text: + Many elementary functions of complex values have the same derivatives as those for real functions: for example D z2 = 2z. 1 Δ sin {\displaystyle |f(z)-(-1)|<\epsilon } Then the contour integral is defined analogously to the line integral from multivariable calculus: Example Let {\displaystyle \Omega } two more than the multiple of 4. The complex number a + b i a+bi a + b i is graphed on this plane just as the ordered pair ( a , b ) (a,b) ( a , b ) would be graphed on the Cartesian coordinate plane. → , . 0 By Cauchy's Theorem, the integral over the whole contour is zero. Today, this is the basic […] Generally we can write a function f(z) in the form f(z) = f(x+iy) = a(x,y) + ib(x,y), where a and b are real-valued functions. P���p����Q��]�NT*�?�4����+�������,_����ay��_���埏d�r=�-u���Ya�gS 2%S�, (5��n�+�wQ�HHiz~ �|���Hw�%��w��At�T�X! for all + 1. c FW Math 321, 2012/12/11 Elements of Complex Calculus 1 Basics of Series and Complex Numbers 1.1 Algebra of Complex numbers A complex number z= x+iyis composed of a real part <(z) = xand an imaginary part =(z) = y, both of which are real numbers, x, y2R. 2. i^ {n} = -1, if n = 4a+2, i.e. , and let ( In single variable Calculus, integrals are typically evaluated between two real numbers. Limits, continuous functions, intermediate value theorem. One difference between this definition of limit and the definition for real-valued functions is the meaning of the absolute value. lim z Complex numbers are often represented on the complex plane, sometimes known as the Argand plane or Argand diagram. Another difference is that of how z approaches w. For real-valued functions, we would only be concerned about z approaching w from the left, or from the right. γ t The theorem tells us a little more: Suppose that F(z) is a complex antiderivative for f(z), i.e. Δ This is a very condensed and simplified version of basic calculus, which is a prerequisite for many courses in Mathematics, Statistics, Engineering, Pharmacy, etc. x ) z Then, with L in our definition being -1, and w being i, we have, By the triangle inequality, this last expression is less than, In order for this to be less than ε, we can require that. Assume furthermore that u and v are differentiable functions in the real sense. z {\displaystyle f(z)=z^{2}} is holomorphic in the closure of an open set Ω − e Online equation editor for writing math equations, expressions, mathematical characters, and operations. Then we can let ζ z >> {\displaystyle \gamma } In the complex plane, however, there are infinitely many different paths which can be taken between two points, , with 1 6.2 Analytic functions If a function f(z) is complex-di erentiable for all points zin some domain DˆC, then f(z) is … Given the above, answer the following questions. We can’t take the limit rst, because 0=0 is unde ned. z Equation of a plane A point r (x, y, z)is on a plane if either (a) r bd= jdj, where d is the normal from the origin to the plane, or (b) x X + y Y + z Z = 1 where X,Y, Z are the intercepts on the axes. This curve can be parametrized by ( , {\displaystyle f(z)} Here we have provided a detailed explanation of differential calculus which helps users to understand better. ( In the complex plane, there are a real axis and a perpendicular, imaginary axis . 0 All we are doing here is bringing the original exponent down in front and multiplying and … You can also generate an image of a mathematical formula using the TeX language. → 2. ) z 1 f Suppose we have a complex function, where u and v are real functions. = i Variable substitution allows you to integrate when the Sum Rule, Constant Multiple Rule, and Power Rule don’t work. where we think of %���� ) ¯ Ω  {\displaystyle {\bar {\Omega }}} It says that if we know the values of a holomorphic function along a closed curve, then we know its values everywhere in the interior of the curve. ) ( {\displaystyle \ e^{z}=e^{x+yi}=e^{x}e^{yi}=e^{x}(\cos(y)+i\sin(y))=e^{x}\cos(y)+e^{x}\sin(y)i\,}, We might wonder which sorts of complex functions are in fact differentiable. This is a remarkable fact which has no counterpart in multivariable calculus. The symbol + is often used to denote the piecing of curves together to form a new curve. I've searched in the standard websites (Symbolab, Wolfram, Integral Calculator) and none of them has this option for complex calculus (they do have, as it has been pointed out, regular integration in the complex plain, but none has an option to integrate over paths). ] 4. i^ {n} = 1, if n = 4a, i.e. i 2 A function of a complex variable is a function that can take on complex values, as well as strictly real ones. + ( 2 ( 1 The following notation is used for the real and imaginary parts of a complex number z. t We can write z as 3. i^ {n} = -i, if n = 4a+3, i.e. Imaginary part of complex number: imaginary_part. Creative Commons Attribution-ShareAlike License. 3 Hence, the limit of Complex formulas involve more than one mathematical operation.. Complex analysis is a widely used and powerful tool in certain areas of electrical engineering, and others. This formula is sometimes called the power rule. ϵ z )  ( 1 z : If z= a+ bithen a= the Real Part of z= Re(z), b= the Imaginary Part of z= Im(z). z z Before we begin, you may want to review Complex numbers. = e ζ Thus, for any The basic operations on complex numbers are defined as follows: (a+bi)+(c+di)=(a+c)+(b+d)i(a+bi)–(c+di)=(a−c)+(b−d)i(a+bi)(c+di)=ac+adi+bci+bdi2=(ac−bd)+(bc+ad)i a+bic+di=a+bic+di⋅c−dic−di=ac+bdc2+d2+bc−adc2+d2i In dividing a+bi by c+di, we rationalized the denominator using the fact that (c+di)(c−di)=c2−cdi+cdi−d2i2=c2+d2. − f e {\displaystyle \gamma } How do we study differential calculus? . . In Algebra 2, students were introduced to the complex numbers and performed basic operations with them. *����iY� ���F�F��'%�9��ɒ���wH�SV��[M٦�ӷ����)�G�]�4 *K��oM��ʆ�,-�!Ys�g�J���$NZ�y�u��lZ@�5#w&��^�S=�������l��sA��6chޝ��������:�S�� ��3��uT� (E �V��Ƿ�R��9NǴ�j�$�bl]��\i ���Q�VpU��ׇ���_�e�51���U�s�b��r]�����Kz�9��c��\�. {\displaystyle f(z)=z^{2}} as z approaches i is -1. The fourth integral is equal to zero, but this is somewhat more difficult to show. [ The complex number calculator allows to perform calculations with complex numbers (calculations with i). − Complex numbers can be de ned as pairs of real numbers (x;y) with special manipulation rules. Ω −  {\displaystyle |z-i|<\delta } + Now we can compute. γ In advanced calculus, complex numbers in polar form are used extensively. γ ϵ f {\displaystyle \Omega } 0 The students are on an engineering course, and will have only seen algebraic manipulation, functions (including trigonometric and exponential functions), linear algebra/matrices and have just been introduced to complex numbers. '*G�Ջ^W�t�Ir4������t�/Q���HM���p��q��OVq����濜���ל�5��sjTy� V ��C�ڛ�h!���3�/"!�m���zRH+�3�iG��1��Ԧp� �vo{�"�HL$���?���]�n�\��g�jn�_ɡ�� 䨬�§�X�q�(^E��~����rSG�R�KY|j���:.3L3�(����Q���*�L��Pv�͸�c�v�yC�f�QBjS����q)}.��J�f�����M-q��K_,��(K�{Ԝ1��0( �6U���|^��)���G�/��2R��r�f��Q2e�hBZ�� �b��%}��kd��Mաk�����Ѱc�G! This is useful for displaying complex formulas on your web page. x = = 3 With this distance C is organized as a metric space, but as already remarked, Simple formulas have one mathematical operation. This can be understood in terms of Green's theorem, though this does not readily lead to a proof, since Green's theorem only applies under the assumption that f has continuous first partial derivatives... Cauchy's theorem allows for the evaluation of many improper real integrals (improper here means that one of the limits of integration is infinite). {\displaystyle \Omega } These two equations are known as the Cauchy-Riemann equations. i ) Does anyone know of an online calculator/tool that allows you to calculate integrals in the complex number set over a path?. 0 ( endobj | 2 Cauchy's integral formula characterizes the behavior of holomorphics functions on a set based on their behavior on the boundary of that set. Generally we can write a function f(z) in the form f(z) = f(x+iy) = a(x,y) + ib(x,y), where a and b are real-valued functions. /Filter /FlateDecode z The differentiation is defined as the rate of change of quantities. cos z The important vector calculus formulas are as follows: From the fundamental theorems, you can take, F(x, y, z) = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k Fundamental Theorem of the Line Integral , → ) {\displaystyle \zeta \in \partial \Omega } | y ) Introduction. | We also learn about a different way to represent complex numbers—polar form. Complex analysis is the study of functions of complex variables. formula simpli es to the fraction z= z, which is equal to 1 for any j zj>0. Because I wanted to make this a fairly complete set of notes for anyone wanting to learn calculus I have included some material that I do not usually have time to cover in class and because this changes from semester to semester it is not noted here. z It would appear that the criterion for holomorphicity is much stricter than that of differentiability for real functions, and this is indeed the case. z . ) min In fact, if u and v are differentiable in the real sense and satisfy these two equations, then f is holomorphic. to If − a If z=c+di, we use z¯ to denote c−di. z y A frequently used property of the complex conjugate is the following formula (2) ww¯ = (c+ di)(c− di) = c2− (di)2= c2+ d2. With the help of basic calculus formulas, this is easy to solve complex calculus equations or you can use a calculator if they are complicated. §1.9 Calculus of a Complex Variable ... Cauchy’s Integral Formula ⓘ Keywords: Cauchy’s integral formula, for derivatives See also: Annotations for §1.9(iii), §1.9 and Ch.1. ) Its form is similar to that of the third segment: This integrand is more difficult, since it need not approach zero everywhere. b {\displaystyle i+\gamma } Therefore f can only be differentiable in the complex sense if. , an open set, it follows that {\displaystyle f(z)=z} Also, a single point in the complex plane is considered a contour. Suppose we want to show that the ( ) lim {\displaystyle f} Conversely, if F(z) is a complex antiderivative for f(z), then F(z) and f(z) are analytic and f(z)dz= dF. one more than the multiple of 4. e Note that we simplify the fraction to 1 before taking the limit z!0. y C �v3� ��� z�;��6gl�M����ݘzMH遘:k�0=�:�tU7c���xM�N����zЌ���,�餲�è�w�sRi����� mRRNe�������fDH��:nf���K8'��J��ʍ����CT���O��2���na)':�s�K"Q�W�Ɯ�Y��2������驤�7�^�&j멝5���n�ƴ�v�]�0���l�LѮ]ҁ"{� vx}���ϙ���m4H?�/�. = three more than the multiple of 4. is an open set with a piecewise smooth boundary and ) {\displaystyle \zeta -z\neq 0} z f Use De Moivre's formula to show that \sin (3 \theta)=3 \sin \theta-4 \sin ^{3} \theta Δ ( Δ {\displaystyle z_{0}} Calculus is a branch of mathematics that focuses on the calculation of the instantaneous rate of change (differentiation) and the sum of infinitely small pieces to determine the object as a whole (integration). Euler's formula, multiplication of complex numbers, polar form, double-angle formulae, de Moivre's theorem, roots of unity and complex loci . {\displaystyle \gamma } This function sets up a correspondence between the complex number z and its square, z2, just like a function of a real variable, but with complex numbers. ϵ z Cauchy's Theorem and integral formula have a number of powerful corollaries: From Wikibooks, open books for an open world, Contour over which to perform the integration, Differentiation and Holomorphic Functions, https://en.wikibooks.org/w/index.php?title=Calculus/Complex_analysis&oldid=3681493. t i ( Δ I'm searching for a way to introduce Euler's formula, that does not require any calculus. + Δ {\displaystyle z:[a,b]\to \mathbb {C} } in the definition of differentiability approach 0 by varying only x or only y. Hence the integrand in Cauchy's integral formula is infinitely differentiable with respect to z, and by repeatedly taking derivatives of both sides, we get. ϵ , then. Since we have limits defined, we can go ahead to define the derivative of a complex function, in the usual way: provided that the limit is the same no matter how Δz approaches zero (since we are working now in the complex plane, we have more freedom!). 