Means and Standard Deviations as Lengths

When we talk about quality control we hear about distributions, such as the poisson, hypergeometric, binomial, normal, “t”, chi-squared and “F”. How complicated! And we are told to worry about things being independent, are inundated with words like variance, mean, median, mode, standard deviation, whether the standard deviation is homo or hetroscedastic (whether the standard deviation is constant or not), confidence limits, and such things as Type I error, Type II error, null hypothesis etc. It cannot be denied that all of these have their place. However, to get to the basics, all we are really trying to do is measure lengths. Statistics is really simply analytical geometry or linear algebra, depending on one’s outlook. Let’s look at the mean and standard deviation.

Mean (one type of average). We are told that it is the first moment around the origin.

Mathematically it is the integral of xf(x)dx between some limits where f(x) is some distribution  function. Yet it is still length.

Consider a set of “n” data points, X= (x1, x2, —, xn). Then visualize a graph of n dimensions with a single location, X, representing those data. Also visualize a line in that n dimensional space that is equidistant from each axis, i.e. It goes through (1,1,—–,1) etc. Drop a line perpendicular from X to that equidistant line. Call that point M=(µ, µ,—-, µ).  Divide every point by the square root of n, the number of data points to introduce the number of tests into our considerations.

The line (δ ) from the X to M would be the vector (x1– µ, x2– µ, —, xn– µ) while the line (µ) from the origin to M would be the vector (µ, µ,—-, µ). Since the two lines are perpendicular, their scalar (or inner or dot) product would be zero:

((µ, µ,—-, µ))·((x1– µ, x2– µ,—, xn– µ)/ )= 0

x1, + x2, +—-,+ xn – nµ = 0

µ= (x1, + x2, +—-, + xn)/n, which is identical to the form for the mean.

That is, the length of the line µ from the origin to M is equal in value to the mean of the data points.

Standard Deviation. The length of the line, δ, from X to M is the square root of (1/n)*((x1)2+ (x2)2+—-,+ (xn)2 – nµ2). (1/n)*(x1)2+ (x2)2+—-,+ (xn)2 is the square of the length of the line from the origin to the data, X,  while (1/n)*(nµ2) is the square of the length from the origin to the point of M.

δ = ((1/n)*((x1)2+ (x2)2+—-, + (xn)2 -nµ2))0.5

Thus the equation of the length of the line δ is identically to one of the equations used for calculating standard deviations (where the standard deviation is not a random variable. If the sample standard deviation (s) is a random variable, 1/n would be replaced with 1/(n-1)).

Rulers. To measure lengths we need a ruler. We use miles in the United States, in Canada they use kilometers while in Russia, the Verst may be used. In statistics the ruler used is the length, “δ”, if the standard deviation is known or, “s” if the standard deviation is a random variable.

The many terms mentioned above and the sophistication of the mathematics are important in establishing the reliability of the data, still, basically we are only measuring lengths.


Asphalt Compositions Vary.

Those skilled in the art of asphalt technology have known that the composition of an asphalt depends primarily on the crude source. Secondary effects are oxidation and modification either by the addition of polymers or air blowing, which is controlled oxidation to make roofing, pond linings etc. The properties of an asphalt therefore can also vary according to the crude source. Back in the 1960s Rostler, White and others compiled a list of properties and compositions of a very large number of asphalts. It turns out that the properties of blends of asphalts from different sources are sometimes not predictable.

Blending Predictions

The plot of the loglog(viscosity) vs. log(absolute temperature) of an asphalt generally is a straight line. Special graph paper has been available for decades. It turns out that in blending petroleum products, including asphalt, using that graph paper with 0% of an oil at 100° F and 100% at 300° will generally be linear also. At times the X axis may be assumed to be linear rather than the log(absolute temperature). (In ASTM D4887, the X axis is linear.) The resulting plot is not always linear, however, depending upon the composition of the second ingredient. As an example, when blending recovered asphalt from RAP with an aromatic oil, such as Dutrex® 739 or Reclamite® base stock, the viscosity may drop faster than predicted. On the other hand, if a paraffinic oil is used, the actual viscosity may be higher than that predicted from the plot.

We had found that blending 50% 85/100 asphalt from California costal crude with 50% 85/100 asphalt from San Joachim Valley crude resulted in an asphalt with a penetration in the 130s. The same thing was found with a blend of Dubai asphalt with LA Basin asphalt. There are thermodynamic reasons for this based upon non-electrolyte solution chemistry.

Recycled Shingles (RAS)

Roofing asphalt is manufactured by air blowing fluxes containing added lube stock. This changes the composition. An asphalt shingle contains two different air blown products. One is used to saturate the felt or fiberglass while the other is a more viscous asphalt (more air blown) and used in the coating. These two asphalts might be incompatible as the coating asphalt, though harder, contains more oil. If the oil from the coating migrates to the felt or fiberglass the coating might slide off. There is a test used to measure compatibility. Also ferric chloride or phosphorus pentoxide might be used as a catalyst. As the use of air blown asphalt in paving has been correlated with non-load associated cracking, care should be taken in recycling such asphalt. Cracking occurs when the asphalt cannot relax stresses as fast as they build up. A low temperature ductility test is valuable in detecting asphalts that are prone to crack.

Recycled asphalt shingles (RAS) are now being used in paving. In recovering the asphalt from shingles the saturant asphalt and the coating asphalt are blended. It will be interesting in following the performance of pavements using RAS and RAS/RAP added asphalt. As mentioned above, historically, air blown asphalts in pavements are more prone to crack.


It is therefore important to understand that the terms “asphalt” or “bitumen” describe a broad set of materials as does the word “vehicle” in describing a set of transportation equipment. Just because two asphalts are black does not mean that they are compatible. And just because two asphalts are of the same grade, does not mean that a blend will be the same grade. Also, the oxidation process that occurs over time in the pavement is not the same as that which happens in the hot plants, and which is mimicked by the Rolling Thin Film Oven test (RTFO). The RTFO oxidation is the same process that occurs in air blowing. That implies that the chemistry of the oxidation of the asphalt in RAP is different than the chemistry of the asphalt in RAS.