

Carina Nebula [Courtesy NASA]  Palau de la Musica Catalana, Barcelona, Spain [Photo by DHB, (c) 2011] 
Some of the most commonly used radiometric schemes are [Dalrymple1991, pg. 111; Dalrymple2004, pg. 55]:
In mathematical terms, radioactive decay is governed by a simple exponential formula, taught in many high school math classes:
P_{1} = P_{0} e^{L t}
where e = 2.71828... is the wellknown math constant, P_{0} is the original amount of the radioactive material, P_{1} is the amount after time t, and L is the decay constant for the radioactive isotope. This decay constant L can be expressed in terms of the half life T (the time it takes for onehalf of the material to decay) as L = log(2) / T, where log(2) = 0.693147... is the basee logarithm of 2. In other words, if we know P_{1} and P_{0}, or even merely their ratio, we can solve the above equation for the time t. However, usually it is not possible to apply this formula directly, because, for instance, in many cases we do not know the original amount of the radioactive isotope when the rock was solidified. Also, such a calculation does not provide us with any statistical error margin to doublecheck the result.
Fortunately, scientists have developed several methods that not only circumvent the difficulty of not knowing the original amounts, but also provide a very reliable means of statistical validity checking. For example, the rubidiumstrontium isochron method, one of the most widely used schemes, is based on the radioactive decay of rubidium87 into strontium87 by the emission of a highenergy electron. The halflife T of this decay has been measured in careful laboratory measurements as T = 48.8 billion years. On the other hand, strontium86 is a stable isotope. By some simple algebraic manipulation of the basic radioactivity formula above, one can show that the following formula must hold at any time t:
(Sr87/Sr86)_{t} = (Sr87/Sr86)_{0} + (e^{L t}  1) (Rb87/Sr86)_{t},
where (Sr87/Sr86)_{t} is the ratio of the amounts of these two elements in the sample material after time t; (Sr87/Sr86)_{0} is the same ratio at time = 0; and (Rb87/Sr86)_{t} is the ratio of these two isotopes at time t. Note that this equation is in the simple form y = b + m x, namely the formula for graph of a straight line with slope m and with yintercept b: here y = (Sr87/Sr86)_{t}, b = (Sr87/Sr86)_{0}, m = (e^{L t}  1) and x = (Rb87/Sr86)_{t}. If all we have is one data point, the formula above doesn't help much more than the original formula. But if we have multiple data points  multiple measurements of different samples say within a single igneous rock, then these should all lie on a straight line, whose slope m is simply related to the age of the specimen by the formula m = e^{L t}  1. Note that it doesn't matter that we don't know the original ratio (Sr87/Sr86)_{0}; instead, this original ratio actually comes out as a result of the calculation!
Of course, in real scientific research, scientists do not rely on manually drawing points on graph paper to determine a bestfit straight line or to determine the line's slope or yintercept. Instead, they use a statistical technique known as linear regression, which computes the leastsquares best fit of a straight line through a sequence of points. This technique, which is used in virtually all disciplines of modern social science, physical science and engineering, is entirely straightforward, and computer software is widely available to do the requisite calculations and, in fact, is built in to most spreadsheet programs. An important fact is that linear regression, in addition to giving the best fit of the slope of the line (which then leads immediately to the date), also gives a statistical confidence interval as to the possible error in the determination of the slope. Details about linear regression are available in any elementary statistics book, or online  see [Linear2013].
Here are just four examples of isochron graphs, which are entirely typical among the tens of thousands of examples that could be mentioned. Note how breathtakingly close these points are to the fitted lines (thus confirming with high statistical confidence the validity of the resulting dates):
The data for the first graph (upper left) is a set of measurements of basaltic achondrites (meteorites) in [Basaltic1981, pg. 938]; the data for the second graph (upper right) is from early Archaean gneisses rocks near Isua, Greenland [Morbath1977]; the data for the third graph (lower left) is from ancient gneiss rocks in Swaziland [Carlson1983]; the data for the fourth graph (lower right) is from lunar dunite rocks gathered during Apollo 17 [Papanastassiou1975]. The corresponding dates obtained from these isochrons (based on the slopes of the lines), together with statistical standard deviations, are: 4.396 ± 0.18, 3.673 ± 0.014, 2.991 ± 0.15, and 4.478 ± 0.034 (each figure is in billions of years). See also [Dalrymple1991, pg. 149, 185, 248, 328].
