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Further Reading


Sure Signs

How to Understand the Degree of Certainty in Key Discoveries

by Hugh Ross

Article originally appeared in
Salvo 35

I have devoted my life to researching and presenting scientific ­evidence for the Christian worldview and for the trustworthiness of the Bible's message. People in my public audiences express great excitement and joy about the new scientific evidence, but many of them are reluctant to use it in their own engagements with skeptical friends and associates. One reason for their hesitation concerns their inability to evaluate how certain they can be of the results I present—especially when it comes to how scientists use numbers. How can you be sure?

It is not that I fail to express the level of certainty in the results I present. Whenever I express a value for a particular result, I consistently append the "error bars," that is, a graphical representation of the variability of the data I'm presenting. Wherever possible, I also add corroborating results from independent means of investigation. My wife and many friends, however, have pointed out to me that most lay American audiences have little or no understanding of what error bars mean and of the importance of multiple independent means of investigation.

Hopefully in what follows, readers will gain a better understanding of how to evaluate the scientific claims made here and elsewhere and will become equipped to communicate that understanding to others. Let me use for my illustrative example a new measurement that provides (additional) evidence for a cosmic creation event and implies the existence of a cosmic Creator.

An international team of astronomers has recently determined the rate of expansion for the universe based on studies of HII regions (star-forming gaseous nebulae) in 69 galaxies over a broad range of distances. This seems to be a straightforward scientific determination, but it is loaded with philosophical implications. Here's why: Since the cosmic expansion rate measured is at least approximately constant, the inverse of that rate: (1) establishes that the universe had a beginning, (2) yields the amount of time that has transpired since the cosmic beginning, and (3) implies that a cosmic Initiator must exist to have set the cosmos in motion.

Thus, if we can put confidence in the astronomers' measurement, we will have strong scientific corroboration for the biblical view of creation. So just how much confidence can we place in it? How do we determine that?

What Error Bars Mean

To answer these questions, we first need to understand the error bars associated with the team's measurement. The cosmic expansion rate they measured was 74.3 ± 3.1 kilometers/second/megaparsec.1 (A megaparsec = 3.258 million light years.) The 74.3 figure is the mean, or averaged value, of all the measurements taken, and the ±3.1 figure is the error bar, or what statisticians call the standard deviation. Together, they tell us that the astronomers determined that the cosmic expansion rate probably lies somewhere between 71.2 and 77.4 kilometers/second/megaparsec.

If we apply the formula for deriving probability from the mean and standard deviation, the ±3.1 error bar tells us that there exists a 68 percent probability or certainty that the rate truly lies somewhere in this range. Conversely, it means that there is a 32 percent probability that the true cosmic expansion rate lies outside that range of values.

Sometimes scientists will quote a range of possible values based on two or three standard deviations; i.e., a doubling or tripling of the standard deviation figure. For the above measurement, two standard deviations would be ±6.2, meaning that the range of possible expansion rate values would be between 68.1 and 80.5 kilometers/second/megaparsec. But in this case, the probability that the true cosmic expansion rate lies outside that range of values is only 4.6 percent. The application of three standard deviations widens the range of possible values even more, to between 65.0 and 83.6 kilometers/second/megaparsec, but it also lowers the probability that the true cosmic expansion rate falls outside that range of values all the way down to 0.3 percent.

Thus, as more standard deviations are applied, the range of possible values grows proportionally, but the probability that the true value falls within that range grows almost exponentially.

The error bars described above, however, account only for possible random or statistical errors. These errors refer to what is called the "statistical scatter" in the values a scientist observes in his measurements. As mentioned above, the result of 74.3 ± 3.1 kilometers/second/megaparsec was based on measurements of 69 nearby galaxies that contained HII regions that could be accurately measured. The ±3.1 error bar designates the scatter about the mean manifested in the measurements, what we might loosely call the margin of (random) error.

Accounting for Systematic Errors

Random errors, however, are not the only kind of error that can directly impact the certainty of a published result. An indirect but equally important factor is what scientists term "systematic" errors. These are errors resulting from flaws, biases, or inadequacies in the instruments, in data selection, or in the assumptions held about the phenomenon being measured. Any of these things can skew all the measurements to either higher or lower values, so scientists must take them into account.

The team that produced the cosmic expansion rate we have been discussing exhaustively examined all possible instrumental problems and biases, as well as all the assumptions about the properties of HII regions that could conceivably be mistaken or that could change if one were to sample more-distant galaxies. This examination led them to conclude that, in addition to the ±3.1 random error in their cosmic expansion rate determination, there also existed a ±2.9 systematic error.

