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Business writing: Don’t Lose Etiquette Just Because its Email


Like your mother most probably taught you, respect others.

Many of us have been taught how to effectively write letters (snail mail) but the transition to email was rapid and it was an exploratory journey whereby not all of us carried forward the lessons taught to us before.

Whether you are prospecting for business or communicating with your team, it is important to keep emails professional. In as much as the medium is readily available and we happen to live in the social media era, it is still required of you to keep the high standards of business writing.

Thanks to Avi Megiddo whose matrials I am going to use to emphasize the importance of etiquette when writing business emails.

There are a lot of interactive exercises in this post that I hope you will take advantage of and eplore the subject under review, the subject of etiqutte when writing business emails.

Lets start by understanding what etiquette is all about in the interactive video below:

Like any other compositin or essay it is important to have a well defined structure that follows a logical flow of ideas so that it is easy for the recipient to comprehend.

Now lets look at structure of a business email:

By the way, it is important to note that being formaland being rude are two distinct things. Actually a formal email is conveyed in the most polite terms permissible in the circumstance. It is premised on the idea of mutual civility. So as a word of advise, you must make sure that your email gets to the point without being brash and rude. Politeness is by no means a sign of weakness, it is power actually.


So, if you received a request for information, how do you respond to such an email? What elements should you include in the email so as to optimal convey your message?

What about an email to notify the different parties that the meeting has changed? Below is an illustration of best practices:

…and lastly

MathsGenius selected for the 2018 Tony Elumelu Foundation Entreprenuership Programme

edzai zvobwo mathsgenius

Johannesburg, South Africa

Zimbabwean mathematician and author, Edzai Conilias Zvobwo, affectionately known as “The MathsGenius” is among the 1 250 entrepreneurs who have been selected for the prestigious Tony Elumelu Foundation (TEF) programme.


More than 150,000 Africans from 114 countries worldwide applied to join the 4th cycle of The Tony Elumelu Foundation’s 10-year, $100 million TEF Entrepreneurship Programme. Today, the Foundation announced the African entrepreneurs with the most innovative, high-potential business ideas.

The 2018 cohort includes an additional 250 entrepreneurs to the standard selection of 1,000 – thanks to a \$1,000,000 partnership with the International Committee of the Red Cross (ICRC) to support 200 entrepreneurs in conflict and fragile zones of Nigeria (the North East where the Boko Haram scourge is felt and the Niger Delta region which suffers environmental degradation from oil spillage) a \$200,000 agreement with the United Nations Development Programme (UNDP) to support 40 pan-African entrepreneurs and a \$50,000 partnership with Indorama to support 10 Nigerians.

TEF Founder, Tony O. Elumelu, CON, commented: “The number and quality of applicants, 151,000  in total, was outstanding – it illustrates the strength and depth of entrepreneurial promise and commitment on our continent. Selection is never easy, and we profoundly regret that we cannot help all. Our partnerships with the Red Cross, UNDP and Indorama, alongside ongoing discussions with other international organizations, reflect the growing global recognition of what we have known all along – that entrepreneurship is the most effective path to sustainable development on our continent and our programme is the model to follow.”

Tony Elumelu, a Nigerian business tycoon has committed to empowering African small business people through his ideology called africapitalism. Africapitalism is an economic philosophy that is predicated on the belief that Africa’s private sector can and must play a leading role in the continent’s development.

The principles of Africapitalism are:

  • Entrepreneurship. Unlock the power of individuals to create and grow their business ideas into successful companies
  • Long-term Investments. Deploy patient capital that creates greater and broader economic value as opposed to merely the extraction of resources
  • Strategic Sectors. Invest in sectors delivering a financial return as well as broader economic and social value – agriculture, power, healthcare, and finance
  • Development Dividend. Conduct investments and business activity in a manner that delivers financial returns to shareholders as well as economic and social benefit to stakeholders
  • Value-Added Growth. Leverage locally available human and financial capital, raw material and other inputs that create longer, more integrated, and higher value regional supply chains
  • Regional Connectivity. Facilitate intra-regional commerce and trade through the development of national and cross-border physical infrastructure, and the harmonization of policies and practices
  • Multi-Generational Development. Focus on investments and economic growth strategies that build value for future generations
  • Shared Purpose. Foster collaboration between businesses, investors, governments, academia, civil society, philanthropists, and development institutions to create conditions that will empower the African private sector to thrive

Africapitalism is a call-to-action for businesses to make decisions that will increase economic and social wealth, and promote development in the communities and nations in which they operate. Such a decision will ultimately help businesses become more profitable as the communities they serve become well-off consumers, healthy and better educated employees, and even entrepreneurs who go on to become suppliers and service providers.