3 < y + 3 f z < x��ZKs�F��W���N����!�C�\�����"i��T(*J��o ��,;[)W�1�����3�^]��G�,���]��ƻ̃6dW������I�����)��f��Wb�}y}���W�]@&�$/K���fwo�e6��?e�S��S��.��2X���~���ŷQ�Ja-�( @�U�^�R�7\$��T93��2h���R��q�?|}95RN���ݯ�k��CZ���'��C��Z(m1��Z&dSmD0����� z��-7k"^���2�"��T��b �dv�/�'��?�S�ؖ��傧�r�[���l�� �iG@\�cA��ϿdH���/ 9������z���v�]0��l{��B)x��s; x 0 Although calculus is usually not used to bake a cake, it does have both rules and formulas that can help you figure out the areas underneath complex functions on a graph. We parametrize each segment of the contour as follows. %PDF-1.4 ) Use De Moivre's formula to show that \sin (3 \theta)=3 \sin \theta-4 \sin ^{3} \theta This difficulty can be overcome by splitting up the integral, but here we simply assume it to be zero. On the real line, there is one way to get from | {\displaystyle \Delta z} y γ being a small complex quantity. Because Solving quadratic equation with complex number: complexe_solve. 3 You will need to find one of your fellow class mates to see if there is something in these notes that wasn’t covered in class. {\displaystyle z-i=\gamma } 2 δ , the integrand approaches one, so. If you enter a formula that contains several operations—like adding, subtracting, and dividing—Excel XP knows to work these operations in a specific order. Note that, for f(z) = z2, f(z) will be strictly real if z is strictly real. BASIC CALCULUS REFRESHER Ismor Fischer, Ph.D. Dept. (1 + i) (x − yi) = i (14 + 7i) − (2 + 13i) 3x + (3x − y) i = 4 − 6i x − 2i2 + 6i = yi + 3xi3   This function sets up a correspondence between the complex number z and its square, z2, just like a function of a real variable, but with complex numbers.Note that, for f(z) = z2, f(z) will be strictly real if z is strictly real. A function of a complex variable is a function that can take on complex values, as well as strictly real ones. x > i We now handle each of these integrals separately. , and For example, suppose f(z) = z2. ranging from 0 to 1. Two popular mathematicians Newton and Gottfried Wilhelm Leibniz developed the concept of calculus in the 17th century. ) Here we mean the complex absolute value instead of the real-valued one. For example, suppose f(z) = z2. So. . Differentiate u to find . ) {\displaystyle z\in \Omega } z e . ( δ γ z 2 As an example, consider, We now integrate over the indented semicircle contour, pictured above. Γ = γ 1 + γ 2 + ⋯ + γ n . z ≠ , then z {\displaystyle \delta ={\frac {1}{2}}\min({\frac {\epsilon }{2}},{\sqrt {\epsilon }})} The complex numbers z= a+biand z= a biare called complex conjugate of each other. As distance between two complex numbers z,wwe use d(z,w) = |z−w|, which equals the euclidean distance in R2, when Cis interpreted as R2. Let e Note that both Rezand Imzare real numbers. ( = If f  (z) is continuous within and on a simple closed contour C and analytic within C, and if z 0 is a point within C, then. z {\displaystyle f} Therefore, calculus formulas could be derived based on this fact. be a complex-valued function. is a simple closed curve in Math Formulas: Complex numbers De nitions: A complex number is written as a+biwhere aand bare real numbers an i, called the imaginary unit, has the property that i2 = 1. = Sandwich theorem, logarithmic vs polynomial vs exponential limits, differentiation from first principles, product rule and chain rule. The process of reasoning by using mathematics is the primary objective of the course, and not simply being able to do computations. F0(z) = f(z). y {\displaystyle \epsilon \to 0} z �y��p���{ fG��4�:�a�Q�U��\�����v�? 2 1. i^ {n} = i, if n = 4a+1, i.e. the multiple of 4. is holomorphic in This page was last edited on 20 April 2020, at 18:57. − ( , and z This is implicit in the use of inequalities: only real values are "greater than zero". Every complex number z= x+iywith x,y∈Rhas a complex conjugate number ¯z= x−iy, and we recall that |z|2 = zz¯ = x2 + y2. Note then that z 2 {\displaystyle x_{2}} 2 Ω Declare a variable u, set it equal to an algebraic expression that appears in the integral, and then substitute u for this expression in the integral. = = e , then. x x sin z x This indicates that complex antiderivatives can be used to simplify the evaluation of integrals, just as real antiderivatives are used to evaluate real integrals. 5 0 obj << 0 {\displaystyle t} i Ω cos ) z In this unit, we extend this concept and perform more sophisticated operations, like dividing complex numbers. For this reason, complex integration is always done over a path, rather than between two points. z z If f (x) = xn f (x) = x n then f ′(x) = nxn−1 OR d dx (xn) =nxn−1 f ′ (x) = n x n − 1 OR d d x (x n) = n x n − 1, n n is any number. In the complex plane, if a function has just a single derivative in an open set, then it has infinitely many derivatives in that set. If such a limit exists for some value z, or some set of values - a region, we call the function holomorphic at that point or region. {\displaystyle \Gamma =\gamma _ … The complex number equation calculator returns the complex values for which the quadratic equation is zero. Ω The complex numbers c+di and c−di are called complex conjugates. , if f {\displaystyle z_{1}} In advanced calculus, complex numbers in polar form are used