Johanan Collins

A covariate is a confusing term in statistics. (References 1 & 2) For example, SPSS describes a continuous independent variable as a Covariate in its General Linear Models and as an Independent Variable in Regression. Minitab describes a continuous independent variable as a Covariate in its General Linear Model.
 
In research, one is looking for relationships between explanatory variables and the response variable.
 
Covariates are the variables that the researcher is not interested in but it affects the response variable.
 
Example 1
Which training methodology out of the three methodologies available is most effective in improving exam scores for the ICSE Exam?
 
The Explanatory Variable is the training methodologies and the exam score is the response variable.
 
Covariates, in this case, could be the current grade of the students, education level of parents, the income of parents, or studying abilities of the students within the three groups of the training methodologies. The variability on account of these covariates needs to be taken into consideration in order to see the correct relationship between the training methodology and the exam score.
 
For example, in this case, we could use the current grade of the students as a covariate since the current grade would most likely be highly correlated with the exam score.
 
Example 2.
What is the relationship between the Square Foot Area of a house and its price?
 
The Explanatory Variable is the Square Foot Area of the house and the price is the response variable.
 
Covariates, in this case, could be the age of the house, the location of the house, or even the distance from the bus stop. The variability on account of these covariates needs to be taken into consideration in order to see the correct relationship between the Square Foot Area of the house and the price.
 
For example, in this case, we could use the age of the house as a covariate since older houses are likely to be cheaper.
 
The most common settings for Covariates are ANOVA and regression.
 
ANOVA. Analysis of Variance is used to find if there is any statistical difference between three or more independent groups. However, in Example 1 above, we use the current grade as a covariate and would perform an ANCOVA (Analysis of Covariance). In this case, the students’ current grade, a continuous variable, is included as a covariate
 
Regression. In regression, we are attempting to quantify a relationship between one or more independent explanatory variables and the response variable. We run a Multiple Linear Regression by including Square Footage (Variable of Interest) and House Age (the covariate) as an explanatory variable and House Price as the response variable. The regression coefficient of the Square Footage would give the change in House Price per unit change in Square Footage after taking into account the House Age.
 
Thus, by adding House Age in the regression model in Example 2, the noise (variation on account of the covariate) is reduced. References
1.     https://www.theanalysisfactor.com/confusing-statistical-terms-5-covariate/
2.     https://www.theanalysisfactor.com/series-on-confusing-statistical-terms/
3.     https://www.statology.org/covariate/

Definitions of Reliability
 
Manufacturing & Service
-       Change of Quality over time.
-       The ability of a product, service, or system to maintain the original level of quality over time and under different conditions.
-       The probability that is inherent in a system, service, or product that ensures it effectively carries out it’s designed function over a specified time, in a defined environment without failure.
 
Research
-       Reliability is the consistency of a research study or measuring test. https://www.simplypsychology.org/reliability.html).
-       Reliability is the consistent replication of findings from research.  A high correlation coefficient indicates high reliability.
 
Psychometrics/Statistics
-       The overall consistency of a measure. A highly reliable measure repeatedly produces similar results under similar conditions.
-       "It is the characteristic of a set of test scores that relates to the amount of random error from the measurement process that might be embedded in the scores” https://en.wikipedia.org/wiki/Reliability_(statistics)
 
Components of Reliability
 
Manufacturing & Service
These are the likelihood of success, the designed function, performance within given specification limits, a time period, and the environment/conditions such as the temperature, speed, or pressure. ASQ defines these elements as the probability of success, durability, dependability, quality over time, and availability to perform the function.  
Psychometry/Statistics.
The components of reliability are precision, reproducibility, and consistency. Measurements are influenced by factors that contribute to consistency and factors that contribute to inconsistency. The factors that contribute to consistency are the stable characteristic of the person or the attribute that is being measured whereas the factors that contribute to inconsistency are attributes of the person or circumstance that might alter the measurement. Examples of these are things such as health, tiredness, motivation, emotions, comprehension of the task, attention, distractions, instructions, and even luck.
Four Main Types of Reliability
Test-retest. This is conducting the same test over a period of time. Interrater Reliability. This is conducting the same test but by different individuals Parallel Forms Reliability. This is conducting alternate versions of tests that are designed to measure the same attribute. Internal Consistency Reliability. This is the correlation between multiple items of the test and the construct it is supposed to measure.
Examples of Reliability
Electric Scooter under a warranty of 24,000 Kms or 2 years whichever is earlier Television. Life time guarantee. Ecommerce website uptime is 99.997% Once data entry error per one million keystrokes References
https://asq.org/quality-resources/reliability
https://www.simplypsychology.org/reliability.html
https://en.wikipedia.org/wiki/Reliability_(statistics)
https://www.scribbr.com/methodology/types-of-reliability/

Burn-in Testing is the testing of the components of a system before it is put into service or before its assembly into the system. It is done to establish the load capacity of a component by forcing it into failure under supervised test conditions.
 
The plot of the Failure rate against time takes a shape of a Bathtub curve. This shows that the failure rate is high early in the product life cycle. The burn-in testing is done in a stressful and accelerated environment in order to determine the failure rate during this early stage of the product life cycle when the failure rate is high. The failed components would be improved upon and replaced thus preventing premature failure of the system. Since the Burn-in testing generally destroys the components, a trade-off needs to be made to ensure a high yield.
 
Cost Implications. Burn-in testing is not done for all components. It will generally depend upon the cost of the test, the cost of the component since it is damaged during the test, the cost of failure, the cost of warranty claims, and the cost of goodwill lost.
 
Advantages of Burn-in Testing
Higher reliability of the product. Proactive prevention such as planned preventive maintenance. Fewer customer returns leading to fewer service and call centers. Better estimated lifespan of the product Examples of Burn-in Testing Electronic Components testing is done at high voltage, current or temperature. Audio Equipment is tested by leaving it on for a large number of days. Stress testing of ropes, wires, and jigs is done to assess the safe working load.  
References
https://en.wikipedia.org/wiki/Burn-in
https://www.techopedia.com/definition/2161/burn-in-testing#:~:text=Burn-in%20testing%20is%20the%20process%20of%20increasing%20the,be%20replaced%20before%20the%20device%20reaches%20the%20consumer.