Africapitalism means that the business of development cannot be left to governments, donor countries and philanthropic organizations alone. The private sector must be involved in the business of development.

Edzai Zvobwo was ecstatic about his selection and is looking forward to the courses and the conference to be held in Nigeria later on in the year. Speaking to 1873FM, he stated, “I am very happy to be part of this select group of people who have gone through a rigorous process and have been given the nod that they have something tangible that will assist in pushing Africa forward. I resonate with the idea of africapitalism and hope to grow so as to able to support future entrepreneurs in the mother continent.”

The full list of selected entrepreneurs can be found HERE.

How is South Africa preparing its learners for the Nth Industrial Revolution?


How is South Africa preparing its learners for the Nth Industrial Revolution?

Education is meant to prepare learners for jobs of the future. Politicians and economists the world over are talking of the 4th Industrial Revolution and how it’s changing the idea of work as we know it. It is a given that countries that have or are making great strides at preparing their workforce for these changes are thriving or will thrive in this New World Order.

During the debate on President Cyril Ramaphosa’s first State of the Nation Address, Basic Education Minister Angie Motshekga said “pupils will need new skills to meet the demands of a world using advanced technologies, including artificial intelligence, robotics and driverless cars.” She also told the gathering that her department has been engaging with international and local experts on the implications of the fourth industrial revolution on education.

She further alluded to the fact that these advanced technologies such as automation, artificial intelligence, robotics, nano-technology, 3D printing and autonomous vehicles will demand non-routine inter-personal and analytical skills, social skills such as persuasion emotional and social intelligence and will demand creativity, agility and adaptability which are not necessarily being taught or nurtured in our current schooling system.

I applaud the minister for her statement in parliament as this should set the tone for education reform in South Africa. The question that I have for Minister Angie is: How are you preparing the learners for the Nth Industrial Revolution? N is a natural number greater than 3 in this context.

I do not doubt the minister’s passion for empowering the learners to meaningfully contribute to the success of the country but what I doubt is her willingness to be radical in overhauling the ailing system into a vibrant and value adding entity.

As an education support practitioner I can give a testimony of the improvements that have taken under her stewardship but are not necessarily known to the public because the department does not celebrate its successes enough. These small wins have been the result of unrelenting efforts from both the public and private sectors especially the NGOs. The business sector has also heeded her call for a multi-stakeholder approach and are contributing immensely to the addition of value in the system.

The biggest constraint in the ecosystem is lack of Radical Education Reform (RER) to get rid of any remnants of apartheid era education and truly usher in a quality and fair education ecosystem. The reforms that have been implemented to date have been infinitesimally small and have not reconfigured the education system as platform for the nurturing of future STEM leaders who will be ready to take on the challenges of tomorrow. The changes have been cautious and accommodative of the weakest links in the system.

When I talk about weak links, I am referring to both teachers and learners who are being carried by the system and are a hindrance to progress. I am a firm believer in “No Child Left Behind” policy which is fair and conforms to the human rights values that we subscribe to, but this should not be a reason to deny the bright and gifted children a chance to be the best they can be within a system that rewards diligence. For example the mathematics curriculum is shallow and was designed with the weakest learner in mind as opposed to designing it for the best learner. This is anti-development and has to be rectified.

From a STEM perspective, this lack of RER is the reason why we are lagging behind in the race for the Nth Industrial Revolution. As a mathematician, I am well aware of the fallacy of hasty generalization whereby one cannot conclude an outcome based on one instance but for emphasis and illustration I will employ an example of one the many reasons why we are not at the forefront in fields like robotics, artificial intelligence, optics, space travel and all the emergent industries that rely heavily on STEM. My example will be posed as a question. Why are South African mathematics learners not taught matrices in high schools?

Given that the topics included in the matric curriculum are not rigorously dealt with, a big void exists whereby crucial topics are omitted. Statistics was not in the curriculum and was introduced recently, the reason given was that most teachers had not done the topic in Bantu schooling system so there were no skills to pursue this subject, I am glad that it’s now included. What about matrices? Students who do not pursue STEM subjects in university will never get to know what a matrix is.

