Category: Big Data & Analytics

Statistics – Individual Series Arithmetic Median ”; Previous Next When data is given on individual basis. Following is an example of individual series − Items 5 10 20 30 40 50 60 70 In case of a group having even number of distribution, Arithmetic Median is found out by taking out the Arithmetic Mean of two middle values after arranging the numbers in ascending order. Formula Median = Value of ($frac{N+1}{2})^{th} item$. Where − ${N}$ = Number of observations Example Problem Statement − Let”s calculate Arithmetic Median for the following individual data − Items 14 36 45 70 105 145 Solution − Based on the above mentioned formula, Arithmetic Median M will be − $M = Value of (frac{N+1}{2})^{th} item. \[7pt] , = Value of (frac{6+1}{2})^{th} item. \[7pt] , = Value of 3.5^{th} item. \[7pt] , = Value of (frac{3^{rd} item + 4^{th} item}{2})\[7pt] , = (frac{45 + 70}{2}) , = {57.5}$ The Arithmetic Median of the given numbers is 57.5. In case of a group having odd number of distribution, Arithmetic Median is the middle number after arranging the numbers in ascending order. Example Let”s calculate Arithmetic Median for the following individual data − Items 14 36 45 70 105 Given numbers are 5, an odd number thus middle number is the Arithmetic Median. ∴ The Arithmetic Median of the given numbers is 45. Calculator Print Page Previous Next Advertisements ”;

Statistics – Inverse Gamma Distribution ”; Previous Next Inverse Gamma Distribution is a reciprocal of gamma probability density function with positive shape parameters $ {alpha, beta } $ and location parameter $ { mu } $. $ {alpha } $ controls the height. Higher the $ {alpha } $, taller is the probability density function (PDF). $ {beta } $ controls the speed. It is defined by following formula. Formula ${ f(x) = frac{x^{-(alpha+1)}e^{frac{-1}{beta x}}}{ Gamma(alpha) beta^alpha} \[7pt] , where x gt 0 }$ Where − ${alpha}$ = positive shape parameter. ${beta}$ = positive shape parameter. ${x}$ = random variable. Following diagram shows the probability density function with different parameter combinations. Print Page Previous Next Advertisements ”;

Statistics – Relative Standard Deviation ”; Previous Next In probability theory and statistics, the coefficient of variation (CV), also known as relative standard deviation (RSD), is a standardized measure of dispersion of a probability distribution or frequency distribution. Relative Standard Deviation, RSD is defined and given by the following probability function: Formula ${100 times frac{s}{bar x}}$ Where − ${s}$ = the sample standard deviation ${bar x}$ = sample mean Example Problem Statement: Find the RSD for the following set of numbers: 49, 51.3, 52.7, 55.8 and the standard deviation are 2.8437065. Solution: Step 1 – Standard deviation of sample: 2.8437065 (or 2.84 rounded to 2 decimal places). Step 2 – Multiply Step 1 by 100. Set this number aside for a moment. ${2.84 times 100 = 284}$ Step 3 – Find the sample mean, ${bar x}$. The sample mean is: ${frac{(49 + 51.3 + 52.7 + 55.8)}{4} = frac{208.8}{4} = 52.2.}$ Step 4Divide Step 2 by the absolute value of Step 3. ${frac{284}{|52.2|} = 5.44.}$ The RSD is: ${52.2 pm 5.4}$% Note that the RSD is expressed as a percentage. Print Page Previous Next Advertisements ”;

Statistics – Scatterplots ”; Previous Next A scatterplot is a graphical way to display the relationship between two quantitative sample variables. It consists of an X axis, a Y axis and a series of dots where each dot represents one observation from a data set. The position of the dot refers to its X and Y values. Patterns of Data in Scatterplots Scatterplots are used to analyze patterns which generally varies on the basis of linearity, slope, and strength. Linearity – data pattern is either linear/straight or nonlinear/curved. Slope – direction of change in variable Y with respect to increase in value of variable X. If Y increases with increase in X, slope is positive otherwise slope is negative. Strength – Degree of spreadness of scatter in the plot. If dots are widely dispersed, the relationship is consider weak. If dot are densed around a line then the relationship is said to be strong. Print Page Previous Next Advertisements ”;

