Dimensional Analysis is the art of converting from any two units to another. It can also be called unit transformation. Unit analysis is an advanced form of statistical proportional reasoning in which a given data sample can be transformed by a known geometric mean or percentage to give a corresponding result with a different unit, dimension or measure. Some examples of dimensions that can be measured by dimensional analysis include volume, mass, length, height and weight.
For instance, the volume of an apple varies with its size. Atoms are tiny particles of gas, and they come in various sizes. The atomic weight of an apple is therefore given by the specific weight of the particle, measured in grains of sand. Since atoms are of different sizes, the atomic weight of even identical units of the same size can differ by an insignificant amount in any comparison. Unit analysis is therefore used in the context of measuring amounts, measurements and volumes.
A graphical example of dimensional analysis would be a conversion chart such as the one shown below. In the top frame is a graph of weight loss for a person, as measured by BMI. The second frame has the corresponding graphs of percentage body fat, the percentage of calories burned and the corresponding derived units of fat. Below the second frame is another graph, this time of percentage body fat concentration. The bottom frame shows the conversion of all the variables to their standardized units.
The first frame is a logistic regression, showing the log of the measured quantity as it changes from one measured quantity to another measured quantity. This is an example of the first derivative. The rate of change of the original quantity is represented by a negative number on the right-hand side of the logistic regression equation. The other three derivatives are equal to one another.
The most common type of dimensional analysis used is the arithmetic mean. Here, the distance between any two points plotted against the mean (the arithmetic average) is plotted as a function of time t. The slope of this function is graphed as a function of the dimension being plotted, the mean, the value of the dimension at time t and the number of observations used in the calculation. A similar kind of geometric analysis can also be done using the normal curve or parabola. Here, the plotted function is tangent to a parabolic function of measure. The normal curve is tangent to a cylindrical curve whose slope is constant over time. Another form of dimensional analysis is to plot a function of measure on a horizontal bar and its correlation with the known value, expressed as a function of time t and the number of data points used in the calculation.
A final kind of dimensional analysis is performed using the conversion factors of measurements. The conversion factors can be measured in units other than the units that have been used in the calculation, depending on whether the measurement is in inches or millimeters, or some other number. For instance, if we want to find the ratio of the square footage of footage to the square footage of the area measured, the conversion factors will be in inches to feet.
One more kind of dimensional analysis involves the ratios of the atomic weight of an element or material to its weight in grams. This is called the concentration of atoms or mass in a substance. The atomic weight of an element or material can be measured in a number of different ways. One such way is to measure the fraction of a molecule that consists of atoms. Alternatively, different kinds of atomic weights can be measured by bombarding substances with high energy electrons, which are sent through small holes to atoms below.
There are two basic kinds of dimensional analysis: primary graphical analysis and secondary graphical analysis. In primary graphical analysis, one compares the total area covered by the elements, and then divides this area into smaller parts by multiplying the individual units of measurement. The parts are then summed, and the resulting value is the percentage of the total area that is covered by each unit of measurement. In secondary graphical analysis, the units are given in whole numbers, and the comparison is made between the area covered by the elements of one substance and the units of another substance.