Matrices are pervasive in the new economy and the lack of exposure by learners to this topic is one of the many drivers why we are not producing top of the range students who become leaders in STEM fields. Matrices are everywhere in our day to day lives and we ought to empower the learners the ability to solve complex and wicked problems using matrices.

Below is a list of some of the areas where matrices are used extensively:

  • In physics related applications, matrices are applied in the study of electrical circuits, quantum mechanics and optics.
  • In the calculation of battery power outputs, resistor conversion of electrical energy into another useful energy, these matrices play a major role in calculations. Especially in solving the problems using Kirchoff’s laws of voltage and current, the matrices are essential.
  • In computer based applications, matrices play a vital role in the projection of three dimensional image into a two dimensional screen, creating the realistic seeming motions.
  • Stochastic matrices and Eigen vector solvers are used in the page rank algorithms which are used in the ranking of web pages in Google search.
  • The matrix calculus is used in the generalization of analytical notions like exponentials and derivatives to their higher dimensions.
  • One of the most important usages of matrices in computer side applications are encryption of message codes.
  • Matrices and their inverse matrices are used for a programmer for coding or encrypting a message.
  • A message is made as a sequence of numbers in a binary format for communication and it follows code theory for solving. Hence with the help of matrices, those equations are solved. With these encryptions only, internet functions are working and even banks could work with transmission of sensitive and private data’s.
  • In geology, matrices are used for taking seismic surveys.
  • They are used for plotting graphs, statistics and also to do scientific studies in almost different fields.
  • Matrices are used in representing the real world data’s like the traits of people’s population, habits, etc.
  • They are best representation methods for plotting the common survey things.
  • Matrices are used in calculating the gross domestic products in economics which eventually helps in calculating the goods production efficiently.
  • Matrices are used in many organizations such as for scientists for recording the data for their experiments.
  • In robotics and automation, matrices are the base elements for the robot movements. The movements of the robots are programmed with the calculation of matrices’ rows and columns. The inputs for controlling robots are given based on the calculations from matrices.

Just these illustrations will show you that we are robbing our learners of an opportunities to be at the fore-front of the technology-driven new era.

How will our learners compete with learners from Russia, China, Estonia, Finland and other countries that have these life-changing topics in the primary and high school curriculum? Even a child from Zimbabwe, Zambia or Kenya is introduced to matrices in Form 3, so why can’t the South African government radically change the education system so that it brings meaningful value to the country and continent? How long shall we wait cautiously as the world moves forward at a fast pace and we are surely being condemned to the zenith of the STEM advancement and breakthroughs.

My message to Minister Angie is that she should not settle for little gains but be radical and really change the course of history through education. She should be bold to call out those who are impeding progress whether its officials, or unions like SADTU who in their bid to protect their own, undermine the improvement of the whole system. I hope this little message of mine reaches the minister. I salute people like Panyaza Lesufi who work tirelessly to improve education for all, they are many out there, we just need the minister to lead the revolution.

The Eduation Support Forum (TEDSF) needs your help to push for Radical Education Reform (RER) in South Africa.

Have a look at this great cause, The Education Support Forum at https://www.payfast.co.za/donate/go/theeducationsupportforum

