Fibonacci Numbers and the Golden Ratio: Unlocking the Secrets of √5

                                                                                                                           By Vishnu Vinjamuri

Abstract:

The relationship between the Fibonacci sequence and the golden ratio, particularly in relation to the number 5, illustrates how the irrational number can be approximated through simple arithmetic involving Fibonacci numbers. As the sequence progresses, an accuracy of up to 15 digits is attainable, a level of precision commonly utilized in applications such as Microsoft Excel. By leveraging the error ratio generated at each stage, a weighted average of the results from subsequent stages is computed, facilitating faster convergence. The outcomes are not only promising but also intriguing, with convergence achieved at a Fibonacci number as low as 6765. 

Introduction:

Fibonacci Sequence and Fibonacci Numbers:

It is well known that Fibonacci sequence is a sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted by F_n.

The list of Fibonacci numbers include:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,144,233,377,610,987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141, .......

Golden Ratio:

The Golden Ratio, commonly denoted by the Greek letter phi (φ), is a significant mathematical constant with an approximate value of 1.6180339887. It is algebraically defined as the positive solution to the equation:                                   

- Equation 1

The positive solution to Equation 1 is, .

Fibonacci Sequence and Golden Ratio:

The ratio of two consecutive Fibonacci numbers approaches the golden ratio as the numbers increase towards infinity.

This is represented by the following equation:

                                      -Equation 2

Equation 2 can be re-written as:

Which means, for a large n,

                                        - Equation 3

To obtain the value of  , Equation 3 can be re-written as:

-1                                     -Equation 4

Using Equation 4, we can develop approximations for

Computation Using the Known Relation:

Let us start with the number 3, and present the results in the below table.

We will compare the results with the established value of 2.23606797749979. In the Python program, an accuracy of up to 32 digits is implemented for reasons that will be elaborated upon later.

The results are presented in the below Table 1.

Table 1 – Table 1 – Estimation of  from the known relation between  and Fibonacci numbers

Fibonacci Number	Estimate	% error	Error Ratio
0
1
1
2
3	3.000000000000000	34.1640786499874%
5	2.000000000000000	-10.5572809000084%	-30.9017%
8	2.200000000000000	4.3498389499900%	-37.0820%
13	2.250000000000000	-1.6130089900093%	-38.6271%
21	2.230769230769230	0.6230589874905%	-38.0329%
34	2.238095238095230	-0.2369671577017%	-38.2592%
55	2.235294117647050	0.0906618499902%	-38.1727%
89	2.236363636363630	-0.0346080647157%	-38.2057%
144	2.235955056179770	0.0132222663539%	-38.1931%
233	2.236111111111110	-0.0050499949535%	-38.1979%
377	2.236051502145920	0.0019289937405%	-38.1961%
610	2.236074270557020	-0.0007368002241%	-38.1968%
987	2.236065573770490	0.0002814340751%	-38.1965%
1597	2.236068895643360	-0.0001074980423%	-38.1966%
2584	2.236067626800250	0.0000410606287%	-38.1966%
4181	2.236068111455100	-0.0000156837602%	-38.1966%
6765	2.236067926333410	0.0000059906636%	-38.1966%
10946	2.236067997043600	-0.0000022882301%	-38.1966%
17711	2.236067970034710	0.0000008740258%	-38.1966%
28657	2.236067980351190	-0.0000003338485%	-38.1966%
46368	2.236067976410650	0.0000001275185%	-38.1966%
75025	2.236067977915800	-0.0000000487078%	-38.1966%
121393	2.236067977340880	0.0000000186045%	-38.1966%
196418	2.236067977560480	-0.0000000071067%	-38.1966%
317811	2.236067977476600	0.0000000027141%	-38.1966%
514229	2.236067977508640	-0.0000000010371%	-38.1966%
832040	2.236067977496400	0.0000000003958%	-38.1966%
1346269	2.236067977501080	-0.0000000001516%	-38.1966%
2178309	2.236067977499290	0.0000000000577%	-38.1966%
3524578	2.236067977499970	-0.0000000000223%	-38.1966%
5702887	2.236067977499710	0.0000000000081%	-38.1966%
9227465	2.236067977499810	-0.0000000000036%	-38.1966%
14930352	2.236067977499770	0.0000000000009%	-38.1966%
24157817	2.236067977499790	-0.0000000000009%	-38.1966%

 

The error ratio is negative due to the oscillation of the estimate around its true value, transitioning from one Fibonacci number to the next.