extensively. As with real-valued functions, we have concepts of limits and continuity with complex-valued functions also – our usual delta-epsilon limit definition: Note that ε and δ are real values. i . be a path in the complex plane parametrized by → A calculus equation is an expression that is made up of two or more algebraic expressions in calculus. In this course Complex Calculus is explained by focusing on understanding the key concepts rather than learning the formulas and/or exercises by rote. {\displaystyle z(t)=t(1+i)} The Precalculus course, often taught in the 12th grade, covers Polynomials; Complex Numbers; Composite Functions; Trigonometric Functions; Vectors; Matrices; Series; Conic Sections; and Probability and Combinatorics. lim {\displaystyle \epsilon >0} Thus we could write a contour Γ that is made up of n curves as. i + f Calculus I; Calculus II; Calculus III; Differential Equations; Extras; Algebra & Trig Review; Common Math Errors ; Complex Number Primer; How To Study Math; Cheat Sheets & Tables; Misc; Contact Me; MathJax Help and Configuration; My Students; Notes Downloads; Complete Book; Current Chapter; Current Section; Practice Problems Downloads; Complete Book - Problems Only; Complete … Cauchy's theorem states that if a function Khan Academy's Precalculus course is built to deliver a comprehensive, illuminating, engaging, and Common Core aligned experience! In Calculus, you can use variable substitution to evaluate a complex integral. → and Viewing z=a+bi as a vector in th… /Length 2187 Complex formulas defined. Powers of Complex Numbers. , and let For example, let {\displaystyle \lim _{z\to i}f(z)=-1} ?����c��*�AY��Z��N_��C"�0��k���=)�>�Cvp6���v���(N�!u��8RKC�' ��:�Ɏ�LTj�t�7����~���{�|�џЅN�j�y�ޟRug'������Wj�pϪ����~�K�=ٜo�p�nf\��O�]J�p� c:�f�L������;=���TI�dZ��uo��Vx�mSe9DӒ�bď�,�+VD�+�S���>L ��7��� ∈ In a complex setting, z can approach w from any direction in the two-dimensional complex plane: along any line passing through w, along a spiral centered at w, etc. = Δ ∈ The order of mathematical operations is important. Differential Calculus Formulas. 1 i Recalling the definition of the sine of a complex number, As  ( Continuity and being single-valued are necessary for being analytic; however, continuity and being single-valued are not sufficient for being analytic. 1 0 obj γ of Statistics UW-Madison 1. 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Argand plane or Argand diagram, which is equal to zero, here... A function that can take on complex values for which the quadratic is! You can use variable substitution allows you to integrate when the Sum Rule, and Common Core aligned!! Can only be differentiable in the use of inequalities: only real values are greater! We have provided a detailed explanation of differential calculus which helps users to better. Remarkable fact which has no counterpart in multivariable calculus in the real sense complex numbers—polar form parametrize each of. The process of reasoning by using mathematics is the basic [ … ] basic calculus Ismor..., like dividing complex numbers = z 2 { \displaystyle f ( z ) = z2 semicircle,! Of a complex variable is a remarkable fact which has no counterpart in multivariable calculus therefore, calculus formulas be., complex numbers x ; y ) with special manipulation rules the concept of calculus in the complex values the! 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Using the TeX language provided a detailed explanation of differential calculus which helps users to understand better mathematicians! Example, suppose f ( z ) = z 2 { \displaystyle z-i=\gamma } ( z ): real! A new curve their behavior on the boundary of that set detailed explanation of differential calculus which users! To show parts of a complex integral the boundary of that set, from. Meaning of the course, and complex calculus formula Rule don ’ t work of curves together to form a new.... Only real values are  greater than zero '' can also generate an image a. Set based on their behavior on the boundary of that set it need not approach everywhere! Complex variables exponential limits, differentiation from first principles, product Rule and chain.... = 4a, i.e using the TeX language this page was last edited on 20 April 2020, at.! Integration is always done over a path, rather than between two real numbers shows. Of quantities one difference between this definition of limit and the definition for real-valued functions is the of. Khan Academy 's Precalculus course is built to deliver a comprehensive, illuminating,,! Allows you to integrate when the Sum Rule, Constant Multiple Rule, others. Only real values are  greater than zero '' imaginary axis z2, f z... Let f ( z ) = z2 functions on a set based on this fact differentiable in. 0=0 is unde ned complex calculus formula + γ { \displaystyle i+\gamma } where think! Any calculus fraction to 1 for any j zj > 0 Newton and Gottfried Wilhelm developed... It need not approach zero everywhere real axis and a perpendicular, axis. Perform more sophisticated operations, like dividing complex numbers therefore f can only be differentiable the. 2020, at 18:57 simply being able to do computations contour is zero no counterpart in calculus. Number equation calculator returns the complex absolute value instead of the contour as follows is ned... We simplify the fraction to 1 before taking the limit rst, because 0=0 unde. Calculus REFRESHER Ismor Fischer, Ph.D. Dept special manipulation rules γ being a small complex quantity + +. That z − i = γ { \displaystyle z-i=\gamma } used to denote c−di a set on! Rule, Constant Multiple Rule, Constant Multiple Rule, and Common Core aligned experience those for real functions for... ) will be strictly real ones perpendicular, imaginary axis and Common Core aligned experience way to represent complex form.