Linear Algebra Exercise

linear algebra

Matrix Method: Problems

  1. 4x + 4y + 4z = 12

    x + 5y + 6z = -16

    3x + y + z = 17

  2. 4x + y + 4z = 36

    3x + 4y + z = 24

    1000x + 3y + 2z = 28

  3. 2x + 6y + z = 16

    -x – 4y + 6z = 18

    x + y + z = 10

  4. 3x + 4y – z = 1

    -2x + y + z = 3

    12x + 12y + 12z = 0

  5. 2x – y + z = 17

    1/2x + 2y – z = 10

    -x + 5y – z = 1

  6. 2x + 2y + 2z = 13

    4x + y – z = 0

    -2x + 4y – z = 3

  7. 5/2x + 1/3y – 3/2z = 0

    2x + 3y – z = 9

    -2x + y + 2z = 7

  8. 2x + y + z = 49

    3x – z = 0

    2x – y = 0

  9. 23x + y + z = 6

    46x + 2y – z = 3

    69x – y – 2z = -5

  10. 2x + y – z = 15

    -2x – y + z = -5

    7x + 2y + 2z = -4

  11. 2x + 3y – z = 25

    x + 4y + z = 25

    x – 6y + 5z = 25

  12. -2x + 2y + 2z = 120

    3x + 3/4y + 3/5z = 150

    1/3x + 1/4y + 1/5z = 30

  13. 2x + y – z = 12

    x + 2y – 2z = 6

    3x – y + 4z = 36


  1. x = 7, y = 1, z = -3
  2. x = 0, y = 4, z = 8
  3. x = 6, y = 0, z = 4
  4. x = -1, y = 1, z = 0
  5. x = 10, y = 2, z = -1
  6. x = 1/2, y = 2, z = 4
  7. x = 2, y = 3, z = 4
  8. x = 7, y = 14, z = 21
  9. x = 1/23, y = 2, z = 3
  10. x = 2, y = 1, z = -10
  11. x = 12, y = 2, z = 5
  12. x = 30, y = 40, z = 50
  13. x = 6, y = 6, z = 6

Amazing Computer Science/Mathematics Support for GIS Analysts


I am always looking for material that enhances my people’s mathematics and computer science proficiency so I stumbled onto some information compiled by Alex Tereshenkov which is very helpful. Hope you enjoy it and also visit Alex’s Blog.

A lot of people who are studying GIS at school or already working as GIS analysts or GIS consultants often wonder what kind of competence will help to be attractive for employers and what domains of expertise are going to be in demand in the foreseeable future.

Usually the kind of questions GIS professionals ask is how much a GIS analyst should learn from other domains. So, we are wondering how much math, statistics, programming, and computer science should GIS analysts learn. Naturally, knowing what kind of GIS specific expertise is in demand is also very helpful. I have several posts on how get better at GIS here, here, and here.

To know what kind of GIS tools can do what kind of job is definitely helpful. This is much like a woodworker should know what kind of tools he has in his toolbox and what tools are available in the woodworking shop. Finding an appropriate tool for a certain job is not so hard nowadays with the Internet search engine and QA sites. However, the ability to understand both how data processing tools work and what happens behind the scenes to be able to interpret the analysis results is indispensable.

What is often true for many GIS analysts is that during their studies the main focus was on the GIS techniques and tools while math and CS courses were supplementary. This makes sense and the graduates are indeed most often competent GIS professionals capable of operating various GIS software suites, provide user support, and perform all kind of spatial analysis. However, it is also possible that in a career change, a person who hasn’t done any studies on GIS, is working as a GIS analyst and needs to catch up a bit. For those people who feel that they lack background GIS competence that they should had a chance to learn during their studies, or for you who just want to learn something that could help to have a broader view and give a deeper understanding of the GIS, I have compiled a list of useful links and books. Please enjoy!

There are lots of great questions answered on the GIS.SE web site; here is just a few:

Great books:

Spatial Mathematics: Theory and Practice through Mapping (2013)
This book provides gentle introduction into some mathematical concepts with focus on mapping and might be a good book to start learning math in GIS. No advanced background in math is required and high-school math competence will be sufficient.

Table of contents

  • Geometry of the Sphere
  • Location, Trigonometry, and Measurement of the Sphere
  • Transformations: Analysis and Raster/Vector Formats
  • Replication of Results: Color and Number
  • Scale
  • Partitioning of Data: Classification and Analysis
  • Visualizing Hierarchies
  • Distribution of Data: Selected Concepts
  • Map Projections
  • Integrating Past, Present, and Future Approaches

Mathematical Techniques in GIS, Second Edition (2014)

This book gives you a fairly deep understanding of the math concepts that are applicable in GIS. To follow the first 5 chapters, you don’t need any math except high school math. Later on, the book assumes that you have good knowledge of math at the level of a college Algebra II course. If you feel that it gets hard to read, take an Algebra II course online at Khan Academy or watch some videos from MIT to catch up first and then get back to the book. What I really liked about this book is that there are plenty of applicable examples on how to implement certain mathematical algorithms to solve the basic GIS problems such as point in polygon problem, finding if lines are intersecting and calculating area of overlap between two polygons. This could be particularly useful for GIS analysts who are trying to develop own GIS tools and are looking for some background on where to get started with the theory behind the spatial algorithms.