Overall Equipment Effectiveness / Overall Equipment Efficiency
OEE measures the utilization of a machine. It is a ratio of the parts produced vs the theoretical capacity.
Causes of OEE
·       Availability. Availability Losses can be due to planned maintenance, unplanned repairs, breakdowns, material shortage, changeovers, lack of transport for finished goods, absence of operator, etc.
·       Performance. Speed Losses are reduced rate of production due to wear and tear, poor maintenance, idling, lack of training leading to incorrect operation, wrong settings, etc.
·       Yield. Quality Losses are losses due to defects. These wastages can be in the form of scrap, rework, and start-up losses.
Example.
·       Theoretical OEE – 100 parts per hour
·       Availability Losses – 5 parts per hour
·       Speed Losses – 10 parts per hour
·       Quality Losses – 15 parts per hour
·       Good parts produced – 70
·       OEE – 70%
TAKT Time
Takt is a musical term. In German it means beat. It is the rhythm/rate at which production needs to be paced in order to meet customer demand. It is measured in Time/Unit. It helps in planning the resources such as employees, machines, raw material, transport, storage, etc with respect to the required production. Customers, transporters, employees, etc., can be given an accurate estimate of when to expect the finished goods. The Takt time changes as the business grow, hence necessary action such as increasing capacity will need to be taken to stay ahead of demand.
Takt Time = Total Available Production Time/Customer Demand
Example
Total Available Production Time/day = 480 Minutes

Cycle Time
Cycle Time is the time spent on a specific task for one unit. Time/Unit
Example 1. Unbalanced Line Cycle Time.
·       Cycle Time Task A = 30 sec/part,
·       Cycle Time Task B = 60 sec/par - Bottleneck,
·       Cycle Time Task C = 15 sec/part,
·       Cycle Time Task D 15 sec/part
·       Throughput without OEE = 60 parts/hour
·       Target Cycle Time = 60 Sec x 0.8 OEE = 48 Sec
·       Throughput with OEE (80%) = 3600/48 = 75 Parts Per Hour
Example 2. Balanced Line Cycle Time.
·       Cycle Time Task A = 30 sec/part,
·       Cycle Time Task B1 = 30 sec/part,
·       Cycle Time Task B2 = 30 sec/part,
·       Cycle Time Task CD = 30 sec/part
·       Throughput without OEE= 120 parts/hour
·       Target Cycle Time = 30 Sec x 0.8 OEE = 24 Sec
·       Throughput with OEE (80%) = 3600/24 = 150 Parts Per Hour
The target cycle time for a process is Takt Time multiplied by the OEE
If the Target Cycle Time for a process is optimized without taking into consideration the reduced production due to OEE losses, it is likely that the customer demand is not going to be met.
For the customer demand to be met taking into consideration OEE losses the
·       Average Output >= Average Customer Demand or
·       Target Cycle Time <= Takt Time x OEE
This is illustrated in the examples above
References
https://www.benchmarksixsigma.com/forum/topic/36096-overall-equipment-effectiveness-oee/#comment-50835
https://www.allaboutlean.com/oee-definition/
https://www.allaboutlean.com/line-balancing-3/

Simply put a number of related projects managed together form a program and a number of unrelated projects/programs managed together form a portfolio.
Project Management
A Project is a temporary endeavor that has a start date/time and finish date/time and consumes resources. It is broken down into various tasks. Hence, we can say that a project has a charter, the deliverables, timeframe, budget, commitment of resources right at the beginning of the Project.
Program Management
A program is generally a long-term endeavor that consists of a number of concurrently executed projects. Hence, we can say that a program will align with the organizational strategic goals. The benefits of a program are ongoing and hence it does not have a specific timeframe. It moves from one level of improvement to the next level. The budget for a program is generally earmarked in the annual budget of the organization. Programs are managed at the higher levels of management. A program would accomplish multiple organizational goals, have multiple small deliverables over time, have mid-course corrections, change in plans, have ambiguity in the output, and will be ongoing.
Program management helps the organization select projects based on its long-term strategic goals by constantly addressing the business risk. It helps maintain constant communication with all stakeholders thereby ensuring that the entire organization and its stakeholders are on the same page. By looking at the bigger picture, program management helps to effectively prioritize resources. Overall, it encourages an environment of continuous improvement towards a common strategic goal in the organization.
Portfolio Management
To begin with, a Portfolio has a bigger scope and objectives than program or project management. It deals with macro-management of the organization as against micromanagement in a Project. PMI defines portfolio management as it “ensures that an organization can leverage its project selection and execution success. It refers to the centralized management of one or more project portfolios to achieve strategic objectives.” This centralized management helps the organization achieve its overall long-term strategical objectives by setting project priorities based on the overall business objectives to make sure that the benefits to the business are optimized. It may however be noted that portfolio management does not oversee each individual project or program. PMI Study Circle defines a portfolio as a group of unrelated or related programs/projects. Hence it can consist of multiple programs or projects. As against project management, which consists of similar projects, a portfolio generally consists of dissimilar programs/projects. The KRA for a portfolio is the combined performance of its components which could be projects/programs.
Portfolio Management supports the programs and projects through the optimum allocation and utilization of resources based on the organization’s long-term strategic goals. It results in better economies of scale and fewer risks. It ensures constant communication and better coordination among the stakeholders of the programs and projects leading to lesser conflicts and greater overall success of the programs and projects.
Example
Portfolio. The goal of a political party is to come to/remain in power. In order to come to/remain in power, the party has determined that it needs to focus on primary education, health care and women empowerment. These three focus areas would form the portfolio of the political party.
Program – Primary Education. In the area of primary education, the party may decide on a program to reduce illiteracy in the country. It could break down this program into various projects such as building new schools, increasing the infrastructure in existing schools, training, and induction of new teachers.
Program Health Care. In the area of health care, the party may decide on a program to increase the outreach of health care at the village level. It could break down this program into various projects such as building new hospitals/clinics, increasing the infrastructure in existing hospitals/clinics, training, and induction of new doctors and nurses.
Program Women Empowerment. In the area of women empowerment care, the party may decide on a program to increase the percentage of women in the workforce. It could break down this program into various projects such as encouraging parents to educate daughters, encouraging parents to get daughters married after completing their education, creating a women friendly work environment.
References
https://www.clearpointstrategy.com/difference-between-program-and-project-management/
https://www.pmi.org/learning/featured-topics/portfolio
https://pmstudycircle.com/project-management-vs-program-management-vs-portfolio-management/#:~:text=%20Program%20management%20manages%20similar%20projects%20while%20portfolio,changes%20with%20the%20strategic%20objectives%20of%20the%20organization.

Single Factor Experiments - Unbalanced Data. This is when the number of observations taken within each treatment is different.
Advantages of using a Balanced Design / Disadvantages of Unbalanced Design
The test statistic is relatively insensitive to small departures from the assumption of equal variances for the treatments if the sample sizes are equal. (This is not the case with unequal size samples)
The Power of the Test is maximized if the samples are of equal size
In an Unbalanced Design ANOVA, a modification is made to the Sum of Squares formulas.

