The term mathematics is often used generically to include statistics, operational research and computing as well as conventional mathematics. Statisticians are involved in answering questions, making decisions and solving problems in the face of uncertainty, using limited information collected in experimental observations or surveys. Mathematical models develop, out of abstract concepts of mathematics, a dream world of idealization and imagination useful in real world explanation, prediction, summarization, assumption, speculation, planning and control based on statistical data and its evaluation. In any of these cases, stating initially a problem objective enables problem conversions into mathematical form identifying relations between variables. Simulations of situations being modeled are very complex, especially when there are random effects to be considered.

1. Reports are made based on Mathematical modeling of statistical data.

A basic understanding of Mathematical Modeling and statistical data require a need for more detailed guidelines as to the listing of statistical procedures implemented in articles concerning real-life analysis. Areas of uncertainty can range from the definition, inclusion, and number of data or events to the type of theoretical model used to simulate our problem. The misuse and uncertainty of statistics and statistical data is a vital concern in areas of scientific importance, which include articles published in Science Magazine.

There are two basic types of mathematical models. On one hand, deterministic models are used in cases where the outcome is a direct consequence of the initial conditions of problems, and often times, involving differential equations in which time is the independent variable. On the other hand, stochastic models are reserved for situations where a random effect plays a central role in the problem. The activity of modeling is a process, which involves a number of clearly identifiable stages:

1. Identifying the real problem.

• Establish the purpose and objective.
• List source of facts and data (determining reliability and rendered uniqueness).
• Decide essentially deterministic or stochastic.
• Characterize possible simulations.

1. Formulating a mathematical model.

• Identify and list all relevant factors.
• Collect and examine data for information explaining the behavior of the variables.
• Denote variables choosing appropriate symbols and assign units.
• State any assumptions necessary.
• Draw up relationships and equations connecting the problem variables.

1. Proportionality.
2. Linear and non-linear relationships.
3. Empirical Relations (derived from and based entirely on data).
4. Input/output principles?
5. Laws of motion.
6. Difference and differential equations.
7. Matrices.
8. Probabilities and statistical distributions, etc.

1. Obtaining the mathematical solution to the problem using:

• Algebraic and/or numerical methods.
• Calculus.
• Graphs.
• Writing computer programs or using a prepared package if suitable.

1. Interpreting the mathematical solutions.

• Examine the results obtained from the mathematics (values, correct sign, and size).
• Increasing or decreasing appropriately.
• Certain graphs linear.
• Sensitivity analysis (consider large and small values of variables for sensible behavior).
• Best solution expected or should initial conditions be changed?

1. Comparing results with reality.

• Can your results be tested against real data?
• Do your mathematical solutions make sense?
• Do your predictions agree with the real data?
• Evaluate your model.

1. Has it fulfilled its purpose.
2. Can the model be significantly improved by greater mathematical sophistication.
3. Do the interim results (time between events) suggest that more accuracy is needed with an improved model?

Very often the "modeling cycle’ is traversed a number of times before the results are satisfactory.

1. Writing a report.

• Who is the report for and what do the readers want to know?
• How much detail is required in the report?
• How can the report be constructed so that the important features are clear and the results that we want to be read stand out?

In the second stage of the modeling process list all the factors, which are the building blocks of the model,

that we can identify as being relevant to the problem. Going through the list, the least relevant factors are thrown out, retain only the important ones, using personal judgment and knowledge about the system being modeled including any available data. A factor may be quantifiable (given a numerical value) or it may be a quality (named not measured numerically) or simply a relationship between two other factors.

Quantifiable factors can normally be divided into:

• Constants (fixed values, such as the speed of light).
• Parameters (constant values for a particular problem but can change from problem to problem).
• Variables (can be discrete or continuous).

It is often useful to distinguish between variables (parameters) which are:

• Inputs to a model.
• Outputs from the model.

In some cases, it is possible, after working through the model, to express the output variables in terms of the input variables by mathematical expressions. Within each group of factors look for distinguishing relationships with distinctive input variables and output variables, where some variables are direct consequences of other variables and some variables are clearly independent. The degree of importance of a particular factor cannot be judged at the outset. With the model developed and implemented to produce an answer, investigation determines the effect a particular factor has on the answer by varying the factor and noting the answer’s behavior.

If factors are the building blocks, assumptions provide the cement with which to put the structure together:

1. Assumptions about whether or not to include certain factors.
2. Assumptions about the relative sizes of terms or the relative magnitudes of the effects of various factors.
3. Assumptions about the forms of relationships between variables.

Choosing assumptions to keep the model as simple as possible:

• Note all assumptions.
• Note all consequences of assumptions.
• Test assumptions against data.
• Beware of implicit assumptions (not defined, i.e. gravitational attraction of any two particles).

