gh.linkedin.com/pub/bill-adongo/86/927/2a4

CENTRAL PREDICTION

The gravitational point equation can also used to predict mean value (central values) of two data pair (x_p,y_p).

This equation is developed mainly to enhance the best accuracy for predicting the mean value (Central values) of two data pair.  For example, we can predict the value of y_p when the value x_p is known.

The equation constitutes the variables y_p, x_p, c and k which I named as predictive value, suggestive value, constant and coefficient of a suggestive value respectively. The developed formula for the equation is given as;

y_p = Kx_p +C

Where;

K = Σy/2Σx

C = Σy/2n

PROOF AND ANALYSIS OF THE FORMULARS

For an apparent relationship between x and y values from a sample or population data, the total sum of x is apparently related to the total sum of y (i.e Σy is relative to Σx).

This implies that

Σy = kΣx + nc

But the relationship between k and C are given as;

Nc = Σy - kΣx ---(1)

Or

kΣx = Σy – nc --- (2)

Adding equation (1) and (2) together, we have

nc + kΣx = (Σy - kΣx) + (Σy – nc)

Arranging equal pairs together, we have:

nc – (Σy - kΣx) = (Σy – nc) - kΣx

By principle of equal pairs, we have:

Nc = Σy - nc

2nc = Σy

C = Σy/2n ---- (3)

And:

(Σy - kΣx) = kΣx

Σy - kΣx = kΣx

K2Σx = Σy

K = Σy/2Σx --- (4)

EXAMPLE

The table below contained the apparent relationship between students high school average (HAS) and grade point average (GPA) after the freshman year of college

HAS(X)                               GPA(Y)

80                                        2.4

85                                        2.8

88                                         3.3

90                                         3.1

95                                         3.7

92                                         3.0

82                                         2.5

75                                          2.3

78                                          2.8

85                                         3.1

(a)  If the students have average HAS of 85, find the best estimate of their average GPA

(b)  If they students have average HAS of 92, find the best estimate their average GPA

SOLUTION

    HAS(x)                 GPA(y)

   80                          2,4

   85                          2.8

  88                           3.3

  90                           3.1

   95                           3.7

   92                            3.0

   82                            2.5

  75                            2.3

  78                            2.8

  85                            3.1

ΣX= 850               ΣY= 29

K = Σy/2Σx

K = 29/2(850)

K = 0.0171

And:

C = Σy/2n

C = 29/2(10)

C = 1. 45

Hence, the predictive equation is given as:

y_p = 0.0171x_p + 1.45

(a) y_p = 0.0171 (85) + 1.45

y_p = 2.90

(b) y_p = 0.0171 (92) + 1.45

             y_p = 3.0