Table of contents

  • Characteristics of Geographic Information
  • Numbers and Numerical Analysis
  • Algebra: Treating Numbers as Symbols
  • The Geometry of Common Shapes
  • Plane and Spherical Trigonometry
  • Differential and Integral Calculus
  • Matrices and Determinants
  • Vectors
  • Curves and Surfaces
  • 2D/3D Transformations
  • Map Projections
  • Basic Statistics
  • Correlation and Regression
  • Best-Fit Solutions

GIS: A Computing Perspective, Second Edition (2004)

The book is a bit dated, but it is probably the best book in computer science for a GIS professional. It provides very deep understanding of the computational aspects that are used in GIS.

Table of contents

  • Introduction
  • Fundamental database concepts
  • Fundamental spatial concepts
  • Models of geospatial information
  • Representation and algorithms
  • Structures and access methods
  • Architectures
  • Interfaces
  • Spatial reasoning and uncertainty
  • Time

Practical GIS Analysis (2002)

This book is a unique example of a book for GIS professionals who want to see how the basic GIS algorithms and tools work. The exercises that follow give readers a chance to execute many common GIS algorithms by hand which let truly understand even some complex operations such as generating TIN or finding the shortest path on a street network. The software used as a reference is ArcView GIS 3, but it is still relevant as the GIS concepts haven’t changed much since then.

Table of contents

  • GIS Data Models
  • GIS Tabular Analysis
  • Point Analysis
  • Line Analysis
  • Network Analysis
  • Dynamic Segmentation
  • Polygon Analysis
  • Grid Analysis
  • Image Analysis Basics
  • Vector Exercises
  • Grid Exercises
  • Saving Time in GIS Analysis

Maths for Map Makers (2004)

I haven’t read this book so don’t have anything to comment on this. Sorry!

Table of contents

  • Plane Geometry
  • Trigonometry
  • Plane Coordinates
  • Problems in Three Dimensions
  • Areas and Volumes
  • Matrices
  • Vectors
  • Conic Sections
  • Spherical Trigonometry
  • Solution of Equations
  • Least Squares Estimation
  • References
  • Least Squares models for the general case
  • Notation for Least Squares

Exploring Spatial Analysis in GIS (1996)

I haven’t read this book either. I guess this one might be hard to find, but have listed it here just in case.

How much mathematics does a GIS Analyst have to know?


How much math does a GIS Analyst need to know? This is a question that is asked by many prospective and practicising GIS professionals. SOme ask because they want to become analysts or some are already analysts but are not sure of their competencies.

It is important that we differentiate between the different types of analysts that we have.

We have a GIS theory analyst who knows lots of mathematics, statistics and computer science and is interested in developing new spatial analysis models and then we have a practitioner who does not necessarily need to know how to manipulate complex matrix algebra or know the formula of Euclidean distance but uses these measures through GUIs like ESRI’s ArcGIS and ArcMAp.

SO the question is, what kind of an analysts are you? Are the theorists or just a practitioner?

For the Theorist:

Spatial analysis is a highly quantitative subject and for one to be great at developing models for academic or industry purposes, one has to have a very good foundation of mathematics with at least a degree in a quantitative discipline. If not a full degree then one should have taken some modules at colege level to be able to decipher the abstract language used in coming up with these models. One should think of themselves as a Spatial Data Scientist who has the following skills:

Technical Skills

  • Math (e.g. linear algebra, calculus and probability)
  • Statistics (e.g. hypothesis testing and summary statistics)
  • Machine learning tools and techniques (e.g. k-nearest neighbors, random forests, ensemble methods, etc.)
  • Software engineering skills (e.g. distributed computing, algorithms and data structures)
  • Data mining
  • Data cleaning and munging
  • Data visualization (e.g. ggplot and d3.js) and reporting techniques
  • Unstructured data techniques
  • R and/or SAS languages
  • SQL databases and database querying languages
  • Python (most common), C/C++ Java, Perl
  • Big data platforms like Hadoop, Hive & Pig
  • Cloud tools like Amazon S3
  • Geology, geography, GIS, or a related field;
  • Manipulating vector and raster data,
  • Geographic information systems and visualization software packages, including geologic cartographic production and quantitative spatial analyses e.g. ArcGIS software suite (ArcMap, ArcCatalog, ArcToolbox, ArcGIS server);
  • Ground-based and remotely-sensed data collection methodologies and processing;
  • Geodatabase management, and demonstrated ability to deal with large datasets (Big Data techniques)
  • Written and verbal communications skills, as demonstrated through published papers and presentations;
  • Project management abilities

Business Skills

  • Analytic Problem-Solving: Approaching high-level challenges with a clear eye on what is important; employing the right approach/methods to make the maximum use of time and human resources.
  • Effective Communication: Detailing your techniques and discoveries to technical and non-technical audiences in a language they can understand.
  • Intellectual Curiosity: Exploring new territories and finding creative and unusual ways to solve problems.
  • Industry Knowledge: Understanding the way spatial analysis in GIS and Region Sciences functions and how data are collected, analyzed and utilized.