An accuracy of fifteen digits is attained when F_n+1=24157817 and F_n=14930352. In this context, Equation 4 can be reformulated as:

                              - Equation 5                           

The Fibonacci sequence is also formed by Binet’s formula which is as follows:

                               - Equation 6

It can be inferred and demonstrated that the error estimation  using Equation 4 decreases as  for large n, which is approximately equal to 38.1966% as determined in Table 1.

The Excel-based error estimates were insufficient for accurately producing the aforementioned error for large Fibonacci numbers due to limitations in the number of decimal places in Excel.

Consequently, the calculations were refined using Python programming to achieve an accuracy level of 32 decimal places before being imported back into the Excel table.

Optimizing Error Reduction and Enhanced Solutions:

Given that the errors decrease by approximately 38.1966% from one step to the next, we shall pursue enhanced convergence by employing a weighted average of the solution between successive steps.

We will designate the initial solution from Equation 4 at the n^th step as r5_n,i.

By calculating the weighted average based on the error ratio, the adjusted value of the solution will be presented as below:

      - Equation 7

Where r5_n,w is the modified estimate of  as weighted average of the solution from Equation 4.

The findings from Equation 7 are detailed in Table 2 below. It is evident that the convergence is significantly more rapid, occurring at a value as low as 6765, in contrast to 24157817 as indicated in Table 1.

 Table 2 – Optimized Error Reduction Solution

Fibonacci Number	Modified Estimate for	% error
0
1
1
2
3	2.27639319635939	1.8033986115524%
5	2.24120226788020	0.2296124461366%
8	2.23685242618125	0.0350816115322%
13	2.23618034018203	0.0050250119125%
21	2.23608448454537	0.0007382175206%
34	2.23607037951385	0.0001074213345%
55	2.23606832828112	0.0000156874179%
89	2.23606802866699	0.0000022882670%
144	2.23606798496276	0.0000003337543%
233	2.23606797858983	0.0000000487482%
377	2.23606797765834	0.0000000070907%
610	2.23606797752311	0.0000000010428%
987	2.23606797750312	0.0000000001490%
1597	2.23606797750030	0.0000000000229%
2584	2.23606797749985	0.0000000000029%
4181	2.23606797749980	0.0000000000006%
6765	2.23606797749979	0.0000000000000%

Analysis of Table 2 indicates that the error decreases by approximately 685%. Further calculations involving weighted averages may introduce unnecessary complexity.

Conclusions and Inferences:

• The relationship between and Fibonacci numbers is well-established, allowing us to express with an accuracy of up to 15 decimal places as  .                            
• By utilizing the weighted average of estimates derived from the relationship between and Fibonacci numbers, we can similarly express  with an accuracy of up to 15 decimal places as .
• Enhanced accuracy can be achieved by selecting larger Fibonacci numbers.

Prime Counting - A Self-correcting Approximation

                                                                                                                           By Vishnu Vinjamuri

Abstract:

The famous prime-counting function, known as p(x) = x/log x  is widely used for most practical purposes but underestimates the number of primes. Several other formulas are available that provide more accurate estimates of the number of primes less than a given positive integer. In this article, a unique approximation is presented using the formula  p(x) = x/log x as the base, which results in a very high degree of accuracy while remaining computationally simple.

Introduction:

The prime counting function is denoted by   p(x)  and has several approximations.

The most popular and widely used such approximation is

      

- Equation (1)

Where x is any given real number greater than 2 and p(x)  is the number of primes less than or equal to n.

It is well known that the above approximation runs asymptotic to the actual total number of primes and is a lower bound approximation.