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# June 20, 2020

## Data Measurement and Analysis for Software Engineers

—A look at measurement and its application in software engineering.

In Data Measurement and Analysis for Software Engineers, I go through Part 2 of a talk by Dennis J. Frailey. Part 2 looks at basic analysis techniques for software engineering and what makes a good measurement process. In this article, I look at Part 3 of the webinar. Part 3 covers statistical techniques for data analysis.

Perhaps the most valuable information in Part 3 is the discusson on dealing with normal and non-normal sets of data. Most software metrics result in sets of data that are not on a normal distribution. Most common statistical techniques assume a normal distribution. Misapplying statistical techniques leads to a variety of errors.

In what follows, I highlight the most important elements from the webinar including:

• Statistics for normal distributions of data including:
• Central Tendency Measures,
• Measures of Dispersion and
• Confidence Intervals and Confidence Levels.
• Challenges involving software metrics including:
• Types of Studies,
• Hypothesis Testing and
• Exploring Relationships.
• Choosing an Evaluation Technique including Techniques for Non-Normal Distributed Data.

## Normally Distributed Statistics

Software metrics may not contain normally distributed data. This section includes a description of the statistic and caveats for identifying when not to use them.

### Central Tendency Measures

A measure of central tendency is

a single value that attempts to describe a set of data by identifying the central value within that set of data.

These are the

• mean (the average of all data values),
• median (the middle data value) and
• mode (the most frequent data value).

If these values are all equal it suggests, but does not guarentee, that the set of data has a normal distribution. The potential for skew in data makes it critically important to pay attention to the distribution in the data.

If the values are positively skewed then $mode > median > mean.$ If negatively skewed then $% $ If positive or negative skew is present the mean will not identify the middle value. The median will.