For the Practitioner:

Now there are people withing the GIS community who are truly afraid of mathematics and statistics, they do not have programming skills and heavily rely on the GUIs like ArcGIS and ArcMap to click their lives away. What kind of maths and stats do they need? To help us answer this I looked around the web for similar questions and stumbled onto the following:

I make my living applying mathematics and statistics to solving the kinds of problems a GIS is designed to address. One can learn to use a GIS effectively without knowing much math at all: millions of people have done it. But over the years I have read (and responded to) many thousands of questions about GIS and in many of these situations some basic mathematical knowledge, beyond what’s usually taught (and remembered) in high school, would have been a distinct advantage.

The material that keeps coming up includes the following:

  • Trigonometry and spherical trigonometry. Let me surprise you: this stuff is overused. In many cases trig can be avoided altogether by using simpler, but slightly more advanced, techniques, especially basic vector arithmetic.
  • Elementary differential geometry. This is the investigation of smooth curves and surfaces. It was invented by C. F. Gauss in the early 1800’s specifically to support wide-area land surveys, so its applicability to GIS is obvious. Studying the basics of this field prepares the mind well to understand geodesy, curvature, topographic shapes, and so on.
  • Topology. No, this does not mean what you think it means: the word is consistently abused in GIS. This field emerged in the early 1900’s as a way to unify otherwise difficult concepts with which people had been grappling for centuries. These include concepts of infinity, of space, of nearness, of connectedness. Among the accomplishments of 20th century topology was the ability to describe spaces and calculate with them. These techniques have trickled down into GIS in the form of vector representations of lines, curves, and polygons, but that merely scratches the surface of what can be done and of the beautiful ideas lurking there. (For an accessible account of part of this history, read Imre Lakatos‘ Proofs and Refutations. This book is a series of dialogs within a hypothetical classroom that is pondering questions that we would recognize as characterizing the elements of a 3D GIS. It requires no math beyond grade school but eventually introduces the reader to homology theory.)

    Differential geometry and topology also deal with “fields” of geometric objects, including the vector and tensor fields Waldo Tobler has been talking about for the latter part of his career. These describe extensive phenomena within space, such as temperatures, winds, and crustal movements.

  • Calculus. Many people in GIS are asked to optimize something: find the best route, find the best corridor, the best view, the best configuration of service areas, etc. Calculus underlies allthinking about optimizing functions that depend smoothly on their parameters. It also offers ways to think about and calculate lengths, areas, and volumes. You don’t need to know much Calculus, but a little will go a long way.
  • Numerical analysis. We often have difficulties solving problems with the computer because we run into limits of precision and accuracy. This can cause our procedures to take a long time to execute (or be impossible to run) and can result in wrong answers. It helps to know the basic principles of this field so that you can understand where the pitfalls are and work around them.
  • Computer science. Specifically, some discrete mathematics and methods of optimization contained therein. This includes some basic graph theory, design of data structures, algorithms, and recursion, as well as a study of complexity theory.
  • Geometry. Of course. But not Euclidean geometry: a tiny bit of spherical geometry, naturally; but more important is the modern view (dating to Felix Klein in the late 1800’s) of geometry as the study of groups of transformations of objects. This is the unifying concept to moving objects around on the earth or on the map, to congruence, to similarity.
  • Statistics. Not all GIS professionals need to know statistics, but it is becoming clear that a basic statistical way of thinking is essential. All our data are ultimately derived from measurements and heavily processed afterwards. The measurements and the processing introduce errors that can only be treated as random. We need to understand randomness, how to model it, how to control it when possible, and how to measure it and respond to it in any case. That does not mean studying t-tests, F-tests, etc; it means studying the foundations of statistics so that we can become effective problem solvers and decision makers in the face of chance. It also means learning some modern ideas of statistics, including exploratory data analysis and robust estimation as well as principles of constructing statistical models.

Data Exploration Using R For Beginners


We will use R to explore the iris dataset. (loaded in R by default). Below are functions to help you with the dataset.