Many other approximations using this format are of the form:

  - Equation (2)

Where B is a constant, whose value is 1.08366 as proposed first by Legendre and later changed to 1.0. From Wikipedia (source unknown), we see further modifications to these constants between different values of n, for better approximation and these constants seem to be working well.

Pierre Dusart suggested formulas using similar and more complex constants, with lower and upper bounds.

In 2024, Timothy Ganesan proposed the below formula for prime numbers which is a better approximation than 1.1.

 

- Equation (3)

The various approximations coming under the family of Equation (2), provide a very good estimate of the number of primes, but the constants need to be adjusted as the number n increases. Several other equations exist which are complex in nature.

These equations and varying constants make the computation more complex.

In this article, let us approximate the number of primes by looking at the very Equation (1) and closely examining the distribution of primes in large intervals, aiming to keep the solution simple.

From now onward, let us call Equation (1) the Prime Counting Function.

All logarithms in this article are natural logarithms.

Minimum Shear Wall Thickness - ACI318 or IS456? - A Study from Fundamentals

Introduction 

The Indian Concrete code IS 456-2000 recommends a minimum shear wall thickness of h/30, while the American concrete code ACI318 places it at h/25. 

Which one is correct? Which approach could be more prudent.

Let us discuss from fundamentals.

Problem Visualization

A shear wall is expected to have a lot of strength in its one plane. It's out of plane resistance is negligible compared to its in-plane resistance. This is due to its huge in-plane stiffness compared to its out of plane stiffness.

Its out of plane behavior is more or like that of a slab.

But low out of plane resistance does not mean we can have a thin concrete sheet with minimum cover to the rebar as our shear wall.

Let us analyze from first principles which approach could be good.

Mathematical Modeling

For a compression member's load at buckling failure, we have the following Euler's formula:q

P=p²EI/l²

Where P is the critical load at buckling

E is the modulus of elasticity

I is the modulus of elasticity of the section

l is the effective length of the section

For a shear wall, the effective length is its height h, or 0.75h, depending on the end conditions.

When it reaches the state of buckling, let us assume it loses its rotational restraint at its ends. The the effective length l = height h.

Then the above equation becomes

P=p²EI/h²

Then the critical buckling stress becomes:

P=p²EI/Ah²

Where p=critical compressive stress at buckling

A=cross-section area of the shear wall

This critical buckling stress needs to be more than the design compressive strength of the section.

From IS:456-2000, the Indian Concrete Code, treating the shear wall as a compression member, the maximum compressive strength in limit state design is 0.4f_ck.

If the critical buckling occurs before reaching the design compressive strength, it will escape the design check as the software does not do any buckling analysis of the wall element and at the most checks the code specified height to thickness ratio.

The term EI needs to be multiplied by a stiffness modifier in the above equation.

IS 16700 suggests a modifier of 0.7 under factor loads and 0.9 under un-factored loads. This may be OK for an in-plane moment causing some tension and cracking.

For out of plane, it will be more like a slab action, but the element will still be primarily under compression.

ACI 318-14 suggests a stiffness modifier of 0.35 for a cracked wall and 0.7 for an un-cracked wall under ultimate conditions. Under service conditions this can be enhanced 1.4 times.

Buckling being an extreme case, let us use the serviceability modifier. This means a cracked stiffness modifier of 0.49. 

This means, instead of EI in the above formula 0.49EI will be used.

Results

Considering a unit height of 1m or 1000mm, the thickness of the wall is h/30=33.33mm or h/25=1000/25=40mm.

For any height, same ratio of height by thickness continues and the results will be the same.

Below are the critical buckling stress results vs critical compressive stress for various grades of concrete and for a wall thickness of h/30 and h/25.

Graphically, they can be shown as below:

Inference

• For grades above M30, the buckling strength of the wall is less than its compressive strength for a thickness of h/30, leading to the possibility of a buckling failure before compression failure.

• From M25 thru M60,  the buckling strength of the wall is much higher than its compressive strength for a thickness of h/25, avoiding a failure in the wall before the failure in compression for which it is designed.

