§ 63.01 – What are the basic elements of selecting a sample at random?

(a)    Choosing a method with which you select a sample at random.

(b)   Developing lists of service units that you expect to operate:

(1)   A single list of all services if your sampling plan is not based on service grouping.

(2)   One list for each service group if your sampling plan is based on service grouping.

(c)    Selecting a sample at random from each list with the chosen method.

§ 63.03 – What method may I use for random sampling?

(a)    You may use the traditional method based on a table of random numbers.

(b)   You may use any other method for random sampling as long as it meets these two criteria:

(1)   sampling under the method is random. 

(2)   sampling under the method is without replacement.  Without replacement means that the method will not select the same service unit more than once.

§ 63.05 – What is a table of random numbers?

(a)    It is a list of integers whose frequency and sequence of appearance in the list have been determined entirely by chance. 

(b)   For convenience and simplicity in use, published tables of random numbers usually appear in the form of separate columns of five-digit numbers.  Both rows and columns may be consecutively numbered for easy reference.  Table 63.01 shows an example.

(c)    Appendix 98 is a comprehensive table of random numbers you may use for random sampling if you choose to use this method.

Table 63.01. Example of a Table of Random Numbers

Rows

(1)

(2)

(3)

(4)

(5)

1

10480

15011

01$536

02011

81647

2

22368

46573

25595

85393

30995

3

24130

48360

22527

97265

76393

4

42167

93093

06243

61680

07856

5

37570

39975

81837

16656

06121

§ 63.07 – What is the list of all service units that I expect to operate?

 

(a)    It is the amount of revenue service that you expect to operate. 

(b)   It is measured with the following characteristics:

(1)   It is in the unit of sampling and measurement of the sampling plan you have chosen.

(2)   It is for the duration corresponding to the sampling frequency that you have chosen for your sampling plan.  The duration would be a day for interval-based sampling but a week, a month, or a quarter for period-based sampling.

(3)   It is for each of the service groups you have defined if your sampling plan is based on service grouping.

(c)    For scheduled services, the list must include:

(1)   all service units that are listed on the schedule, and

(2)   all service units that are not on the schedule but are expected to be operated, such as trippers, shuttles, and other special operations.

(d)   It would be the number of one-way bus trips you expect to operate in a week by your MB express routes, for example, if:

(1)   your sampling plan is in terms of one-way bus trips,

(2)   you have chosen weekly sampling for your MB service,

(3)   your sampling plan involves service grouping, and

(4)   you use express routes as one of the groups.

§ 63.09 – How do I develop the list of all service units for using a table of random numbers?

(a)    Suppose that:

(1)   you have chosen weekly sampling for your vanpool service,

(2)   your sampling plan requires 2 vanpool-days per week,

(3)   you want to select a random sample for next week, and

(4)   you have 101 vanpools and each of them is expected to operate every day next week.

(b)   One way to develop the list would be to assign a serial number of four-digits to each vanpool-day for all 707 combinations of vanpools and service days.  The first digit would represent the day of week with 1 for Monday and 7 for Sunday.  The other three digits would represent vanpool numbers ranging from 001 through 101.

(c)    The vanpool numbers may differ from what you use for your internal purposes.  Make sure that you have a one-to-one list between the internal vanpool numbers and the new serial numbers if they differ.

(d)   The serial numbers on a given day must be consecutive without any gaps.

(e)    Write down just the first and the last for each day of week so that you will know the range of numbers.  For example, the range for Monday is 1001-1101.

(f)    You should summarize these serial numbers in a summary table as in Table 63.03.

Table 63.03.  Example of a Summary Table of Serial Numbers

Day of Week

Range of Serial Numbers

1 (Monday)

1001-1101

2

2001-2101

3

3001-3101

4

4001-4101

5

5001-5101

6

6001-6101

7

7001-7101

§ 63.11 – How do I use a table of random numbers?

 

(a)    Suppose that:

(1)   you have 101 vanpools that you expect to operate every day of next week,

(2)   you have numbered the vanpool days for all vanpools as shown in Table 63.03, and

(3)   your sampling plan for your vanpool service requires 2 vanpool days per week. 

(b)   You must work with a constant length of digits from a table of random numbers for sampling.  This constant length is given by the number of digits in the longest serial number you have assigned to members of the list of all service units. 

(1)   This constant length would be 4 for the example in Table 63.03 because all serial numbers are four-digits long. 

(2)   This constant length would be 5 if your serial numbers vary in length, ranging from one to five digits.

(c)    You must combine adjacent digits from the table of random numbers as needed to obtain a two-, three-, or four-digit number, or any other length number from the table. 

(1)   You may choose to work with rows or columns in combining adjacent digits.  You may start with any row if you choose to work with rows.  You may start with any column if you choose to work with columns.  You may also choose to work from right to left or from left to right if you choose to work with rows. 

(2)   If you choose to work with rows and to start with row 1, for example, the first 9 four-digit numbers from Table 63.01 would be:

 

1048, 0480, 4801, 8015, 0150, 1501, 5011, 0110, and 1101.

 

(d)   You must continue forming four-digit numbers until you find two four-digit numbers from the table of random numbers that are in the summary table of serial numbers (Table 63.03).  In the above example, these two numbers are 1048 and 1101.  That is, you should select vanpools 48 and 101 and collect sample data from them on Monday of next week.

(e)    Once you are done sampling for a week, you should mark where you stopped forming four-digit numbers in the table of random numbers.  For example, the symbol, $, has been inserted between digits 1 and 5 in row 1 and column 3 of Table 63.01.  You should start the above process for a later week after that mark.

(f)    Once you are done sampling for a year, you should mark where you stopped forming four-digit numbers in the table of random numbers if you plan to use this method of random sampling in the future.  

(1)   If you number your service units exactly as for a previous sampling year, you should start the above process for a new sampling year after the last mark of your previous sampling year.

(2)   If you number your service units with a different approach for a new sampling year, you may start the above process anywhere in the table of random numbers.

§ 63.13 – What are the pros and cons of using a table of random numbers?

 

(a)    Using a table of random numbers for random sampling has the following advantages:

(1)   It is applicable to all situations.

(2)   You do not need to list explicitly all service units.

(b)   It has the following disadvantages:

(1)   You must number your service units consecutively without gaps, at least within each subset of your service units.  Subsets are formed by route, type of service days, etc.

(2)   You may not be able to take advantage of the serial numbers you have already assigned to your service units for internal purposes.

(3)   It can be difficult to use if the serial numbers have a large number of digits.

§ 63.15 – What information should I keep from my sampling process?

 

(a)    You should have an auditable record of your sampling process.  That record should cover the following:

(1)   A written description of the sampling plan.

(2)   A written procedure for your method of selecting a sample at random.

(3)   The list of all service units from which you selected a sample at random for each act of sampling.  For example, you should have 52 such lists if you did weekly sampling for an entire year.

(4)   The random sample.