The value of the cumulative frequency table values at a given point is the number of elements within the data set above the lowest value of that interval of the frequency table constructed in accordance with the open left boundary convention.

Example

Within this example we work through an illustration in which the cumulative frequency table from above using the open left boundary convention is evaluated.

Consider the set of boundaries { 1, 2, 3, 4, 5 }, which divide the real line into six sub-intervals. Now if we use the open left boundary convention then the real line will be divided into the sub-intervals:

(-infinity, 1], (1,2], (2,3], (3,4], (4,5], (5, infinity)

Note that, each point on the real line can be assigned to one of these sub-intervals and therefore when assigning a data point to one of these intervals there will only be one sub-interval in which it belongs.

Therefore, if we consider the data set { 0.5, 1.4, 1.3, 2.0, 2.3, 4.5, 5.5}, if the assign this data set in accordance with the above the conventions then we will have: using Open Left Boundary (OLB) convention:

• Within the interval (-infinity, 1], we assign the data element 0.5; and hence the frequency of this interval is 1.
• Within the interval (1, 2], we assign the data element 1.4, 1.3, 2.0; and hence the frequency of this interval (wrt OLB convention) is 3.
• Within the interval (2, 3], we assign the data element 2.3, and hence the frequency of this interval (wrt OLB convention) is 1.
• Within the interval (3, 4], we assign no data elements, and hence the frequency of this interval (wrt OLB convention) is 0.
• Within the interval (4, 5], we assign the data element 4.5, and hence the frequency of this interval (wrt OLB convention) is 1.
• Within the interval (5, infinity), we assign the data element 5.5, and hence the frequency of this interval (wrt OLB convention) is 1.

Now in follows that the associated values of the cumulative frequency table are given by:

• Cumulative frequency table above -infinity is: 1 + 1 + 0 + 1 + 3 + 1 = 7
• Cumulative frequency table above 1 is: 1 + 0 + 1 + 3 + 1 = 6
• Cumulative frequency table above 2 is: 0 + 1 + 3 + 1 = 5
• Cumulative frequency table above 3 is: 1 + 3 + 1 = 5
• Cumulative frequency table above 4 is: 3 + 1 = 4
• Cumulative frequency table above 5 is: 1

Hence, for this case the series returned by this methods to represent the cumulative frequency table would be: {7, 6, 5, 5, 4, 1}.

Reference

StatisticalEngine Class
StatisticalEngine Members