Price of an Old
Car

or

The way from Correlation to Regression in Mathcad

Translation
in English with the help of Nathan L. Hunsaker tkhunny@velocity.net

Definitions:

*Correlation*
is dependence without completely regular fluctuation. It is not possible to
take into account the influence of all change in all factors simultaneously.

*Regression*
is dependence on central tendencies of one or more variables.

And now the heart of the problem.

By chance the author found in the Internet (http://collab.mathsoft.com/read?8889,34)
the following ** correlation** data[1]
(see the table – Price of an Old Car, Depending on Its Age and Mileage[2]).
Also,

Age |
MileAge |
Price |
Age |
MileAge |
Price |
Age |
MileAge |
Price |

11,5 |
88000 |
1195 |
13,5 |
120000 |
495 |
7,5 |
41000 |
3495 |

13,5 |
103000 |
750 |
11,5 |
101000 |
895 |
13,5 |
124000 |
850 |

10,5 |
82000 |
1295 |
3,5 |
39000 |
4995 |
10,5 |
77000 |
1595 |

10,5 |
65000 |
1495 |
10,5 |
78000 |
1295 |
6,5 |
56000 |
3495 |

12,5 |
97000 |
800 |
6,5 |
52000 |
2695 |
12,5 |
83000 |
895 |

10,5 |
70000 |
1495 |
9,5 |
84000 |
1995 |
9,5 |
67000 |
2495 |

8,5 |
51000 |
2295 |
4,5 |
39000 |
3995 |
4,5 |
38000 |
3990 |

10,5 |
80000 |
1495 |
4,5 |
46000 |
3675 |
6,5 |
43000 |
3400 |

9,5 |
79000 |
1995 |
12,5 |
92000 |
795 |
13,5 |
92000 |
795 |

6,5 |
57000 |
2695 |
11,5 |
108000 |
975 |
11,5 |
78000 |
1295 |

More specifically, the request was to give a formula or graph that could
be used for an appraisal of the price of an old car by the year of manufacture
and the odometer reading (mileage).

This author saw the question as an excellent example of problems that
can be solved using MathCad. [3].

As a rule, one should begin regressive analysis (search for regression)
with visualization of the tabular data – see fig. 1,
fig. 2, fig. 3
and fig. 4, where the parameters of
cars are shown in volume (fig. 1). Three
different views are given (fig. 2, __fig. 3__ and fig. 4).

Lines that describe following laws are lined in fig. 2, __fig. 3__ and fig. 4.

1. Correlation of the mileage of a car and its age is limited by two
considerations: ma

3. The dependence of the car's price on its mileage (fig. 4)
is more difficult – it appears to change exponentially: (a + 10^{b x}).
Under such dependence, (b<0) the car's price is assumed never to reach zero.
We cannot say the same thing about the dependence of price on age (fig. 3).
In year 14-15 the car becomes «invaluable» in either of two ways – the car's
price is zero (most practical outcome) or the car becomes... a museum exhibit.
(Who understands collectors, anyway?)

Conversion from a Plane (fig. 2,
__fig. 3__ and fig. 4)
to a volume (fig. 1)
gives the following regression formula[3].
With this formula, we calculate the minimum price of the car (% of the price of
a new car[4]):

44.98+2.98▪Age+55.02▪10^{-0.00000937▪Mileage}

where: Age is shown in years, but Mileage is shown in kilometers.

On a three-dimensional graph, our formula gives the following surface (fig. 5):

The points from our table are situated close to surface. It is easier to
see these points, in relation to the surface, if we turn the surface view sideways
(fig. 6):

From this view we can see (fig. 6)
that some of the points are above the surface and some are under the surface.
This is the direct result of the least squares criterion: the sum of squares of
deviations from the surface. The same criterion was used for determining lines
near points fig. 3
and fig. 4.

In the fig. 5
and fig. 6
the surface in all its «artistry» (it is semi-transparent – all points are
visible. The colors change from the cold tones (cheaper cars) to warmer tones
(more expensive cars) and «one-cost» lines are shown). It is not very
practical: To fix the price of the car we must use either the formula (see above), or the graph plot (fig. 7):

By the graph in fig. 7
it is possible to determine both the price of a car and to discard cars outside
acceptable limits, as shown in fig.2.

Mathcad-document performing the required calculations: (Car_Price.mcd –
Mathcad 2001 Premium). See: ftp://twt.mpei.ac.ru/ochkov/Auto.

Epilogue

To quote Winston Churchill "There are lies, _______ lies,
and Statistics" – the last being the hero of this article.

The task about the Price of the Old Car on the MAS: http://twt.mpei.ac.ru/MCS/Worksheets/old-car.xmcd

References:

1. **Ochkov
V.F., Pushnachev U.V.** «24
etudes on Basic».

2. **Ochkov V.F., Rakhaev M.A.**
«Etudes on QBasic,
QuickBasic and Basic Compiler».

3. **Ochkov
V.F.. «**Mathcad
8 Pro for students and engineers».

[1] In our table, data about one model of the car
are collected. Other factors can influence the price of an old car: storage conditions
(garage or street), maintenance record, and etc., but we shall take into
account only two the chosen factors – age and mileage.

[2] These two words (age and mileage) flow
together in English – MileAge, age, that are expressed in miles (1 mile ≈
1609 ì).

[3] Here the word 'regress' is an antonym of the
word 'progress': Development of a market for old cars leads to the stagnation
of the automobile industry. [Personal note from Nathan L. Hunsaker: that's
really not an American ideal. Free enterprise, whereever it succeeds, cannot be
a bad thing to a true capitalist.]

[4] In our sampling we have 9037 dollars. This
must be increased 20-30% – the price of a new car decreases abruptly when it
leaves the dealer's lot.