Recommendations

• Based on this study, it is recommended to limit the thickness of planar shear walls to not less than h/25.
• When it is difficult to keep the thickness above h/25, it is recommended to at least have the boundary element thickness greater than h/25 with the remaining wall thickness greater than h/30. A buckling analysis maybe performed on such walls.
• For shear walls having return walls in the perpendicular direction, h/30 may be maintained. Their buckling strength may be ascertained from a buckling analysis.

Undergrad data science projects by non-IT engineering students - A few tips

1.0 Introduction

As some people rightly say, "Data is the new oil". Is does not necessarily mean one can make a lot of money by getting into data sciences, though it is one of the lucrative career options at this point in time. Data is the new oil, or any new ore, for that matter. Like an ore can be explored in many different ways right from manual excavation to the most modern mechanized and automated explorations, data can be explored in many ways, right from finding out the mean and standard deviation of a given small data set up to applying the most modern computational techniques.

In India, there is a lot of gap between the supply and demand of engineers, with many engineering colleges even at the remotest parts of the country. It is practically impossible for all the pass-outs to get jobs in their own fields of study even if there is a significant improvement in the curriculum to make the students industry-ready by the time they pass out.

Like the engineers of the last 2-3 decades had to be computer literates by the time they passed out from the colleges even to survive in their own field, the engineers of present generation and next generation need to be data literate. This data literacy can be achieved without extensive efforts from one's end unless one wishes to be highly specialized in data sciences.

Right from high school, the students of mathematics are exposed to the rudiments of probability and statistics, and so are engineering students at the undergrad level! Applying these concepts in their own field of study to study the data that they deal with, enhances their understanding of the subject matter and enables them to make data-driven decisions. This also opens an additional avenue for job opportunities.

1.1 Target readers

The target readers of this blogspot article are engineering students of non-IT streams who want to improve their data skills and faculty members and managements who wish to improve their campus placements.

2. Developing the skills

A non-IT engineering student has many necessary skills in data sciences as one studies advanced concepts of probability and statistics as part of engineering maths, and acquires a good amount of coding skills. Is that enough?

2.1 Courses

To understand terminology used in and to get acquainted with the state-of-the art data sciences, an engineering undergraduate can do some of the online courses. To enroll for a new course offered by IITs and get certified, one can visit www.swayam.gov.in. To listen to the archived courses, one can visit www.nptel.ac.in. Search using the keywords Data Science and Data Analytics and you will find many courses - introductory to advanced. A beginner can do introductory courses.

One can also go thru the video lectures in Coursea, Udacity etc. 

2.2 Coding

One can use any programming language to write a program for solving the equations. However, to capitalize on the several readily available libraries, one can learn Python and/or R.

2.3 Software packages

Students have the privelege to freely download many software packages from the net by furnishing their credentials. Look for the below:

1. Student versions available for free

2. Free software (see https://www.predictiveanalyticstoday.com/top-free-statistical-software/ for recommendations)

3. Packages available with your academic institution

4. Excel Analysis Toolpak available as an add-in in your MS Exce. See the link below to find how to activate it:

https://support.office.com/en-us/article/use-the-analysis-toolpak-to-perform-complex-data-analysis-6c67ccf0-f4a9-487c-8dec-bdb5a2cefab6

Get in touch with experts in the field for their opinion which can include your teachers in the IT/Computer Science/Mathematics/Statistics departments.

3. Applying the skills in your project work

To apply the skills, one can do their undergraduate project work. To infuse data analytics into one's own project work, the following steps can be taken.

3.1 Define your problem

This can be as visualized by you or your project guide. You may also let your guide know that you are interested in a project that involves some sort of data analysis.

3.2 Gather data

This can be done in several ways. Could be from experimentation, could be from data generated from engineering formulas, could be from literature, or could be from databases available in records, standards, archives etc.

3.3 Visualize the data

This can be achieved by plotting. If the data is too large, you may want to do sampling.

3.4 Analyze the data

How much data? Entire data set or a sample? What is the right tool? How do we test?

Depends on the nature of your problem, no. of factors involved, size of the data etc.! For this, the knowledge gained from coursework will help.

3.5 Interpret the data

After the analysis, interpret what you have observed out of it i.e. the results of the analysis.