Statistics – Process Sigma ”; Previous Next Process sigma can be defined using following four steps: Measure opportunities, Measure defects, Calculate yield, Look-up process sigma. Formulae Used ${DPMO = frac{Total defect}{Total Opportunities} times 1000000}$ ${Defect (%) = frac{Total defect}{Total Opportunities} times 100}$ ${Yield (%) = 100 – Defect (%) }$ ${Process Sigma = 0.8406+sqrt{29.37}-2.221 times (log (DPMO)) }$ Where − ${Opportunities}$ = Lowest defect noticeable by customer. ${DPMO}$ = Defects per Million Opportunities. Example Problem Statement: In equipment organization hard plate produced is 10000 and the defects is 5. Discover the process sigma. Solution: Given: Opportunities = 10000 and Defects = 5. Substitute the given qualities in the recipe, Step 1: Compute DPMO $ {DPMO = frac{Total defect}{Total Opportunities} times 1000000 \[7pt] , = (10000/5) times 1000000 , \[7pt] , = 500}$ Step 2: Compute Defect(%) $ {Defect (%) = frac{Total defect}{Total Opportunities} times 100 \[7pt] , = frac{10000}{5} times 100 , \[7pt] , = 0.05}$ Step 3: Compute Yield(%) $ {Yield (%) = 100 – Defect (%) \[7pt] , = 100 – 0.05 , \[7pt] , = 99.95}$ Step 3: Compute Process Sigma $ {Process Sigma = 0.8406+sqrt{29.37}-2.221 times (log (DPMO)) \[7pt] , = 0.8406 + sqrt {29.37} – 2.221 times (log (DPMO)) , \[7pt] , = 0.8406+sqrt(29.37) – 2.221*(log (500)) , \[7pt] , = 4.79 }$ Print Page Previous Next Advertisements ”;

Statistics – Harmonic Resonance Frequency ”; Previous Next Harmonic Resonance Frequency represents a signal or wave whose frequency is an integral multiple of the frequency of a reference signal or wave. Formula ${ f = frac{1}{2 pi sqrt{LC}} } $ Where − ${f}$ = Harmonic resonance frequency. ${L}$ = inductance of the load. ${C}$ = capacitanc of the load. Example Calculate the harmonic resonance frequency of a power system with the capcitance 5F, Inductance 6H and frequency 200Hz. Solution: Here capacitance, C is 5F. Inductance, L is 6H. Frequency, f is 200Hz. Using harmonic resonance frequency formula, let”s compute the resonance frequency as: ${ f = frac{1}{2 pi sqrt{LC}} \[7pt] implies f = frac{1}{2 pi sqrt{6 times 5}} \[7pt] , = frac{1}{2 times 3.14 times sqrt{30}} \[7pt] , = frac{1}{ 6.28 times 5.4772 } \[7pt] , = frac{1}{ 34.3968 } \[7pt] , = 0.0291 }$ Thus harmonic resonance frequency is $ { 0.0291 }$. Print Page Previous Next Advertisements ”;

Statistics – Grand Mean ”; Previous Next When sample sizes are equal, in other words, there could be five values in each sample, or n values in each sample. The grand mean is the same as the mean of sample means. Formula ${X_{GM} = frac{sum x}{N}}$ Where − ${N}$ = Total number of sets. ${sum x}$ = sum of the mean of all sets. Example Problem Statement: Determine the mean of each group or set”s samples. Use the following data as a sample to determine the mean and grand mean. Jackson 1 6 7 10 4 Thomas 5 2 8 14 6 Garrard 8 2 9 12 7 Solution: Step 1: Compute all means $ {M_1 = frac{1+6+7+10+4}{5} = frac{28}{5} = 5.6 \[7pt] , M_2 = frac{5+2+8+14+6}{5} = frac{35}{5} = 7 \[7pt] , M_3 = frac{8+2+9+12+7}{5} = frac{38}{5} = 7.6 }$ Step 2: Divide the total by the number of groups to determine the grand mean. In the sample, there are three groups. $ {X_{GM} = frac{5.6+7+7.6}{3} = frac{20.2}{3} \[7pt] , = 6.73 }$ Print Page Previous Next Advertisements ”;

Statistics – Permutation with Replacement ”; Previous Next Each of several possible ways in which a set or number of things can be ordered or arranged is called permutation Combination with replacement in probability is selecting an object from an unordered list multiple times. Permutation with replacement is defined and given by the following probability function: Formula ${^nP_r = n^r }$ Where − ${n}$ = number of items which can be selected. ${r}$ = number of items which are selected. ${^nP_r}$ = Ordered list of items or permutions Example Problem Statement: Electronic device usually require a personal code to operate. This particular device uses 4-digits code. Calculate how many codes are possible. Solution: Each code is represented by r=4 permutation with replacement of set of 10 digits{0,1,2,3,4,5,6,7,8,9} ${^{10}P_4 = (10)^4 \[7pt] = 10000 }$ Print Page Previous Next Advertisements ”;

