as well as the value range and the number of elements in each stratum.

The above table shows the following: with a small number of strata, there is a wide range of values within each stratum. This means that many stock management units would have to be counted to obtain statistically correct results for a given stratum. As the number of strata increases, the value range in each stratum will become narrower. That is, strata will be generated which contain elements with values that only vary slightly.

The system calculates the stratification for each number of strata within the variation interval. The following applies for the stratifications:

• If there is only a small number of strata, they will have wide value ranges. This means that many elements must be counted.
• If there is a large number of strata, each stratum will contain relatively few elements with a narrow value range. Therefore, all elements of a stratum may have to be counted if the minimum sample size has been set accordingly.

Among the stratifications generated, there is one which requires relatively few stock management units to be counted in total. This particular stratification is called the optimum stratification of the inventory sampling.

The optimum stratification may well include strata that contain fewer elements than the minimum sample size specified. For these strata, all elements must then be counted.

You can control how many elements the system chooses for each stratum by choosing Goto ® List ® Stratification ® Optimum variant.

There should always be at least 30 elements in each stratum, as only then does the selection meet the standard statistical distribution. Extrapolation for sample-based physical inventory is based on a mean value estimate. A prerequisite for this is that the mean values of possible samples per stratum form a standard distribution. This is probably the case when there are 30 or more elements. The system therefore expects an entry of 30 or more elements.

For optimum stratification, in the random selection you should choose a maximum of 25% to 30% of the stock management units per stratum in the last few strata (especially in the last stratum).

If this criterion has not been fulfilled, you should reduce the upper value limit of the sampling area by choosing Goto ® Parameter. You then have to form a new stock population and carry out a new stratification. The system re-forms the strata with reference to the new upper value limit.