3.6 Develop solution

If your problem definition is to find the trend, you may not have to execute this step and conclude your project at the above step itself. If you have found what is the cause of the problem, you will have to present a solution. Note that it is generally expected in engineering profession that the person who find a problem also needs to find its solution!

3.7 Test the solution

Once you have developed a solution, you will have to test the solution by doing some iterations of reanalyzing the new data set and testing if the solution works.

3.8 Conclusion and presenting the report

Like in any other project work, this is the final step!

3.9 DMAIC - A popular technique

Define, Measure, Analyze, Improve and Control (DMAIC) is one popular technique where there is a good scope for applying the data analytics skills. For understanding and appreciating the process, visit:

https://asq.org/quality-resources/dmaic

4. Summary and additional thoughts

• Brush up your probability and statistics basics.
• Do some good data analytics/data science courses online, preferably from swayam.gov.in and/or nptel.ac.in.
• Apply the skills in your project work.
• Interact with your peers in the IT stream who should be doing some data science courses as part of their curriculum.
• Choose your mathematics/statistics/IT faculty member as your co-guide.

5. Note for the academic institutions

Showcase the skills of your students to the prospective recruiters and invite more and more companies from both IT and non-IT streams to your campus!

To attempt or not to attempt - A risk management approach toward negative marking

1.0 Introduction

In some of the competitive examinations, there is negative marking. When equal weightage is given for both right answer and wrong answer, the risk of losing marks when a wrong answer is given is high. In case of multiple choice questions, there will be always some probability of answering the question correctly, which increases when there is more than one right answer.

In risk analysis, we have the following formula for the expected value of a variable X.

E[X]=Sp_iX_i  for all possible values of X=X_i,

where 

E[X]=expected value of X.

p_i = probability of the value of X being X_i

2.0 Joint Entrance Exam Main (JEE Main) - A case study

2.1 Introduction - JEE Main

In India, we have the famous Joint Entrance Examination (JEE) for admission into various engineering, architecture and planning courses across the nation offered by various academic institutions run by or funded by the Government of India. This is conducted in two phases i.e. JEE Main and JEE Advanced. JEE Main is the basic qualifying examination for the said courses.

For 2020, the information bulletin for JEE Main can be found at the following link:

https://jeemain.nta.nic.in/WebInfo/Handler/FileHandler.ashx?i=File&ii=58&iii=Y

From the above bulletin (see pages 6 and 7), we find that every correct answer is awarded 4 marks and every wrong answer is awarded one negative mark (-1).

2.2 Definition: Uncertain question

Let us define an uncertain question as the MCQ for which the examinee doesn't know the right answer.

2.3 Answering uncertain questions at random

Going by the question paper for 2019, there are four options for each multiple choice question (MCQ).

The probability of choosing the right answer at random is 1/4=0.25.

The probability of choosing the wrong answer at random is 3/4=0.75.

So, the expected value of attempting a question by randomly choosing an answer is 0.25x4-0.75*.1=0.25.

This means, one may still score positively by taking a chance. However, the probability of the answer being wrong is very high.

When four questions are wrongly answered and one question is rightly answered out of five questions, the outcome is 4*(-1)+4=0. From this, it is evident that the risk of losing marks happens only when one scores less than 1/5th of the total number of questions where one is uncertain of the answer.

When p is the probability of answering a question correctly, 1-p is the probability of answering it wrong. In this case, p and 1-p are 0.25 and 0.75 respectively.

The probability of answering r answers or less correctly for n questions is,

When n= 1 to 5, negative marks will be scored only when i=0.
The probability of obtaining negative marks for n=1 to 5 is as follows:

n	P
5	0.237
4	0.316
3	0.422
2	0.563
1	0.750

Given that there are 20 MCQs in each question paper, the probability of obtaining negative marks by answering all questions at random occurs when less than 4 answers are right i.e. when the number of right answers is 3 or less.

The probability of obtaining negative marks when 20 questions are answered at random is P=0.225.