Statistics – Gamma Distribution ”; Previous Next The gamma distribution represents continuous probability distributions of two-parameter family. Gamma distributions are devised with generally three kind of parameter combinations. A shape parameter $ k $ and a scale parameter $ theta $. A shape parameter $ alpha = k $ and an inverse scale parameter $ beta = frac{1}{ theta} $, called as rate parameter. A shape parameter $ k $ and a mean parameter $ mu = frac{k}{beta} $. Each parameter is a positive real numbers. The gamma distribution is the maximum entropy probability distribution driven by following criteria. Formula ${E[X] = k theta = frac{alpha}{beta} gt 0 and is fixed. \[7pt] E[ln(X)] = psi (k) + ln( theta) = psi( alpha) – ln( beta) and is fixed. }$ Where − ${X}$ = Random variable. ${psi}$ = digamma function. Characterization using shape $ alpha $ and rate $ beta $ Probability density function Probability density function of Gamma distribution is given as: Formula ${ f(x; alpha, beta) = frac{beta^alpha x^{alpha – 1 } e^{-x beta}}{Gamma(alpha)} where x ge 0 and alpha, beta gt 0 }$ Where − ${alpha}$ = location parameter. ${beta}$ = scale parameter. ${x}$ = random variable. Cumulative distribution function Cumulative distribution function of Gamma distribution is given as: Formula ${ F(x; alpha, beta) = int_0^x f(u; alpha, beta) du = frac{gamma(alpha, beta x)}{Gamma(alpha)}}$ Where − ${alpha}$ = location parameter. ${beta}$ = scale parameter. ${x}$ = random variable. ${gamma(alpha, beta x)} $ = lower incomplete gamma function. Characterization using shape $ k $ and scale $ theta $ Probability density function Probability density function of Gamma distribution is given as: Formula ${ f(x; k, theta) = frac{x^{k – 1 } e^{-frac{x}{theta}}}{theta^k Gamma(k)} where x gt 0 and k, theta gt 0 }$ Where − ${k}$ = shape parameter. ${theta}$ = scale parameter. ${x}$ = random variable. ${Gamma(k)}$ = gamma function evaluated at k. Cumulative distribution function Cumulative distribution function of Gamma distribution is given as: Formula ${ F(x; k, theta) = int_0^x f(u; k, theta) du = frac{gamma(k, frac{x}{theta})}{Gamma(k)}}$ Where − ${k}$ = shape parameter. ${theta}$ = scale parameter. ${x}$ = random variable. ${gamma(k, frac{x}{theta})} $ = lower incomplete gamma function. Print Page Previous Next Advertisements ”;

Statistics – Discrete Series Arithmetic Median ”; Previous Next When data is given along with their frequencies. Following is an example of discrete series − Items 5 10 20 30 40 50 60 70 Frequency 2 5 1 3 12 0 5 7 In case of a group having even number of distribution, Arithmetic Median is found out by taking out the Arithmetic Mean of two middle values after arranging the numbers in ascending order. Formula Median = Value of ($frac{N+1}{2})^{th} item$. Where − ${N}$ = Number of observations Example Problem Statement − Let”s calculate Arithmetic Median for the following discrete data − Items, ${X}$ 14 36 45 70 105 145 Frequency, ${f}$ 2 5 2 3 12 4 Comulative Frequency, ${C_f}$ 2 7 9 12 24 28 Terms 1-2 3-7 8-9 10-12 13-24 25-28 Solution − Based on the above mentioned formula, Arithmetic Median M will be − $M = Value of (frac{N+1}{2})^{th} item. \[7pt] , = Value of (frac{28+1}{2})^{th} item. \[7pt] , = Value of 14.5^{th} item. \[7pt] , = Value of (frac{14^{th} item + 15^{th} item}{2})\[7pt] , = (frac{105 + 105}{2}) , = {105}$ The Arithmetic Median of the given numbers is 2. In case of a group having even number of distribution, Arithmetic Median is the middle number after arranging the numbers in ascending order. Example Let”s calculate Arithmetic Median for the following discrete data − Items, ${X}$ 14 36 45 70 105 Frequency, ${f}$ 2 5 1 4 13 Comulative Frequency, ${C_f}$ 2 7 8 12 25 Terms 1-2 3-7 8-8 9-12 13-25 Given numbers are 25, an odd number thus middle number, 12th term is the Arithmetic Median. ∴ The Arithmetic Median of the given numbers is 70. Calculator Print Page Previous Next Advertisements ”;