1. Reports that are of particular interest are those published in Science Magazine.

Measuring the universe’s expansion rate, the Hubble constant, does involve various approaches. Astronomers estimate distance by comparing the apparent brightness of a star with its true brightness. Judging a star’s true brightness is tricky, too. Instabilities in unusual stars called Cepheid variables flicker in a precise way, and the period of the flickering is related to the star’s true brightness. Watson, Cosmology: The Universe Shows Its Age, Science Magazine page 3, illustrates a developing cosmic yardstick, where knowing a Cepheid’s apparent brightness, its flicker rate, and its red shift; astronomers in principle measure the Hubble constant. Watson explains on page 4 that Wendy Freedman and 26 colleagues of Carnegie Observatories in Pasadena, California used this method to initially claim a Hubble constant of .80 +or-.17 and after measuring distances to a dozen galaxies reduces the estimate to .73+or-.11. Also on page 4, Watson points out that Allan Sandage of the Carnegie Observatories at the Space Telescope Science Institute in Baltimore and at the University of Basel in Switzerland, Gustav Tammann, combined Cepheid and another form of standard candle, type Ia supernovae (exploding white dwarf stars), whose distance is inferred from their apparent brightness alone. This team determined distances to six galaxies containing seven supernovae very close to the same brightness, implying that this kind of supernovae can serve as "standard candles". Sandage’s team analyzed 30 other supernovae in galaxies far beyond the Virgo Cluster. These supernovae gave an average value for the Hubble constant of .58+.07or-.08. Watson on page 4 adds that Brian Schmidt of the Mount Stromb and Siding Spring Observatories in Australia observed type II supernovae, giant stars collapsing into neutron stars, hurling their outer layer of hydrogen and helium out into space, glowing like giant light bulbs. Measuring the speed at which this envelope flies outward and using the observations in theoretical methods determine absolute brightness hence the all-important distance in the expanding photosphere, where their best Hubble constant estimate is .73+or-.07.

Ann Finkbeiner in her article, Science Magazine, Astronomy: Hubble Telescope Settles Cosmic Distance Debate—Or Does It? Page 1, points out that astronomers have been trying to measure the Hubble constant since Edwin Hubble found evidence that the universe is expanding in 1929. The range of data collected to support these estimates is comparatively very small to the time predictions. Making a prediction for a value of time outside the range of the data is a risky thing to do and it is usually better and safer to use a model only over the range from which it was derived, when possible. Random variables are basic in the development of useful probability distributions. When modeling random variables, randomness is not chaos but it has a pattern of randomness, which is being modeled. Although incomplete, however a fundamental description, randomness may include uncertainty about or sensitivity to initial conditions. Making a general assumption based on very few data in effect treats the relative frequency distribution as a probability function. Confidence intervals are interval estimators capable of generating narrow intervals of sample measurements contained within upper and lower confidence limits. The probability that a confidence interval will enclose the parameter is called the confidence coefficient. Correlation coefficients would be used to consider a comparison of dependent variations to independent estimations, and developed regression lines enable the assertion of deviations and results as ordinary algebraic expressions. If you have a reasonable number of data, you can calculate the standard deviation of the errors in your model’s predictions so that you can give an approximate confidence interval for future predictions. This gives rise to the question, what is a reasonable number? The requirements for the recommended number of random samples for a standard normal distribution to achieve a reasonable and consistently small error deviation is satisfied by only Sandage’s team with some 30 plus observations. A further problem is that stars in our own galaxy or even nearby ones cannot give a true reading of the Hubble constant, because of surrounding stars and galaxies generating motions that are hard to separate form cosmic expansion.

In page 5, Watson goes on to include the use of data collected by the Hipparcos satellite, 120,000 stars have been mapped 100 times more accurately. Where investigations of Michael Feast of University of Cape Town in South Africa and Robin Catchpole of the Royal Greenwich Observatory in Cambridge, U.K. have determined the distance to nearby Cepheid variable stars by parallax. Tracking changes in the apparent position of a star relative to the background carpet of stars as Earth moves in its orbit around the sun exhibits the farther away the star the less it appears to move. Distance measurement, which is independent of the brightness-flicker relationship, shows that the Cepheids in the large Magellanic Cloud are about 10% farther away than was previously thought. Cepheids in general are then brighter and hence farther away than astronomers had realized. These considerations bring Freedman’s value of .73 down to .66 and Sandage’s constant value down to .54 or less. On page 6, Watson includes that Bruce Peterson, at Mount Stromlo, and colleagues on the MACHO project, considering Hipparcos data, focus on RR Lyraes pulsating stars in the large Magellanic Cloud. Relying on a quirk in pulsation of some of the stars that allows the team to tie down the actual star brightness very accurately, compared with observed brightness, yields an accurate distance for the large Magellanic Cloud that matches the new Cepheid distance. Applying the same calibration scheme to RR Lyraes in the globular cluster M15 reduces the globular cluster age by about 30%, and his best estimation of 12.6+or-1.5 billion years. Watson also inserts here that John Fenley, of Britain’s University of Sussex, and colleagues, using Hipparcos parallax measurements to check RR Lyrae distances found that the traditional distance scales held up well. But the lower cluster ages are consistent with another set of stellar ages from the ancient stars called white dwarfs.

