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, regression analysis was requested.

 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).

### Fig. 4. Correlation of Price and Mileage

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: maximum and minimum average speed of the car. In our sample, we have: 1.004 km/h (7.5 years and 41,000 miles) and 2.046 km/h (3.5 years and 39,000 miles). Hence, we can conclude – if you receive an offer to buy an old car, and the parameters do not fall into our high-speed «wedge» (1-2 km/h), then either the car was operated too actively (speed is more than 2 km/h), or the car was parked for some reason for a very long time (speed is less than 1 km/h), OR someone illegally tampered with the odometer. The car's age is more difficult to hide.

2. A car's price appears to have linear dependence on age (a + b x – fig. 3). It is difficult to define more elaborate dependence from the points left on the graph fig. 3. In our case, after every year of use, the price of a car, on average, decreases 393 dollars.

3. The dependence of the car's price on its mileage (fig. 4) is more difficult – it appears to change exponentially: (a + 10b 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):

### Fig. 5. 3D Plot of Correlation and Regression of the Car's Price from its known Parameters – Age and Mileage

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):

### Fig. 6. «Correlation» of the Points around «Regression»

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):

### Fig. 7. Graph of The Decreasing Price of a Car

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.

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/mas/worksheets/Old_Car.mcd

References:

1. Ochkov V.F., Pushnachev U.V. «24 etudes on Basic». Moscow: Finances and statisticsà, 1988

2. Ochkov V.F., Rakhaev M.A. «Etudes on QBasic, QuickBasic and Basic Compiler». Moscow: Finances and statistics, 1995

3. Ochkov V.F.. «Mathcad 8 Pro for students and engineers». Moscow: Computer Press, 1999

[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.