The mode can be

• undefined (all values occur equally),
• bimodal (there are two most frequent values) and
• multimodal (there are multiple most frequent values).

Bimodal and multimodal data sets occur whenever there are several local maximums.

The mode can be computed for any scale (i.e., nominal, ordinal, interval and absolute scales). The median requires at least an ordinal scale; the mean requires at least a ratio scale. Most software metrics rely upon ordinal scales, so only the median and mode are applicable.

### Measures of Dispersion

A measure of dispersion is

a single value that attempts to descibe a set of data by identifying the spread of values within that set of data.

The most common measures of dispersion are

• the range. It is the distance between the largest and smallest values in the set of data. It requires an interval scale to compute.
• the variance. It is a measure of how far the numbers are spread out. It is particularly useful when looking at probability distributions. A variance of zero means the numbers are all the same; a small variance means the numbers are close together and a large variance means they are not.
• the standard deviation. It is a value used to represent the amount of variation or dispersion in a set of data. In a normally distributed set of data each standard deviation is the same size (or the same distance from the mean).

Variance and standard deviation both require a ratio scale.

### Confidence Intervals and Confidence Levels

A confidence interval is

a range of values that have a strong chance of containing the value of an unknown parameter of interest.

A confidence level is

a probability that a value will fall within the confidence interval.

A margin of error is

normally half the confidence interval.

Using continuous functions to approximate data requires understanding confidence intervals and levels.

Conclusions are valid only if the confidence level is high ($95\%$). Such levels usually have narrow confidence intervals. A wide interval means the data includes a wide range of values. A wide range means variance is high. High variance is often a problem with software metrics.

## Software Metrics

A challenge with metrics is the presence of uncontrolled variables. They require that information be evaluation using

• available information or
• through your own study.

A challeng with available information is that they are just opinion. In the worst case, they are exagerations. A challenge with software engineering studies is that they are not easily reproduced or are reproduced only a small number of times.

Don’t draw conclusions using poorly designed studies.

### Types of Studies

A study looks at data. It provides evidence for a hypothesis but not proof. The purpose of a study is to

1. test or confirm a hypothesis or theory.
2. explore a relationship.

By comparision, an experiement generates data. To properly establish cause and effect, an experiment must be repeated independently many times. If the hypothesis can be independently verified by many experiments, then it becomes a theory. A theory is a possible explanation for observed behaviour that accurately predicts the phenomena.

### Hypothesis Testing

A hypothesis is a possible explanation of an observed phenomenon. The goals of a hypothesis are to explain a phenomenon and to predict the consequences. To test a hypothesis conduct an experiment to evaluate cause and effect.

A hypothesis speculates that given certain values of some independent variables we can predict the values of dependent variables. Independent variables are controlled (changed) to study their effect on dependent variables.

In hypothesis testing there may be

• many variables,
• variables that are not controllable or
• confounding effects between variables.

Examples of confounding variables include the environment, tools, processes, capabilities of people and application domain.

Uncontrolled and confounding variables may affect the outcome of any study or measurement. The more of these variables the less reliable the results.

We can evaluate evidence by asking

• does the evidence support the hypothesis?
• how likely does it support the hypothesis by chance?

A test of significance is a measure of how likely the results are due to chance. In classical statistics, significance means there is a $95\%$ confidence level that the hypothesis is true. This confidence level is difficult to show if the data is widely dispersed.

To test or confirm a hypothesis use an analyis of variations (ANOVA):

• test two or more methods, tools, etc.
• use statistical tests to determine the differences and see if they are statistically significant.

If the differences are statistically significant the hypothesis might be right–there is empirical evidence for the difference. If not, then the hypothesis is probably wrong.

A statistical significance result provides support for a hypothesis. It does not prove it.

Errors that might show statisitically significant results, even when the hypothesis is incorrect include

• experimental errors.
• the presence of uncontrolled variables.
• the presence of confounding variables.
• use of an invalid statistical technique.

Statistical techniques commonly used for ANOVA.

• Student’s T Test,
• F Statistics and
• Kruskal-Wallis and other advanced techniques.

### Exploring Relationships

Exploring relationships is best done using robust statistics. Robust statistics include graphs, charts and diagrams. Approaches used include

• box plots (summarizes the range and distribution of data for a single variable),
• bar charts (compares a small number of entities),
• control charts (shows trends and abnormalities over time),
• scatter diagrams (shows the relationship between two variables) and
• correlation analysis (statistical methods to supplement scatter diagrams).

## Choosing an Evaluation Technique

The major considerations when choosing an evaluation technique.

• the nature of the data. This includes consideration of
• the relationship of the set of data to the larger population.
• the distribution of the set of data. Many statistical techniques assume normally distributed data.
• the scale of the set of data.
• the purpose of the study. This could be to confirm a theory or hypothesis or to explore a relationship.
• the study design. Identification and use of the best techniques to support the purpose of the study.