Factorial Design – Unbalanced Data
Reasons.
Designed as a balanced design initially, however, due to unforeseen problems in running the experiment, may result in loss of some observations
Designed as an unbalanced experiment intentionally.
This may be the case when certain treatment combinations may be more expensive or more difficult to run, hence fewer observations may be taken in these treatment combination cells.
This may be the case when some treatment combinations may be of greater interest to the experimenter as they may represent new or unexplored conditions, so the researcher may do more replication in these cells.
Unbalanced Design Examples
Proportional Data. Here the number of observations in any two rows or columns is proportional. In this case, normal ANOVA works with minor modifications for the sums of squares formula

Approximate Methods
When the unbalanced data is not far away from the balanced data, an approximation can be done to convert the unbalanced data to a balanced one. Some of the ways approximation are done is given below
Estimating the mission Observations. If only a few observations are different, a reasonable procedure for estimating the missing values can be done. For a model with interaction, the estimated value should reduce the Error Sum of Squares. This can be done by taking the average of the observations in Cell (2,2) having 3 observations (1 observation missing).

 
Setting Data Aside. In this case Cell (2,2) has one data point more than the other cells, we set aside one observation from Cell (2,2) in order to obtain a balanced design

Method for Unweighted Means. This method was introduced by Yates (1934) in which the cell averages are treated as data and subjected to standard balanced data analysis to obtain the Sum of Squares for rows, columns and interactions. This is an approximate procedure because the sums of squares of the rows, columns and interactions are not distributed as chi-square random variables.
Weighed Squares of Means Method. Also proposed by Yates (1934). In this method, the terms of the sums of squares are weighted in inverse proportions to their variance.
Exact Method
This is done when empty cells occur (nij = 0) or when nij are very different. Here we develop the sums of squares for testing the main effects and interactions by representing the ANOVA model as a regression model.
References
Design and Analysis of Experiments by Douglas C Montgomery, International Students Edition, Eight Edition

John Wilder Tukey an American statistician and mathematician developed the Rootogram. He is also known for the Fast Fourier Transform algorithm, the Tukey Lambda distribution, Tukey test of additivity, Tukey range test and the Teichmeller-Tukey lemma.
 
Oxford Reference Definition
Oxford Reference defines a rootogram as “a diagram suggested by Tukey in 1971, for comparing an observed bar chart or histogram (with equal-width categories) with a theoretical probability distribution. The comparison is made easier by ‘hanging’ the observed results from the theoretical curve so that the discrepancies are seen by comparison with the horizontal axis rather than a sloping curve. As in the rootogram, the vertical axis is scaled to the square root of the frequencies so as to draw attention to discrepancies in the tails of the distribution.”
 
R Package Documentation Definition
The R Package documentation describes the rootogram function to “graphically compare (square roots) of empirical frequencies with fitted frequencies from a probability model.” “Rootograms graphically compare frequencies of empirical distributions and fitted probability models. For the observed distribution, the histogram is drawn on a square root scale (hence the name) and superimposed with a line for the fitted frequencies. The histogram can be “standing” on the x-axis (as usual), or “hanging” from the fitted curve, or a “suspended” histogram of deviations can be drawn. “
 
Paper of Use of Rootogram for Count Data
A rootogram is a visual tool that was initially used by Tukey to assess the goodness of fit of univariate distributions. Christian Kleiber of the Universitat Basel and Achim Zeileis of the Universitat Innsbruck in their paper “Visualizing Count Data Regressions Using Rootograms” have used rootograms to look at issues such as overdispersion, excess zeros in regression models for count data. Count data regression plots are done in the form of bar plots of the expected and observed frequencies. Rootograms are used to see the fit of both continuous data and count data.
 
Rootograms compare the observed frequencies using bars (histograms) and the expected frequencies using a curve on a square root scale. Taking the square root scale transforms the date to adjust to the scale differences across the intervals. This makes the deviations across the interval for smaller observed/expected frequencies to be more visible in the plot.
For example, the deviations of 9 as comped to 3600 would only be 1:400, however, the square of the numbers 3 and 60 is 1:20. This is a visual magnification of 20 times. 
 
There are three types of rootograms. The standing rootograms show the bars and a curve. In this, the deviations are not aligned. The standing rootogram is the least used as it just plots the bars and the curve representing the model, however, the fit is not shown. The hanging rootograms align all the deviations along the horizontal axis. The bars are hanging from the curve representing the expected frequencies whereas the suspended rootogram shows mainly the deviations as against the observed frequencies. The hanging and suspended use the horizontal reference line which shows the deviations between the observed and expected frequencies.
Example of Rootogram for Poisson Distribution and Negative Binomial Distribution

Analysis of above Rootograms
Rootograms are used to detect patterns such as runs of positive or negative deviations. The top row of the figure above shows only small deviation when fitting a Poisson model to Poisson data. The expected frequencies and observed frequencies show minimum deviation. In the bottom row of the figure above shows large deviations when fitting a negative binomial distribution. The expected frequencies do not track the observed frequencies.
 
References
Kleiber, C, Zeileis, A. (2016). Visualizing Count Data Regressions Using Rootograms. American Statistician, Volume 70, Issue 3, Pages 296 to 303
 
Oxford Reference (https://www.oxfordreference.com/view/10.1093/oi/authority.20110803095919378
 
R Package Documentation
https://rdrr.io/rforge/topmodels/man/rootogram.html

Parametric tests to compare variances for more than two populations are given below
One Independent Variable – 1-Way ANOVA (Completely Randomized Design).
Subjects are assigned randomly to treatments. This design contains only one independent variable with one or more classifications or treatment levels. In this design, the Total Sum of Squares (the variation in the data - SST) is divided into the variance from the treatment (Between Group Variation i.e the columns - SSC) and the error variance (unexplained by the treatment, i.e. within-group variation.  SSE). ANOVA compares the relative size of these two variances using the F Statistic which is the ratio between the two variances.
SST = SSC + SSE
Assumptions for ANOVA. Data is randomly drawn from a normally distributed population and the variance of the populations is equal.
Use of Confidence Intervals.  In the event of the results being significant (p-value<Alpha) the researcher needs to do a post hoc or posteriori test to determine which group is statistically significant from the rest. In order to determine this the Confidence Intervals are plotted and significance is determined by examining the overlap between the confidence intervals.
In addition, Tukey’s Honestly Significant Difference (HSD) test for Equal Sample Sizes, or the Tukey-Kramer Procedure for Unequal Sample Sizes.
Example. The sample consist of the pipe diameter of 4 operators.