From this, though the probability of scoring positive marks or zero out of all uncertain questions is higher except when the no. of MCQS are one or two, there is still a good amount of risk involved. Whether one can take this much risk or not, depends on the future plans one has.

2.4 Making intelligent guesses for MCQs

The risk of scoring negative can be minimized by guesstimating the answers that appear to be the most right.

Assumption: Let us assume the probability of choosing the right answer increases to p=0.5 when one makes an intelligent guess combined with subject matter knowledge.

In this case, the estimated value of attempting one uncertain answer is 0.5*4-0.5*1=1, which is much higher than before.

Now, the probability of a negative score when 8 uncertain questions are answered is,

P=0.035 which is statistically insignificant, as statistical signifcance is often attached to a confidence level of 95% (probability of 1-0.05) and more. This means, we can say with a confidence level of 0.965(1-0.035) or 96.5% that will either improve one's score or there is no harm at least, by making intelligent guesses when the no. of uncertain questions is 8 or more. 

Even when there is only one uncertain question, the probability of scoring a negative mark is 0.5 only if the assumption made above is right. For a higher number of uncertain questions, it will be less than 0.5 which means the probability of net benefit or no loss out of the whole exercise is higher.

3.0 Applicability of the case study

The above case study is applicable in similar cases when the following conditions are met

1.  When the options provided are four out of which only one is correct. When more than one option is correct, however, the risk comes down.
2. The ratio of positive marking to negative marking is 4:1.

4.0 Conclusions

1. For the above case study, it is shown that the expected value from taking the risk of attempting uncertain questions is always positive.
2. The probability of scoring positive is higher than the probability of scoring negative, when more than two multiple choice questions are answered at random.
3. The probability of scoring positive is higher than the probability of scoring negative when more than one multiple choice question is answered by making an intelligent guess, when one can guesstimate correctly with a probability of 0.5.
4. The risk of scoring negative marks comes down when one makes an intelligent guess of the answer rather than selecting the answer at random.
5. When 8 or more questions are answered by an intelligent guess, there is a very good possibility of scoring positive or suffering no loss from the guess work, when the probability of guesstimating the right answer is 0.5.

5.0 Path forward

1. One can do similar math when the number of options and/or the marking scheme are different.
2. The confidence level of scoring no negative mark can be recalculated by varying the probability of success for an intelligent guess.
3. Coaching centers can find out the probability of their students guessing the answer right in each subject separately and advise the students individually. As they have math and stat experts, they can even come up with better models (Good if they are already doing it, but I have not heard of any).
4. Coaching centers also need to teach and demonstrate how to make better guesses. Those preparing themselves for competitive exams on their own need to develop their own strategies.
5. When one has multiple options for pursuing higher studies, or multiple career options, one can take bigger risks, as bigger rewards may come from bigger risks. 

Importance of data literacy for non-IT engineers - Some random thoughts

The big question(s)

Often one may question - why do we get into all this statistics, six sigma, data analytics and all such jargon to do my day-to-day engineering work? I know my stuff and have years of experience behind me. Can a statistical model interpret or predict better than me? Don't we lose focus by doing all this? Will it not require extra efforts?

Well, my article may not provide answer to each and every question a traditional engineer like many of us may have, but will try to address the concerns and apprehensions at a broader level.

What is data analytics and where is it applied?

I do not want to provide any standard textbook or classroom definition of what is meant by data analytics, but one can always find it by googling or referring to the right resources available aplenty online. Many are aware of it by now, and many aren't aware too!

With or without saying, analytics is applied everywhere. When you have a numerical data, as an engineer, you will at least try to find what is its mean and what is its range (maximum and minimum).
This is a very elementary application of data analytics. As you further probe, you will find other parameters like standard deviation, median, mode etc. which statistically describe your data. For large populations, you will do sampling. This is followed by finding the probability distributions associated with your data, making inferences, and the journey continues...

From the above examples, it is evident that the concept of data analytics is not at all new. Analytics at its most elementary level does not require a computer at all!

I am a non-IT engineer. Why do I need it?

In engineering, a lot of data gets generated. One good example is traffic studies. For a structured review and for drawing meaningful conclusions, one needs to apply analytics.