The ±2.9 systematic error means that there exists a 32 percent probability that the ±3.1 random error centers on a value for the cosmic expansion rate outside of the range of 71.4-to-77.2 kilometers/second/megaparsec. As with random or statistical errors, increasing the range of values based on multipliers of the systematic error yields greater certainty in the result. For example, tripling the systematic error to ±8.7 means that there is only a 0.3 percent possibility that the ±3.1 random error centers on a value outside the range between 65.6 and 83.0 kilometers/second/megaparsec.

Independent Measurements

The most accurate published value for the cosmic expansion rate is 69.6 ± 0.7 kilometers/second/megaparsec.2 Is this discordant with the value based on HII regions (i.e., 74.3 ± 3.1)? Not at all. If the two values both reflect accurate measurements of the cosmic expansion rate, then, based on the error bars, there is a high probability (a little greater than 32 percent) that the two values will differ from one another by 6.0 kilometers/second/megaparsec. Since they actually differ by only 4.7, they are not discordant.

Moreover, these are not the only two measurements of the cosmic expansion rate that scientists have made so far. To date, astronomers have made accurate measures of this rate based on nine independent methods, namely: (1) HII regions, (2) angular sizes of temperature differences in the cosmic background radiation, (3) Cepheid variable stars, (4) type Ia supernovae, (5) asymptotic giant branch stars, (6) direct distance measures to masers in galaxies, (7) gravitational lensing of distant quasars, (8) gamma-ray burst events, and (9) cooling of the cosmic background radiation.

One reason why astronomers are so confident about the age of the universe, a cosmic beginning, and the implications of this age and the beginning is that all nine methods yield the same result if one takes into account both the statistical and systematic error bars associated with each method.

Blocking Out Noise

Sometimes scientists are challenged to detect a signal of a real effect in the presence of a lot of environmental or instrumental noise. How do they figure out whether their detection is real or just an artifact of the noise? They do so by measuring how much the signal rises above the noise, specifically by ­determining the signal-to-noise ratio. Barring systematic effects, if the signal-to-noise ratio is two, the probability that the signal is a real effect and not just noise is 68.3 percent. A signal-to-noise ratio of three raises the probability to 95.4 percent; a ratio of four raises it to 99.7 percent; of five to 99.994 percent; and of six to 99.99994 percent.

In the physical sciences, a measured effect is not considered to be a real detection unless the signal-to-noise ratio exceeds five. Thus, physical scientists are not permitted to declare a discovery unless they can demonstrate a signal-to-noise ratio that exceeds five.

Here's an example: Recent attempts have been made to establish that the universe experienced a hyper-inflation expansion event when it was only a tiny fraction of a second old. Such an inflation event is an important prediction of astronomers' cosmic creation model. Therefore, a verification of the cosmic inflation event would serve to increase astronomers' (and the lay public's) certainty that the universe indeed experienced a creation event in finite time.

A figure called the scalar spectral index, designated as ns, reveals whether or not the universe had an inflation event. For a universe with no inflation event, ns would be 1.0 or greater. For a universe that had a simple inflation event, ns would be exactly 0.95. For a universe that has had a complex inflation event, ns would be between 0.96 and 0.97.

The South Pole Telescope research team recently published its latest measurements of ns.3 These measurements, in combination with the best results from the WMAP and Planck satellites, have yielded the most accurate determination of ns to date. The value they published shows ns to be 0.9593 ± 0.0067.4

The research team proved the reality of cosmic inflation to 6.075 standard deviations. That number of standard deviations translates into a certainty measure for the reality of cosmic inflation. Thanks to the team's measurements, we now can be 99.9999999 percent certain that the universe indeed went through a hyper-inflation expansion event when it was only about 10-34 seconds (a super-brief moment) old. However, this certainty measure does not take into account possible systematic effects. Consequently, astronomers are pursuing independent methods of determining cosmic inflation measurements.

Confidence in the Numbers

That latter point is important. Before putting any long-term confidence in a scientific discovery, one should always look for corroboration. Has any other independent research team, using different detection equipment and/or different detection methods, confirmed the result? Is the result confirmed both by observations over time and by experiments? Is there a theory that successfully integrates and explains all the observations and experiments?

Personally, I do not put a lot of confidence in a scientific result unless it has been established by experiments, observations, and theory, and I see consistency among all the observations and experiments. In addition, I expect to see the results becoming progressively better as the error bars, both random and systematic, shrink.

What pleases me about the biblical creation model we have developed and continue to develop at Reasons to Believe is that the march of scientific advances over the past century has yielded an improving consistency among the theory, observations, and experiments that undergird the model. That improving consistency emboldens me, and hopefully you as well, to include scientific data among the reasons we present for our hope for eternal life through Jesus Christ to an unbelieving world. 

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