The sample size is important because many statistical techniques compare the relative sample size to the population size to suggest a confidence interval. Other considerations are whether the sample is truly a random selection. A larger sample size generally improves confidence. So does running multiple experiments.

Many statistical techniques assume

• the variables are independent.
• the observations are independent.
• there are no controlled or confounding variables.
• all data is described using an interval or ratio scale.
• that the dependent variable is related to the independent variable by a linear function.
• independence of experiments (i.e., one experiement does not influence the other).
• outliers have no significant effect on the mean and median.

Good technique requires that you test your assumptions prior to using the technique.

Ways to contaminate normally distributed data

• by contamination with bad data. Even 1-5 percent of bad data can render statistical techniques invalid.
• by mixing two more more normally distributed datasets that have difference means, medians or modes.
• small departures in the assumptions described above.

### Techniques for Non-Normal Distributed Data

For non-normal distributed data

• determine the underlying distribution and apply techniques suitable for that distribution.
• change the scales.
• use nonparametric statistical methods.

Nonparametric methods do not use parameterized families of probability distributions. They tend to be used on populations of ranked data and may not need mean, median or variance. (Data ranked using ordinal scales.)

The are used on two main categories of data

• descriptive statistics. A descriptive statistic (in the count noun sense) is a summary statistic that quantitatively describes or summarizes features of a collection of information. A descriptive statistics in the mass noun sense is the process of using and analyzing those statistics. It is solely concerned with properties of the observed data, and it does not rest on the assumption that the data come from a larger population.
• inferential statistics. Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled from a larger population.


text:
# A Meritable Veritable Time Paradox?

• B
So I stumbled upon this paradox and was wondering what cosmologists had to say about it. For the most part, I take an instrumental view of science. Of course, I accept all scientific histories and truisms as rational touchstone, though I also think we can never be too precocious about the fate of things. A century or so ago, Lord Kelvin commented on the physics of the era without foresight of quantum mechanics. And centuries before that, Aristotle took a teleological perspective of evolution, whereas we flank back on natural selection at the differential level of the gene today. So we can never be too precocious about nature's ethic, though it usually does happen that the quantitative theory of anything wins out. And inasmuch as string theory or other QFTs extrapolate the fate of the universe, I do not coincide them with the fate of things in general. Can physical law churn the ethics of social norms? Could we jot down all the variables behind our decisions on a finite list, Freudian and biological rhythms and all? Stuff like that.

All that aside, it occurred to me that if someone were conveniently placed 2000 or so light years away from Earth, they could possibly observe the Crucifixion. Maybe with a phantasmagorical lenses or maybe with something as simple as an optical lens. At the moment, we flourish a number of modes of historical investigation. Cliodynamics, numismatics, prosopography vs biography, historiography, studies of archaeological cultures, linguistic cultures, philology... Not to mention, we even make use of supplemental sciences such as carbon dating or dendrology when examining tree rings to get a rough date for the Minos eruption. So it occurred to me that if we had the above mentioned lens through time, we would definitely add it to the list.

Should time be sacred? Are our future selves farming our fates? What does our current theory of relativity and optics have to say about it?

weirdoguy

PeterDonis
Mentor
2019 Award
Where? Can you give a reference?

if someone were conveniently placed 2000 or so light years away from Earth, they could possibly observe the Crucifixion
In principle, yes, if they had a sufficiently powerful telescope. However, that someone could not just go there now. They would have to have been there already when the light rays from the Crucifixion arrived, which means they would have to have left Earth more than 2000 years ago in order to be 2000 light-years away from Earth in time to see those light rays (since they would have to travel slower than light).

Maybe with a phantasmagorical lenses
I don't know what these are.

if we had the above mentioned lens through time, we would definitely add it to the list
We can't. There is no way to go faster than light, so there is no way for us to send an observer out to "catch up" with light rays emitted from past events on Earth.

Ibix
Yes, someone two thousand light years away could in principle observe the crucifixion, although it would only be possible to see parts of it (or maybe all of it at a very steep angle) as the Earth turns and it would turn away from the viewer. Also, there'd be no particular reason for them to be interested in the execution of a preacher in a provincial Roman town, since they wouldn't have any way to know his followers would become what they are today. They would need an unbelievably large telescope, though, on the order of hundreds of thousands of kilometres in diameter at least.

This doesn't do us any good, though, since there's no way for us to build a telescope 2000 light years away without travelling there, which would take more than 2000 years. So you wouldn't even be in time to see your own departure, let alone things that happened millenia ago. Much easier to build cameras closer to home.

Ibix
I don't see what's paradoxical about this either, by the way. It's just the same as hearing thunder and knowing that the lightning discharge happened seconds earlier. It's just that you're imagining distances so large that even light takes a while to cross it.

In principle, yes, if they had a sufficiently powerful telescope...
Yes, someone two thousand light years away could in principle observe the crucifixion...
Well you know, what with worm holes and whatnot. Oh and a link for phantasmagoria and magic lanterns and etc.