Well, I design a structure or machine. I have my formulas and software. When I use the formulas rightly, I will get the right output. Why do I need analytics? One may say.

In engineering, many of the formulas are either complex or empirical. When we use such formulas, it is always essential to study the variation in the output as the inputs vary.

When the formulas are not derived from first principles but were empirically arrived at from observations, regression etc., it will be to one's own advantage to study and analyze the data generated by those formulas. This enables one to have control over one's designs and take informed decisions. 

If an equation is empirically arrived at using regression, then studying it further generates additional insights. One can make new models, re-validate the equation, or may even come up with a better equation!

These types of empirical equations we find often in fluid mechanics for example. Unfortunately, in most of the text books I studied in my engineering in the 90s, almost nowhere the rationale behind the empirical formulas was explained.

Even when you have a derived complex formula, the formula will be based on several assumptions. Also the inputs like material properties have their own confidence level attached to them. A statistical study will bail us out from the complexities of the equation as we get to study the numerical data and the trends it follows.

Needless to say that the data from experimentation and observations needs to be studied statistically, which many do, with or without calling it analytics. If a statistical study is not performed, the findings are just not right and it is high time one realized that.

In any case, one can always find out what are the inputs that influence the output most. This can be achieved by design of experiments and sensitivity analyses.

Two types of engineering data

Without getting into the standard technical terms in data analytics, I would classify the numerical data into two types from an engineer's perspective. The first one is visible data, abundantly available everywhere. The second type of data is invisible data, an example of which is hidden in the form of formulas and gets generated only when you calculate the output using those formulas and varying the inputs.

The visible data needs to be handled, using relevant data analytics techniques.

The invisible data can also be handled the same way, by generating data, arranging it in tables, plotting it graphically for visualization and then analyzing.

Sources of the visible data are, data from experimentation, published and validated data, data gathered from observations and so on.

Sources of invisible data not only those from the equations, but also those from computer simulations. For example, the output data generated by a finite element software can be put in this category.

Should I test my data when there is a clear formula?

As people say "seeing is not believing", seeing is not always believing. Hence the data needs to be put to statistical tests. This helps in decision making for those who are unsure of the direction in which one needs to go. Even for those who are "sure" by experience, it helps in re-validating their experience based knowledge.

Examples of some applications

In manufacturing engineering, one often uses the principles of six sigma. There are many statistical techniques that are common between six sigma and data analytics. By exposing oneself to data analytics, one will have a wider range of tools to handle one's data.

Design of experiments is often conducted in engineering, particularly mechanical engineering.

I see the term sensitivity analysis often being used in the analysis of high rise buildings. There are some who are loosely using this term just because the standards are using it. Maybe the standards and leaders of the industry in developing them  need to elaborate more on what it means by sensitivity analysis in line with its standard definition. It will be of great help if a standardized definition is included in the Indian tall concrete buildings code IS 16700.

As one is aware, advanced concepts like artificial intelligence, machine learning etc. are used in all industries.

Epilogue

One may say what I have written is nothing but statistics and hence old wine in new bottle! I would rather call it old nectar in new bottle, often left untasted by many.

The techniques that I have discussed and even more complex techniques are often used by researchers traditionally in engineering. But knowledge and application of the basic concepts at working level will be of great help as discussed and will elevate the society to the next level of a data literate society from a compute literate society.

Path forward

Below is the path forward I would suggest for promoting data literacy among engineers.

1. Engineers should consider refreshing their undergraduate level statistics knowledge as part of their Continuous Professional Development.
2. Guidelines should be framed to include a minimum number of questions on statistics and probability in exams like Professional Engineer's exams.
3. A fixed number of minimum Professional Development Units (PDUs) in statistics and probability/data analytics/data science should be prescribed for retaining a Professional Engineer's or a project manager's certification.
4. Non-IT engineering students interested in analytics with the intention of using it in their core field or making it their career, can consider doing a project in analytics (wait for my next article).
5. Authors of engineering textbooks should include brief notes on how the empirical formulas have been arrived at, and provide reference to the original sources.