The p-value < 0.05 indicates that one or more one operator is statistically significantly different from the other operators.

Examining the 95% Confidence Interval of the Operators, it can be observed that the Confidence Interval of Operator 3 does not Overlap with other Operators.
This is also evident from the Fisher Individual 95% Cis and Tukey Simultaneous 95% CIs and the Box Plot. Thus, we can conclude that Operator 3 is significantly different from the other 3 Operators.

 
Further, the Fisher Individual 95% Cis and Tukey Simultaneous 95% CIs have also bet the Operator 3 is a separate group from the remaining 3 Operators.


One Independent Variable + 1 Block Variable – Randomized Block Design.
The Randomized Block Design is similar to the Completely Randomized Design; however, it has a second variable, referred to as the Blocking Variable that is used to control for confounded or concomitant variables. These variables are not controlled by the researcher but affect the outcome and hence need to be blocked. The SSE (Error Sum of Squares) is segregated into SSR (Sum of Squares Blocks) and SSE (New Error Sum of Squares)
SST = SSC + SSR + SSE
Repeated Measure Design is a randomized block design in which each block level is an individual item/person and that person/item is measured across all treatments. It helps researchers to determine if the means of three or more measures from the same person are similar or different. The repeated measure ANOVA controls the between-subjects variance by removing it from the error term and measuring it separately. 
ANOVA for Latin Square Design
The Latin Square Design utilizes the Blocking principle. It is used to remove two nuisance sources of variability which are in the rows and columns of the square. The rows, and columns are the two restrictions on randomization. The model is totally additive, in that there is no interaction between the treatment, rows and columns. The Analysis of Variance divides the total sum of squares into the sum of squares for the rows, columns, treatment, and error. The F test ratio of MS(Treatment)/MS(Errors) is used to determine if there is no difference in the treatment means. (MS – Mean Sum of Squares)
Randomized Incomplete Block Design.
When it is not possible to run all the treatment combinations in each block, the randomized incomplete block design is used. When all treatments comparisons are equally important, the Balanced Incomplete Block Design is used. This design ensures that each block is selected in a balanced manner so that any pair of treatments are selected the same number of times as any other pair.
 
Two Independent Variables – Two-Way ANOVA
In this design, two or more independent variables are explored at one time. These are also called factorial designs. In this design, each and every level is studied under the conditions of every level of all other treatments. The design can have 3,4, …, n independent variables being studied at one time. For example, the independent variables can be machines, operators, shifts, day of the week, suppliers, and raw materials. All of these independent variables can be done in one study. This will be a Six-Way ANOVA.
The Completely Randomized Design, the Randomized Block Design, and the n-Way ANOVA all have one dependent variable.
Chi-Square Goodness of Fit Test.
This test is used to analyze the probabilities of multinomial distribution trials along a single direction. For example, to study Education with 4 possible outcomes, viz. illiterate, Primary School, Secondary School, Bachelors's and above, the single dimension is Education and the possible outcomes are the levels of education. Also, it is imperative that on one trial only one outcome can occur. The Chi-Square Goodness of Fit test measures the difference between the observed frequencies and expected frequencies. The Chi-Square Distribution is used to measure the significance. It is a one-tailed test since Chi-Square of Zero means the perfect agreement between observed and expected values.
Chi-Square – Test of Independence.
The Goodness of Fit test cannot be used to measure the analyse two variables at one time. For this purpose, the Test of Independence is done. Test of independence can analyse the frequencies of 2 variables with multiple categories to determine if the two are independent. Use Case. To determine if the type of pizza topping is independent of the customer's age or the citizenship is independent of investment in bitcoin.
Non-Parametric tests to compare variances for more than two populations are given below
One Independent Variable – Kruskal Wallis Test.
This test is the nonparametric equivalent to the one-way ANOVA. It is used to check if 3 or more samples come from the same/similar or different populations. It can be used for Ordinal Data and is not based on the shape of the population distribution. It assumes that the groups are independent of each other and random selection of items in the groups.   

The Kruskal Wallis Test for the above data shows a p-value of < 0.05 indicating that one of the operators is statistically significant from the others. Examination of the results shows that Operator 3 is significantly different from other operators.

One Independent Variable – Mood’s Median Test.
It is used instead of the Kruskal Wallis test when there are outliers present in the data. The results from the Mood’s Median Test indicate a p-value of 0.003 with is significant with an Alpha of 0.05.
Confidence Interval Mood’s Median Test. On observing the 95% Midian Confidence Interval of the three operators, it can be seen that the CI of Operator 3 is distinct from the other operators.

One Independent Variable + 1 Block Variable – Friedman Test
This is the nonparametric equivalent to the randomized block design. If the normality of the data cannot be assumed or the data is ranked the Friedman test is used. It assumes independent blocks; no interaction is present between blocks and treatments, and the observations within each block can be ranked. In the Friedman Test below, the Day of the Week is Blocked. The p-value of 0.007 < Alpha of 0.05 indicates that one of the operators is statistically significantly different. On observation, it can be seen that Operator 3 is different from the rest of the Operators.

Multivariate Analysis of Variance (MANOVA). This test is used when there are more than one continuous Dependent Variables. A one-way MANOVA would examine the effect of one independent variable on two dependent variables. For example, to study the effect of education on income and expenditure. A two-way MANOVA would examine the effect of two independent variables on two dependent variables. For example, to study the effect of education and citizenship on income and expenditure.

Multivariate Analysis of Covariance (MANCOVA).
This test includes covariates. It involves finding out the statistical significance between multiple dependent variables (continuous) and an independent (grouping) variable and controlling with additional variables called covariates. Covariates reduce the error term hence the effect of the covariate is removed from the relationship of the independent and dependent variables.

 
References
https://medium.com/nerd-for-tech/everything-about-manova-and-mancova-4c1c237af464

Outliers are part of the real world and need to be investigated before analyzing and interpreting the data. This is, even more, the case with small sample sizes, as the outliers have a greater impact on the results.  Some models such as Principal Component Analysis, Hierarchical Models, K-Means, Linear, and Logistic Regression are very sensitive to outliers. Detection of unusual transactions may be the aim of the operations. This unusual transaction is generally in the form of outliers, such as fraud detection, stock forecasting, etc. Hence understanding outliers is critical because outliers are most likely to bias the entire interpretation or the outliers maybe what we are looking for.
 
Reason for Outliers
 
Error  
The error may be due to Data Entry, Recording, Measurement in Gage, Measurement due Operator, Measurement error due to calibration, Sampling Errors, Data Processing Errors.
 
Part of Normal Process  
Outliers may be present in the data due to Bulk orders, Resellers or Extra Loyal Customers, etc.