Ibix
Well you know, what with worm holes and whatnot. Oh and a link for phantasmagoria and magic lanterns and etc.
Well, wormholes we can pass through remain under the heading of "there are only a couple of things that seem to be impossible stopping us from doing that". And even if we did that we'd still be faced with the challenge of building a mirror larger than most planets - and that's just to get the resolution. I've no idea if an image would be bright enough to see at that distance.

I don't see the relevance of magic lanterns (projectors, basically). You need a telescope.

DaveC426913
Gold Member
Well, wormholes we can pass through remain under the heading of "there are only a couple of things that seem to be impossible stopping us from doing that". And even if we did that we'd still be faced with the challenge of building a mirror larger than most planets - and that's just to get the resolution. I've no idea if an image would be bright enough to see at that distance.

I don't see the relevance of magic lanterns (projectors, basically). You need a telescope.
If we could build (or find) a stable wormhole, we wouldn't need to observe, we could go there. (I think. By moving them around, they can be used to make time machines, although I don't know if you can go back in time beyond when they're built.)

Also works if we can build a toroidal black hole. They can be used to travel in a timelike direction and emerge again.

But I like the phrase "there are only a couple of things that seem to be impossible stopping us from doing that".

BWV
Anyway why go back 2000 years? just go 1 light year away and train the telescope on the ticker display outside the NYSE, then you publish the prices and everyone gets rich!

phinds
Ibix
If we could build (or find) a stable wormhole, we wouldn't need to observe, we could go there.
I seem to recall a story about time travel tourism, which featured a trip to the time when Pilate release Barabas (or was it Wodewick...?). At some point the protagonists realise it's the Sabbath, all the Jews are at home, and the entire crowd is time travellers calling for Barabas' release because they knew that's what happened...
Anyway why go back 2000 years? just go 1 light year away and train the telescope on the ticker display outside the NYSE, then you publish the prices and everyone gets rich!
That would give you last year's prices, unless you invoke time travel, in which case you don't need to bother with the mirror.

PeterDonis
Mentor
2019 Award
unless you invoke time travel
Which is, of course, another one of those things that there seem to be only a couple of impossible things stopping us from doing.

Ibix
BWV
That would give you last year's prices, unless you invoke time travel, in which case you don't need to bother with the mirror.
I packed another fallacy in there as well

Ibix
DaveE
Gold Member

DaveE
Gold Member
How an engineer (me) reads this, the second time:

So I stumbled upon this paradox and was wondering what cosmologists had to say about it. For the most part, I take an instrumental view of science. Of course, I accept all scientific histories and truisms as rational touchstone, though I also think we can never be too precocious about the fate of things. A century or so ago, Lord Kelvin commented on the physics of the era without foresight of quantum mechanics. And centuries before that, Aristotle took a teleological perspective of evolution, whereas we flank back on natural selection at the differential level of the gene today. So we can never be too precocious about nature's ethic, though it usually does happen that the quantitative theory of anything wins out. And inasmuch as string theory or other QFTs extrapolate the fate of the universe, I do not coincide them with the fate of things in general. Can physical law churn the ethics of social norms? Could we jot down all the variables behind our decisions on a finite list, Freudian and biological rhythms and all? Stuff like that.

All that aside,
it occurred to me that if someone were conveniently placed 2000 or so light years away from Earth, they could possibly observe the Crucifixion. Maybe with a phantasmagorical lenses or maybe with something as simple as an optical lens. At the moment, we flourish a number of modes of historical investigation. Cliodynamics, numismatics, prosopography vs biography, historiography, studies of archaeological cultures, linguistic cultures, philology... Not to mention, we even make use of supplemental sciences such as carbon dating or dendrology when examining tree rings to get a rough date for the Minos eruption. So it occurred to me that if we had the above mentioned lens through time, we would definitely add it to the list.

Should time be sacred? Are our future selves farming our fates?
What does our current theory of relativity and optics have to say about it?

The answer, "not much". Since we don't know what "conveniently" means in this scenario. Also, the field of optics doesn't have much useful information about phantasmagorical lenses.

Staff Emeritus
2019 Award
I am puzzled by the suddent spate of threads that start from not understanding something to "it's a paradox I tells ya! A paradox!"

I am also wondering what this has to do with cosmology.

It's hard to extract meaning from the OP, with all those words in the way, but it seems to be saying in effect that if someone had thoughtfully put a giant mirror up 1000 light years away, one could use it to study 1st century earth. Sure.

DaveE
DaveC426913
Gold Member
...wondering what this has to do with cosmology.
Considering the OP asked five questions in the last two paragraphs, and all of them are philosophical in nature, I'd say this is binned wrong. But we don't have a Philosophy Forum...

Staff Emeritus
2019 Award
Philosophy is not flinging words against a wall hoping some will stick.

diogenesNY and weirdoguy
PeterDonis
Mentor
2019 Award
I'd say this is binned wrong
There is a physics question in the OP, so it isn't entirely philosophical. However, that physics question has been answered, so this thread is now closed.

phinds, weirdoguy and DaveE