For many years, fairly large samples were required to produce statistically reliable results. But with the advent of mass spectrometry beginning in the 1970s, even very small samples can now be accurately dated. For example, the "SHRIMP" ion microprobe now in use in numerous laboratories around the world can reliably measure UPb and PbPb ages from spots only 0.02 mm (i.e., 20 micrometers) in size within a zircon crystal [Dalrymple2004, pg. 6062].
It should be emphasized, though, that even relatively unsophisticated equipment can perform radiometric measurements of dates fairly well. For example, as of the present date (see above), numerous used mass spectrometers are available for sale on the auction site eBay.com. Although most items are priced at several thousand dollars, a few are available for as little as a few hundred dollars. Some have said that youngearth creationists will not be convinced of the scientific ages of the earth and fossils until they can measure these dates with their own hands. That day is almost here!
One interesting and timely application of advanced radiometric dating techniques in paleontology is in attempts to disentangle a controversy as to whether or not all dinosaurs (and numerous other species) were extinguished by a giant meteorite impact at a spot just north of the presentday Yucatan Peninsula. In 2011, researchers at the University of Alberta in Canada used the uraniumlead method to date a fossilized dinosaur bone found in New Mexico to be approximately 64.8 million years old. At the time, the CretaceousTertiary meteorite impact was thought to have occurred approximately 65.5 million years ago. This suggested that hadrosaurs such as the researcher's specimen may have survived for roughly 700,000 years after the meteorite event [SD2011b]. But in February 2013, a team of researchers at U.C. Berkeley, using a stateoftheart argonargon scheme that permits a significantly more accurate date determination, found that the impact had occurred 66,038,000 years ago, while the mass extinction occurred 66,043,000 years ago. Given that these dates differ by no more than the statistical error bars of the measurements (11,000 years), they are essentially identical. Thus these new findings offer dramatic confirmation to the theory that the meteorite impact caused the extinction (although climaterelated phenomena prior to that time may have exacerbated stress on these species) [Perkins2013; Choi2013].
In February 2014, a similar study was completed of the TermianTriassic boundary, which marks the largest mass extinction in the past 500 million years. By employing some new uraniumlead dating techniques on specimens taken in China, they were able to establish that the extinction occurred between 251.941 million years (plus or minus 37,000 years) and 251.880 million years (plus or minus 31,000 years), meaning that the extinction took less than about 60,000 years (an eye blink in geological time), thus laying to rest the hypothesis proposed by some that this was a gradual event spanning hundreds of thousands of years [Zimmer2014a].
Another ongoing debate is whether or not Neanderthals persisted in Europe until after humans arrived. In June 2012, researchers announced that some red handprints and dots in a cave in northwestern Spain are more than 40,000 years old, based on a uraniumthorium dating of the calcite covering of the specimens. These results raised the intriguing possibility that the artists who created these images were Neanderthal, since at the time Neanderthals were thought to have remained in the Iberian peninsula region until as recently as 35,000 to 40,000 years ago [Wilford2012]. However, in February 2013 researchers at the University of Oxford in the U.K, using a more sophisticated Carbon14based dating process, found that the latest Neanderthal sites are 10,000 years older than previously thought  i.e., they are 45,000 to 50,000 years old. Thus, for example, the handprints in Spain most likely are human, not Neanderthal [Callaway2013].
These three examples, by the way, underscore the futility in claiming that there is some sort of "conspiracy" or "groupthink" in the field preventing the consideration of youngearth creationist views. Note that each of these three studies have the potential to overthrow the beloved theories of numerous other researchers. If there are fundamental weaknesses in the general class of radiometric dating schemes (or in the particular schemes used in these three studies), why don't the researchers whose results are potentially refuted come forward to publicly identify these weaknesses or flaws? The only believable answer is that there are no fundamental flaws in these schemes  they have withstood decades of rigorous examination and refinement within the scientific community, and well deserve their reputation for reliability, although minor adjustments will be made from time to time as experimental techniques are further refined. For additional discussion, see Conspiracy.
Some valuable (and generally quite readable) references on radiometric dating, including detailed responses to specific issues that have raised by creationists, are the following: Dalrymple1991; Dalrymple2004; Dalrymple2006; Isaak2007, pg. 143157; Miller1999, pg. 6680; Stassen2005; Radiometric2013; Stassen1998; Wiens2002].