How to Detect?
 
Data Visualization  
Outliers can be detected through Data Visualization such as Box plots, Scatter Plots, Histograms, Run Charts, Lag Plots, Line Charts.
 
Statistical Methods  
Outliers can be detected through Statistical Methods such as the Standard Deviation Method, Tukey's Method. Etc. 

What is the strategy to deal with outliers?
 
Keep the outlier and carry out the test with the outliers. Segment the data and carry out a deeper analysis. Imputing outliers and treating them separately.  Set up a filter to do the test without the outliers. Since significant effects are hidden by outliers, it may be appropriate to set up a filter to examine the results without the outliers. Delete the outlier - The outliers may be deleted if there was an error in data or the reason for the outlier is not likely to happen again Delete the outlier after post-test analysis Change the value of the outlier. This may be done by replacing it with a more appropriate value such as the mean or the median. Consider the underlying distribution. An Anderson Darlings or Shapiro Wilk Test may be done to check the normality of the data.  Carry out a Non-Parametric Test in case the underlying distribution is not Normal. Transform the Data. Data can be transformed using the Box-Cox Transformation, Johnson Transformation, log transformations, scaling, cube root normalization, etc. 
Methods and Tests that can be done for data having Outliers
 
Winsorizing or Winsorization  
It is named after Charles P Winsor, who was an Engineer and Biostatistician. In this process the effect of the outliers is reduced by limiting the extreme values. It sets the value of all the outliers to a specific percentile of the sample. Data estimated through the Winsorization method is generally more robust to outliers. Example. A 95% Winsorization would set the bottom 2.5 percentile of the data to the 2.5 percentile value and the top 2.5 percentile of the data to the 97.5 percentile value. 
 
 
Trimming/ Truncation  
This is a method of censoring data. All data above/below a certain percentile is removed. Example. A 95% truncated data would eliminate the bottom 2.5% of the data and the top 2.5% of the data above the 97.5 percentile. TRIMMEAN function in Excel may be used from trimming the data.

Winsorized mean and truncated mean are not the same.
Non-Parametric Tests such as 1 Sample Sign Test, 1 Sample Wilcoxon Test, Mann Whitney, Kruskal Wallis, Moods Median, Friedman, Runs can be done in case of the underlying distributions being not normal.
 
Transformation - Transform Data and carry out Parametric tests.   
Univariate Methods  
Box Plot - The box plot is the easiest method for identifying outliers. It uses the median and the Q1 and Q3 to determine the outliers.

Tukey Method - This method identifies the extreme outliers as being greater than three 3 times the Inter Quartile Range below/above first/third quartile, Mild Outliers as between 1.5 to 3 times IQR.
 
Multivariate Methods 
At times the univariate method may not detect the outliers. Multivariate methods such as multiple linear regression may be used. 
 
Minkowski Error. 
This method can be used to minimize the impact of the outliers on the model. It is a loss index and more insensitive to outliers than the mean square error since in the mean square error the contribution of the outliers increases exponentially.

References
https://en.wikipedia.org/wiki/Winsorizing
https://www.sigmamagic.com/blogs/how-to-handle-outliers/
https://cxl.com/blog/outliers/
https://aichapters.com/how-do-you-handle-outliers-in-data/
https://aichapters.com/how-do-you-handle-outliers-in-data/
https://www.aquare.la/en/what-are-outliers-and-how-to-treat-them-in-data-analytics/

A two-proportion test is a hypothesis test to check if the differences between two population proportions are statistically significant. For example, are the proportion of girls to boys in a school significantly different. The null hypothesis is that there is no difference between the population proportions. It can be tested against an alternate hypothesis that can be two-tailed or left/right-tailed.
The output of a 2 proportions test has two p-values, one is the normal approximation based on the Z statistic and the other is the Fisher’s Exact Test. As their names indicate, the normal approximation test is an approximation and has greater error for smaller sample sizes and becomes more accurate as the sample size increases whereas the Fisher’s Exact test is always exact irrespective of the sample size however is more difficult to calculate as the sample size increases.
The Fisher’s Exact Test is calculated using the hypergeometric distribution. The factorials in the formula make it more and more difficult to calculate the p-value as the sample size increases since it runs every possible combination from the sample, and calculates the total number of successes and failures at that given sample size. It then calculates the p-value from the total successes and failures. 
Thus, for larger samples, it is not only easier to calculate the p-value using the Normal Approximation Test, but the results are closer to Fisher’s Exact test results.
Since we no longer do manual calculations and statistical software have the ability to quickly calculate the p-value from the Fisher’s Exact Test, it makes more sense to use the Fisher’s Exact Test irrespective of the sample size.
For a small number of expected values, when compared to the Chi-Square or G-Test of independence, the Fisher’s exact test is more accurate.
The Normal Approximation Test (Z-test), is not accurate when the number of events/non-events are < 5. This is based on the rule that N*P or N(1-P) should be >5 (where N is the No. of trials and P is the proportion of successes. In other words, the normal distribution can be used in place of the binomial distribution when the sample size is large. If N is small and P is small, the binomial distribution will be skewed and the normal distribution cannot be taken to represent it. This is evident from Table 1 where N is increased and Table 2 where P is increased. It can be seen from these 2 tables, that as N and P are increased the Normal Approximation approaches the Fisher’s Exact Test.
It is also evident from Table 2 that for small samples/P, the Normal Approximation test may indicate that a difference between the population exist when no difference exists.


References
https://stats.stackexchange.com/questions/234010/2-sample-proportions-z-test-vs-fishers-exact-test
https://blog.minitab.com/en/quality-data-analysis-and-statistics/two-p-values-for-a-2-proportions-test-am-i-seeing-double
http://www.biostathandbook.com/fishers.html