Check the dimensionality of the data (how many rows and columns)


[1] 150 5

Variable names or column names of the dataset


[1] “Sepal.Length” “Sepal.Width” “Petal.Length” “Petal.Width” “Species”



‘data.frame’: 150 obs. of 5 variables:
$ Sepal.Length: num 5.1 4.9 4.7 4.6 5 5.4 4.6 5 4.4 4.9 …
$ Sepal.Width : num 3.5 3 3.2 3.1 3.6 3.9 3.4 3.4 2.9 3.1 …
$ Petal.Length: num 1.4 1.4 1.3 1.5 1.4 1.7 1.4 1.5 1.4 1.5 …
$ Petal.Width : num 0.2 0.2 0.2 0.2 0.2 0.4 0.3 0.2 0.2 0.1 …
$ Species : Factor w/ 3 levels “setosa”,”versicolor”,..: 1 1 1 1 1 1 1 1 1 1 …



[1] “Sepal.Length” “Sepal.Width” “Petal.Length” “Petal.Width” “Species”

[1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
[21] 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
[41] 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
[61] 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
[81] 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
[101] 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120
[121] 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140
[141] 141 142 143 144 145 146 147 148 149 150

[1] “data.frame”

Get the first 5 rows


Sepal.Length Sepal.Width Petal.Length Petal.Width Species
1 5.1 3.5 1.4 0.2 setosa
2 4.9 3.0 1.4 0.2 setosa
3 4.7 3.2 1.3 0.2 setosa
4 4.6 3.1 1.5 0.2 setosa
5 5.0 3.6 1.4 0.2 setosa

Get Sepal.Length of the first 10 rows

iris[1:10, "Sepal.Length"]

[1] 5.1 4.9 4.7 4.6 5.0 5.4 4.6 5.0 4.4 4.9

Same as above


[1] 5.1 4.9 4.7 4.6 5.0 5.4 4.6 5.0 4.4 4.9

Distribution of every variable


Sepal.Length Sepal.Width Petal.Length Petal.Width Species
Min. :4.300 Min. :2.000 Min. :1.000 Min. :0.100 setosa :50
1st Qu.:5.100 1st Qu.:2.800 1st Qu.:1.600 1st Qu.:0.300 versicolor:50
Median :5.800 Median :3.000 Median :4.350 Median :1.300 virginica :50
Mean :5.843 Mean :3.057 Mean :3.758 Mean :1.199
3rd Qu.:6.400 3rd Qu.:3.300 3rd Qu.:5.100 3rd Qu.:1.800
Max. :7.900 Max. :4.400 Max. :6.900 Max. :2.500



setosa versicolor virginica
50 50 50


Pie chart




Variance of Sepal.Length


[1] 0.6856935

Covariance of two variables

cov(iris$Sepal.Length, iris$Petal.Length)

[1] 1.274315

Correlation of two variables

cor(iris$Sepal.Length, iris$Petal.Length)
[1] 0.8717538






Scatter plot

plot(iris$Sepal.Length, iris$Sepal.Width)

Pair plot



Elementary Maths for Everyone


Elementary arithmetic is the simplified portion of arithmetic that includes the operations of addition, subtraction, multiplication, and division. … Elementary arithmetic also includes fractions and negative numbers, which can be represented on a number line.

Below are some elementary mathematics problems that you can try at your own time. Tell me what you think about them, are they difficult? Which topics are a problem for you?

Questions: (Place your answers in the comment section)


\item Simplify without using a calculator:
\item ${\left(\frac{5}{4^{-1}-9^{-1}}\right)^{\frac{1}{2}}}$
\item$s^{\frac{1}{2}}\div s^{\frac{1}{3}}$
\item $\dfrac{12m^{\frac{7}{9}}}{8m^{-\frac{11}{9}}}$

\item $(x^3)^\frac{4}{3}$
\item $(s^2)^\frac{1}{2}$
\item $(m^5)^\frac{5}{3}$
\item $(-m^2)^\frac{4}{3}$
\item $(3y^\frac{4}{3})^4$

\item  Add the following sums:


