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Ogive Curves

Ogee Curves
Logistic Regression
Classic Birthday Problem

logistic regression graph
regression equation

This is an example of logistic regression,
a continuous function.  However, note
that a rectangle could be drawn under
any one of the jumps in the curve.
When the pixels are smoothed
to represent continuous data,
the graph is named an ogive curve.

When the second derivative is zero, the slope is zero, and we have a "point of inflection."
Point of inflection graph
Segment at the point of inflection.


Graphing calculator image

Note:  Study the graphing calculator image.  Notice the graph is not smooth and continuous.  It is conventional to define the cumulative frequency for all values of x up to and including the given right end-point value.  Thus, an ordinate represents the sum of all frequencies up to and including  a corresponding frequency in a frequency distribution.   Your calculator or software will calculate a value of y to be mapped with the corresponding value of x  but must "jump" to represent a continuous function.

In mathematics, the name  ogive  is applied to any continuous cumulative frequency polygon and is derived from its resemblance to the shape of architectural molding known as the ogee pattern. 

Another way to express this is to say an ogive curve has the shape of an elongated  S.  Also, it is sometimes called a "double curve" with one portion being concave and the other being convex.

Be careful in drawing a distinction between ogive and ogee, especially when applied to architecture.  Conflict abounds in the literature and common usage.

For example, a 1954 edition of the Funk and Wagnalls dictionary uses the term ogive to describe the arch over doorways and windows as illustrated by the north entrance to Westminster Cathedral, London.  "This sense is wholly arbitrary and is unknown before 1830, when it appears to have been adopted through a misunderstanding, but it is of frequent occurrence since that date."

A mathematician viewing the ogive pattern as seen in the window notes a cusp at the apex.  The slope is undefined at the point of a cusp.

The same dictionary defines  ogee  as a "cyma reverse molding" and having a "section in reverse" or as a "long  S  curve" and continues by  using the examples of "ogee doorway" and "ogee window."
Westminster Abbey window

Proud owners of early American mantle clocks use the term  ogee to describe a weight-driven pendulum mechanism in a rectangular case.  The owners know the internal weight on the left must be wound in a clockwise direction while the weight on the right is hoisted using a counterclockwise direction, thereby aping the pattern of arches in cathedrals.

(The Smithsonian Institution is on the middle panel.)
American "ogee" clock

Other Examples
The Instituto Veneto di Scienze, Lettere ed Arti in Venice.
Venetian arch

The ogive curve is sometimes called the "Venetian Arch."
Another early American clock
with an "ogee" mechanism and an "ogive" case.
Another America "ogee" clock
Classic Birthday Problem ~ An Ogive Curve

What is the probability that at least two people in a given group will have the same birthday?     For this problem we ignore leap years and assume that a person's birthday can fall on any day with the same probability.
Birthday probability
Numerator:  If the  n  persons are to have different birthdays, the the first person can be born on any of the 365 days, but the second person can only be born on one of the remaining 364, the third person on the remaining 363 days, etc.

Denominator:  As there are  n  people and 365 different days, there are 365n possible ways in which the people might have their birthdays.

For a group as small as 23 people, the probability becomes less than one-half, or less than "50-50" that two people will not have the same birthday.  In other words, it is more likely to find that two people will have the same birthday.
Probability                               Ogive Curve

For those who have  MATHEMATICA®, . . .Alfred Gray, Modern Differential Geometry, 2nd ed., CRC Press, 1998, p. 909 has logistic code. Cover of Alfred Gray book
J. F. Kenney and E. S. Keeping, Mathematics of Statistics: Part One, D. Van Nostrand Co., 1954, pp. 29-31.
Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 1999, p. 1267.

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