Economic and social activity facilitated by platforms is broadly defined as Platform Economy. It includes frameworks that facilitate sales, logistics, technology, social activities, etc. These transaction platforms are also called digital matchmakers. Major examples of the platform economy are eBay, Amazon, Facebook, Ola, Udemy, Coursera, etc. The innovation platform offers the technological framework on which other developers can build solutions. Some examples of this are the Microsoft and Ethereum platforms.
Alex Moazed explains in his book Modern Monopolies that “platforms don’t own the means of production; they create the means of connection.” (Moazed, Alex (2016). Modern Monopolies. Macmillan. p. 30.)
The Platform economy was formalized in the year 2000 with many large organizations adopting the technology and disrupting the traditional business models. Firms like Nokia, Blackberry lost out against firms such as Netflix, Amazon, taking leverage of the platform technology. As of today, private, government, and non-government organizations are increasing their presence on the platform economy.
Categories of Platforms. Platforms can be categorized into either investment, transactional, innovation, or integrated. Transactional platforms facilitate different types of online buying and selling. Innovation platforms provide the technological framework on which other developers can create an ecosystem around the platform, Integrated platforms combine both the transactional and innovation aspects into the platform, and investment platforms invest in multiple platform businesses.
Key Drivers and Emerging Key Drivers
The key drivers and emerging key drivers of the platform economy are the network, the ecosystem built around the network, database management systems, machine learning, artificial intelligence, the cloud, the blockchain, decentralized finance, decentralized autonomous organizations, non-fungible tokens, web3, etc.  
Use of Lean Six Sigma Methodology in Platform Economy & Key Drivers
The Platform Economy ties up the entire chain of an eco-system into a single framework. Master Black Belts have knowledge and experience across various domains in both the manufacturing and service industry across the entire spectrum of the chain.  Projects that help in increasing productivity, efficiencies, reducing overall cost, creation of new markets, or even reducing the inefficiencies in the existing markets have all been accomplished by Black Belts. Some of the key drivers and the role played by Master Black Belts are explained below.
Optimizing the Network Effect. The success of a platform depends on the Network Effect. A platform becomes powerful based on the network it is able to bring together. Six Sigma methodology can be used to study the network and create an optimal balance between various components of the network. For example, in a buyer/seller network, there has to be a supply and demand balance. This balance is dynamic and Six Sigma methodology through statistical analysis can be used to maintain the balance of the network so as to be able to monetize it in a profitable manner.
Creating the Ecosystem. The platform economy works around an ecosystem that is not the platform. This ecosystem consists of various developers, who create applications for the ecosystem. These applications could be around any part of the end-to-end supply chain. Lean Six Sigma methodologies can be used in identifying the elements of the ecosystem and creating relevant applications that maximize the platform economy.
Blockchain Technology, NFTs, DeFi, DAOs, etc. The Platform Economy being a centralized technology enables only big players to take advantage of the network, however, the open-source decentralized blockchain technology that enables smart contract functionality and built-in trust has enabled small players, innovators, and entrepreneurs to get on to the platform economy. Take, for example, the Ethereum platform that can attach tangible assets such as property, stocks, art, music, etc to the blockchain. Ethereum is an open-source, decentralized, blockchain technology with built-in trusted smart contracts. Any developer can deploy permanent immutable and decentralized applications. Users can use this technology and enable Decentralized Finance (DeFi), Decentralized Autonomous Organizations (DAOs), Non-fungible Tokens (NFTs), etc. Master Black Belts with their vast experience of working on various projects are uniquely placed to take advantage of the opportunities that are emerging with the decentralized platforms.
References
https://en.wikipedia.org/wiki/Platform_economy
https://en.wikipedia.org/wiki/Ethereum

Fractional Odds (British)
Fractional odds are given in fraction form. The first number to the left is the numerator and the second number to the right is the denominator of the fraction. The denominator is the wager and the numerator is the amount the wager will yield.
Example 1:
Wager on Chennai Super Kings at 10 to 15 fractional odds. For every Rs 15 wagered the potential net profit is Rs 10. A winner will get Rs.25 which is the sum of your original stake of 15 and the net profit of 10.
Fractional Odds - Wagering on an Underdog or High Odds Bet
In a high odds bet, the odds are more in your favour. For Example, when wagering for Kolkata Knight Riders the underdog at a 5 to 1 fractional odds, if you bet Rs. 100 and Kolkata Knight Riders win, you would get Rs.600/- (100 original bet + 500 net profit)
Decimal Odds (European)
This is the most popular used odds and its popularity is spreading from Europe across the world. It is the most economical way to bet, as the true odds are calculated making it the fairest and most accurate odds.
In decimal odds, the pay-out is the stake multiplied by the odds. This means that the original stake is included in the pay-out.
Pay-Out = Stake x Odds
Example 1
Decimal odds for Mumbai Indians to win are 1.82. If you wagered Rs. 100 your potential profit would be Rs. 182 (Rs. 100 + 82)
In Fractional Odds a similar wager on Mumbai Indians would be at 82 to 100.
Example 2
Wager of Rs. 100 on Punjab Kings at 2.75 decimal odds
Pay-Out = Stake x Odds
Pay-Out = 100 x 2.75 = Rs.275/-
In Fractional Odds it would be a wager on Punjab Kings at 175 to 100. It means that your risk of Rs.100 is taken against a chance to win Rs.275 at a net profit of Rs.175/-
American Odds (Moneyline)
This system uses a number system based on $100.
Negative odds mean the amount that needs to be wagered to win $100. Therefore -110 would mean that you need to risk $110 in order to win $100. Positive odds mean the amount that will be won when $100 is wagered. Therefore +110 would mean that you need to risk $100 in order to win $110. The formula for Negative Odds are 100/((100/odds) +1), whereas the formula for positive odds are 100/((odds/100) +1). The breakdown of the formula is
•          100/odds or odds/100 breaks the odds into the decimal amount on the amount to be won on the wager. For example, 100/120= 0.83 means that you will win 83% of the amount you risk
•          The +1 adds the wagered amount to the formula. .83 +1 = 1.83. This means if you wager 100, you will win $83 and get your $100 back.
•          100/1.83 = 53.09. This means if you made 100 wagers, you would need 53.09% wins to get a return of 1.83
Conversion of US Odds to Decimal Odds
Positive US Odds : Decimal Odds = (US Odds/100) + 1
Negative US Odds : Decimal Odds = (100/US Odds) + 1
Conversion of Decimal Odds to US Odds
Decimal Odds >= 2 : US Odds = (Decimal -1) x 100
Decimal Odds < 2 : US Odds = (-100)(Decimal -1)
Breakeven Odds
Breakeven odds is the percentage of time to win the wager in order to breakeven. Breakeven Odds for Decimal Odds are 2.0 as it will double your wager. Decimal odds below breakeven odds (less than even money odds) are when the potential returns are less than your stake. Decimal odds above 2 (plus odds) is when your returns is more than your stake. Breakeven Odds for Fractional Odds are 1 to 1 as it will double your wager.
Comparison of Same Wager
If you wagered Rs. 100 with a potential profit or Rs. 90
Fractional Odds for RCB would be at 90 to 100.
Decimal odds for RCB to win are 1.9
US Odds = (-100)(1.9 – 1) = -90
References
https://www.sbo.net/strategy/fractional-odds/
https://www.bettingpros.com/articles/break-even-win-for-sports-betting/
https://www.sbo.net/strategy/decimal-odds/
 