\item (a) $\left(-2{y}^{2}-4y+11\right) + \left(5y-12\right)$
\item (b) $\left(-11y+3\right)+ \left(-10{y}^{2}-7y-9\right)$
\item (c) $\left(4{y}^{2}+12y+10\right)+\left(-9{y}^{2}+8y+2\right)$
\item (d) $\left(7{y}^{2}-6y-8\right) – \left(-2y+2\right)$
\item (e) $\left(10{y}^{5}+3\right)-\left(-2{y}^{2}-11y+2\right)$
\item (f) $\left(-12y-3\right) + \left(12{y}^{2}-11y+3\right)$

\item Show that the decimal $3,21\dot{1}\dot{8}$ is a rational number.
\item Express $0,7\dot{8}$ as a fraction $\frac{a}{b}$ where $a,b\in \mathbb{Z}$
\item Round-off the following numbers to the indicated number of decimal places:\par
\item $\frac{120}{99}=1,212121212\dot{1}\dot{2}$ to 3 decimal places
\item $\pi =3,141592654…$ to 4 decimal places
\item $\sqrt{3}=1,7320508…$ to 4 decimal places
\item $2,78974526…$ to 3 decimal places

\item Write the following irrational numbers to 3 decimal places and then write them as a rational number to get an approximation to the irrational number. For example, $\sqrt{3}=1,73205…$. To 3 decimal places, $\sqrt{3}=1,732$. $1,732=1\frac{732}{1000}=1\frac{183}{250}$. Therefore, $\sqrt{3}$ is approximately $1\frac{183}{250}$.

\item $3,141592654…$
\item $1,41421356…$
\item $2,71828182845904523536…$

\item Simplify:
\item $2^{3x} \times 2^{4x}$
\item $\dfrac{12p^2t^5}{3pt^3}$
\item $ (3x)^2 $
\item $(3^4 5^2)^3$

\item  Solve for the variable:
\item $ 2^{x+5} = 32 $
\item $ 5^{2x+2} = \frac{1}{125} $
\item $ 64^{y+1} = 16^{2y+5} $
\item $ 3^{9x-2} = 27 $
\item $ 81^{k+2} = 27^{k+4} $
\item $ 25^{(1-2x)}-5^4 = 0 $
\item $ 27^x \times 9^{x-2} = 1 $
\item $ 2^t + 2^{t+2} = 40 $
\item $ 2 \times 5^{2-x} = 5+ 5^x $
\item $ 9^m + 3^{3-2m} = 28 $


\item  The growth of algae can be modelled by the function $f(t) = 2^t$. Find the value of $t$ such that $f(t)=128$.
\item  A type of bacteria has a very high exponential growth rate at $80\%$ every hour. If there are $10$ bacteria, determine how many there will be in five hours?


#Wiki4Her Profile – Fadzayi Chiwandire

Fadzayi Chiwandire


OnePointFour Consulting




Lead Front-end Developer


Web development

What things/people/roles have contributed most to your success?

Prof. Saki Mafundikwa, Erica Van Niekerk, Sheryl Sandberg

If you knew all that you now know, what would you tell your 18 year old self?

To start coding now

When things get tough, how do you keep yourself going?

I have already started i cant put all thay effort to waste

What leadership qualities do you most admire?

Resilience and persistence

Victimhood, does that stance serve women’s causes? Women are still exploited, so what would you tell them?

LEARN a skill, master it and know your worth, when you have a solid skill to fall back and you know your worth they can not help but value and respect you.

Do you mentor any girls as part of your personal and/or corporate responsibility?

I started the diva initiative where i teach young girls, grades 6,7 and 8 how to code www.divainitiative.com

What are some of the leadership challenges/obstacles that are unique to women?

Being under estimated, being objectified. I remember in one meeting one of the guys made it his mission to explain everything that was being said to me in laymen terms.

Why did you choose your career?

I love to solve problem. Naturally i break one huge problem into smaller pieces and tackle one piece at a time…. thats pretty much is the base of a coders attitude

Are the lives of famous women leaders a lot easier than those of the average women today?

I am not sure about this as i have only been avaerage and not yet famous. But i would like to think the struggle is the same. We face some level of struggle in different areas

Gender quotas on boards are a contentious issue, what do you think about it?

Gender balance is important especially when it comes to decision making. Women need to be represented adequately where decisions that affect women are concerned

Any additional message to inspire young girls to succeed?

Be brave, be resilient and use your voice. When we support the growth and empowerment of women and girls, we raise the quality of life for everyone. This is because when women lead they not only lead businesses, they lead in their community, they fight for their children, and they give voice to issues that are important to our collective future — like education and health care.


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