Watson goes on to say that Terry Oswalt of the Florida Institute of Technology in Melbourne explains, basically the ‘dead’ cores of stars are slowly cooling. They shine only because they were initially very hot, and the cooling process takes billions of years. In our galactic neighborhood astronomers see none cooler than 4000 Kelvin. The implication of this abrupt cutoff is that even the oldest white dwarfs have not yet had time to chill out completely. Taking the best guess of the age of the galactic disk of 9.6 billion years, add 2 billion years for the galaxy to collapse from the big bang and the disk to form, and we get an absolute lower limit to the age of the entire universe of about 11 billion-12 billion years. Sandy Legget of the Joint Astronomy Center in Hilo, Hawaii, and his colleagues puts the age of the oldest dwarfs at a younger 8+or-1.5 billion years. These examples are surely lacking descriptive facts of procedure in defense of their determinations.

James Glanz in a Science Magazine article, Astronomy: Holes in the Sky Provide Cosmic Measuring Rod, focuses on shadows instead of flickering lights. The dark silhouettes cast by distant clusters of galaxies against what John Carlstrom, a radio astronomer at the University of Chicago, calls "an amazing backlight" gives us cosmic microwave background radiation (CMBR). The size of the shadows in the CMBR provides a cosmic measuring stick, characteristically independent. Galaxy clusters make dents in the CMBR, Sunyaev-Zeldovich (SZ) effect, holes in the sky in which CMBR’s temperature drops by thousands of degrees Kelvin, caused by gas that collects in the gravitational well of the clusters. Observations of the clusters by x-ray satellites, such as the German Roentgen Satellite and Japanese ASCA satellite, measure the spectrum of the x-rays, which indicates the gas’s temperature and the x-ray brightness. Combined with the depth of the SZ hole, gives astronomers enough information to determine both the gas’s density and its diameter in light-years. By simple geometry, that length can be combined with the angle the gas blob subtends on the sky to give the cluster’s distance. Optical measurements of the speed at which the cluster is hurtling away (red shift) can be incorporated to give us the Hubble constant. Curvature of space and cluster arrangement assumed spherical and cooling flows estimation are subjects of error consideration in this process. Astronomers also need better x-ray pictures of far-off clusters to extract precise Hubble constants.

With estimations for the Hubble constant lying between 0.54 and 0.85, and age predictions of the universe ranging from 8-16 billion years the debate continues. We have a convincing consensus around a median value of 0.68 for the Hubbell constant with the age of the universe averaging around 12 or 13 billion years. When we have another dozen or so credible teams with data defended and modeled estimations we will be approaching an amount of random samples sufficient to create a distribution necessary to achieve confidence intervals and likelihood for the estimation of the predictions of the age of the universe.

John Doyle in his article, Statistical errors in Medicine, Science Magazine, notes that physicians are not, in general, particularly inclined towards mathematics. Doyle concludes that the statistical error rate was "unacceptably high" in some papers containing numerical results published in specific journals—sixty-five (40% of 164) papers in the British Journal of Psychiatry in 1993. The lack of completeness and detailed listings of applied statistics made it difficult to asses the appropriateness of more than half the papers examined in the Clinical Articles section and Transactions of Societies sections of the January through June 1994 issues of the American Journal of Obstetrics and Gynecology (volume 170, numbers 1 to 6). Nearly one-third of the articles contained examples of statistics used inappropriately, suggesting a need for more detailed guidelines as to the listing of statistical procedures used. More recently, a study on the misuse of correlation and regression in three top-rated medical journals found serious problems. Fifteen categories of errors were identified of which eight were significant, including:

• Failure to define clearly the relevant sample number.
• Display of potentially misleading scatterplots.
• Attachment of unwarranted importance to significance levels.
• The omission of confidence intervals for correlation coefficients and around regression lines.

In scientific articles and reports viewed in Science Magazine data very often comes from experiments, observations, and measurements made with imperfect instruments; so errors cannot be avoided. The errors in the data are of course in addition to the errors made during any modeling:

1. Errors due to modeling assumptions.
2. Errors due to collection or definition of samples or events.
3. Errors due to approximate method of solution.
4. Errors due to carrying out the inexact arithmetic (i.e. rounding errors).
5. Errors in the data.

It is clear that predictions cannot be 100% reliable. There is an intrinsic duty however; to try to estimate what the maximum error in predictions is likely to be, support determinations in summary, and then report findings in a manner that the intended audience can understand. The integrity of the people involved in publishing these articles should have the intentions of our scientists and the concerns of their readers equally at heart.