What is Human Centred Design?
Human Centred Design is a problem-solving approach that develops solutions by taking the human perspective and behaviour into consideration at every level of the design. Its goal is to address and incorporate the preferences, pain points, likes, and dislikes of the user. It was first popularised by the global design firm IDEO who found that the Human Centred Design approach led to quicker, increased profits and fewer products failing to market. It starts with the end-user for whom the solution is being designed and ends with a tailor-made solution that is suitable to their requirements. A product may typically meet either a functional, emotional or social need. The functional need is the actual use of the product whereas the emotional and social needs come from the feeling one gets from possessing a product. For example, it could meet an aesthetic or status need.
The Human Centred Design approach is akin to the ‘Jobs to be done’ theory by Harvard Business School Professor Clayton Christensen’s which states that people do not buy a product, but hire it to do a specific job or achieve a particular goal. The Human Centred Design approach looks at the design through this framework of the users’ motivations rather than demographic attributes such as income, gender, age, etc.
The Human Centred Design is so popular in that it has been defined in ISO 9241-210:2019(E) as “an approach to interactive systems development that aims to make systems usable and useful by focusing on the users, their needs and requirements, and by applying human factors/ergonomics, and usability knowledge and techniques. This approach enhances effectiveness and efficiency, improves human well-being, user satisfaction, accessibility and sustainability; and counteracts possible adverse effects of use on human health, safety and performance”.
It has three phases, viz., the inspiration phase, the ideation phase, and the implementation phase. The Inspiration Phase entails interacting with and learning from the end-user. The designer immerses himself totally into the lives of the people he is designing for, so as to understand their context and need. This will remove preconceived notions, biases, misunderstandings about what the customer actually requires. It sort of gets buy-in from the end-user. This stage requires empathy to understand the emotions and experiences of the customer. It is the designer putting himself into the shoes of the customer, getting a feel of the product asking questions that the customer would ask such as where, when, or the purpose for which they use the product. The ideation phase is used to generate various ideas and prototypes based on the experiences from the inspiration phase. The implementation phase would involve bringing the solution to the end-user.
Salient Features
The salient features of Human Centred Design are that it needs empathy, the generation of a large number of ideas through brainstorming or bodystorming, creation of a large number of prototypes along with the end-user.
It caters to the functional, emotional, and social needs of the end-user right from the start.
Product designs that follow the HCD approach are generally successful in the market. On the downside, these products have a long lead time for development.
Benefits
Even though the time to design and produce the product takes time, the likelihood of the product succeeding in the market is very high.
Having been designed from the perspective of the end-user, the product has early acceptance in the market.
Examples
Zoom.
Zoom realized the increased need for videoconferencing solutions not only for businesses but educational institutions, religious organizations, individual trainers, etc. Zoom created the virtual learning system through interaction with teachers and students and designed a system that met the varied end users’ requirements. Zoom ensured security and compliance requirements due to young students using its platform, they further build a whiteboard for easy interaction, dashboards to track student engagement, and integration with various learning management systems
Products used during Commutes
Designers of products used during commute such as coffee, milkshakes, mobile phones, music devices, etc, have interacted with the end-user to make their products more acceptable to the end-user. For example, Bluetooth-enabled hands-free mobile phones, music player controls on the steering wheel, cup holders in the car, increasing the thickness of the drink have been incorporated into the product based on the end-user requirements.
Changing Users Requirements during Covid
Payment and Logistics. With the onset of the pandemic, people had a need for reduced cash payments and increased home delivery of items. Payment and logistics processes have been redesigned to cater to these specific user requirements.
Electronic Tablets. Tablets were basically being used by the designer community. However, during Covid, the customer base extended to teachers, students, managers, etc. Wacom identified this requirement and through interaction with the end-user incorporated various hardware and software changes in their offerings.
References
https://en.wikipedia.org/wiki/Human-centered_design#:~:text=Human-centered%20design%20%28%20HCD%29%20%5Balso%20Human-centred%20design%2C%20as,perspective%20in%20all%20steps%20of%20the%20problem-solving%20process.
https://www.iso.org/standard/77520.html Ergonomics of human-system interaction — Part 210: Human-centred design for interactive systems
 
https://www.designkit.org/human-centered-design
 
https://online.hbs.edu/blog/post/what-is-human-centered-design?tempview=logoconvert
https://online.hbs.edu/blog/post/jobs-to-be-done-examples?tempview=logoconvert

Bodystroming
Bodystorming is a hands-on method used as a creativity technique in interaction design. It helps the user to empathize with the end-user. The designer subjects him/herself to the experience in order to feel and understand it from the perspective of the customer. In doing so it instills a feeling of empathy for the designer. It could include role-play, drama, and simulation. The process also involves brainstorming using the body which helps in bringing the simulation closer to reality and thus generating better ideas.
Advantages
Some of the advantages of bodystorming are it helps with empathy, reflection in the participants, connects the researcher and users, gives a first-hand experience and greater awareness to the designer/researcher
Disadvantages.
At times bodystorming may make some users uncomfortable, and when used in a sensitive context can trigger some bad reactions. In this fast-paced world, some designers find body storming to be time-consuming, when similar results can be achieved using machine learning, artificial intelligence, and simulation models. One of the disadvantages of concern is that bodystorming requires experienced and well-trained designers. In case it is done in a non-professional manner, wrong conclusions can be arrived at.
Use Cases.
Bodystorming is generally used in the design of physical spaces such as the layout of a store or a clinic. It can also be used in the design of software and physical products. Interestingly, it is also used in scientific research. It has been successfully used to teach, learn, and discover new intradisciplinary boundaries.
Example from Scientific Research
In 2009 the dance director, Carl Flink of the American dance company Black Label Movement and a Biomedical engineer of the University of Minnesota got together and created a bodystorming system as a part of the Moving Cell Project. Dance artists and scientists were brought together to rapidly prototype research hypotheses in biomedical engineering using choreographic rules for dancers to follow. Interestingly, bodystorming has proven to increase the speed of scientific research by giving the scientist a psychological sense of empathizing with a molecule.
In 2018, bodystorming was used at the Neuro-Oncological Symposium to model the recent research.
In 2019, bodystorming was used in the PSON Annual Investigators Meeting.
References
https://study.com/academy/lesson/bodystorming-in-design-thinking-definition-purpose-example.html
https://think.design/user-design-research/bodystorming/
https://en.wikipedia.org/wiki/Bodystorming
 

Hick’s Law was named after Britisher, William Edmund Hick, and American Ray Hyman. Their background in psychology helped in framing this law.
 
The law looks very logical in that it states that the more choices a person has, the longer the person will take to reach a decision. The beauty of the law is the mathematical equation that goes with it. That is the time a person takes to reach a decision is a logarithmic function of the number of choices. Since it is a logarithmic function of the number of choices, the increase in time diminishes as the number of choices increase.
 
A picture that comes to mind is a child selecting an ice cream before a Baskin Robbins Ice-cream shop. Considering the implications of the law, it becomes more relevant when designing a short list and loses its significance as the length of the list increases. Hence Hick’s Law can be used in the design of User Experience (UX). Examples of short list are ‘Action Buttons’ or ‘Navigation Menus’ in an App or website. At times the designer thinks it is wise to include more functionality into his website or app, however, he should use Hick’s law to assess the number of functionalities that he should put into his design. A user of the app/website when encountering too many options is likely to get saturated with the choices available and leave the website quickly. This can be measured through various matrices such as the bounce rate, conversion rate, user engagement, time on site, page views, etc., using various analytics software such as Google Analytics.
 
Hick’s Law helps in various design decisions, either in the design of physical products, such as the number of buttons on a TV remote, the number of controls in a washing machine, or in software products as the number of links in the header tab of a website. Hick’s law can be applied to the Tree Structure of a menu, to determine both the horizontal width and vertical depth of the menu.
 
In order to apply Hick’s law, designers should put the choices in categories there by reducing the choices available. Designers can also obscure complexity by breaking down the process into manageable steps with fewer options in each stage.
 
The formula is
RT = a + b log2 (n)
RT – Reaction Time
n – Number of Stimuli
a and b – constants that depend on the task/condition

 
Applicability of Hick’s law to Project Management.
Hick’s law applies to choices that have an equal probability of selection. This means that the user has no previous knowledge of the choices and is making a choice based on what is presented in front of him. If a user is specifically/intentionally looking for a specific choice or has a certain bias, Hick’s law will not be applicable to him. In this case, the time taken to act is likely to be less than the logarithmic function as calculated by Hick’s Law. In such cases, other decision-making tools such as Pugh Matrix or Analytic Hierarchy Process may be used.
 
References
https://www.interaction-design.org/literature/topics/hick-s-law
 

Iceberg Theory
The Iceberg Theory is also called the theory of Omission. It is a writing theory that was made popular by journalist and writer Ernest Hemingway. He used his style of writing as a journalist in the writing of his short stories. In doing so, he focused more on the immediate events which were evident on the surface and less on their context. His writing style caused the reader to understand the context of the story in an implicit manner. His minimalistic style of writing caused each reader to contextualize the story within the readers' framework.
Quoting from Oliver, Charles M “If a writer of prose knows enough of what he is writing about he may omit things that he knows and the reader, if the writer is writing truly enough, will have a feeling of those things as strongly as though the writer had stated them. The dignity of movement of an iceberg is due to only one-eighth of it being above water. A writer who omits things because he does not know them only makes hollow places in his writing.”
The key to the quote by Oliver Charles M is that when the writer omits what he knows the reader is likely to strongly pick up the things omitted by the writer. This adds to the authenticity of the writer. However, when a writer omits things that he does not know, the reader is naturally unable to pick up the things omitted by the writer and it leads to a hollowness and in his writing.
Application of the Iceberg Theory
Leadership/Management. Questions such as “Are leaders born or made” or “Is Management an art or science” have been asked through the ages. Advancement of knowledge and technology has shown that through training, patience, and discipline leaders are made and management is more of a science than an art. Leadership qualities span a vast array of attributes such as professional knowledge, integrity, empathy, etc. This inward focus of character and knowledge and an outward focus on people takes time and effort to build making a leader authentic.
Consultants. Successful consultants are those who have a wide range of knowledge and experience spanning various fields including understanding human behavior. With processes crossing numerous functions, consultants who are experts in a very specific area are unable to optimize solutions. 
Actors. Great Actors spend months or years researching their part. Good examples of this are Tom Hanks who plays the life of a physically disabled and low intelligence man in the movie “Forrest Gump”, Freddie Highmore who plays the part of a young autistic savant surgical resident in the series and Darsheel Safary who plays the part of a dyslexic child in “Taare Jameen Par”.
Teachers, Professors. Similarly, teachers and professors with a deep knowledge of not only their subject but related subjects are generally more successful and popular.
Doctors. With the human body being a diverse spectrum of various systems that interact with each other and with the external environment, a doctor with a deeper knowledge of not only the human body but other external factors is more likely to be successful.
Conclusion.  The Iceberg Theory brings out the fact that when a person with in-depth knowledge over a wide array of subjects speaks, the information appears to be authentic. On the contrary, when a person with superficial knowledge speaks, his lack of in-depth knowledge, of confidence, of authenticity, etc. will betray him and he will be easily found out.
References
https://en.wikipedia.org/wiki/Iceberg_theory
Oliver, Charles M. (1999). Ernest Hemingway A to Z: The Essential Reference to the Life and Work. New York: Checkmark. ISBN 0-8160-3467-2.

The Poisson Distribution is named after the French mathematician Simeon Denis Poisson. It is a discrete distribution that gives the probability of the number of events in a fixed interval of space, time, distance, volume, area, etc. These events should take place independent of each other and at a constant rate. The Poisson distribution is generally used where the number of events is very large and the occurrence of these events is rare. In simple terms it can be described as the average rate at which a specific event happens in a specific time frame, the event will follow a Poisson distribution.
Assumptions
k is the number of times the event can occur (k= 0,1,2,3,…..)
The events are independent of each other
The average rate at which events occur, are independent of the occurrences and may vary with time.
Two events cannot occur at the same time.
Examples
The number of calls received in a Call Centre every hour.
The number of decay events that occur from a radioactive source every day.
No of Accidents in a day on Highway No 4.
The number of meteorites greater than 0.5-meter dia that strike Earth in a year.
The number of customers arriving at a Mall between 8 and 9 am.
Number of Tsunamis that hit the East Coast of India in a decade
Number Goals in a match.
Number of Injuries in a football match.
Number of Hat tricks in a cricket match
Examples that are not Poisson Distribution
The arrival of students for a class
The arrival of spectators for a match.
These are not Poisson Distributions since the rate of arrival is not constant and the events are not independent of each other since students and spectators come in batches.
Occurrences of earthquakes may not follow a Poisson distribution since the aftershocks may not be independent of each other.
Formula
The formula for Poisson distribution is

Where
k is the number of occurrences,
e = 2.71828 (Euler’s Number)
Lamda = E(X) [The mean] = Variance(X)
Lamda can also be taken as the rate for the number of events.
Poisson Distribution in Management
Poisson Distribution is generally used to improve operational efficiency.
Poisson Distribution is used in Queueing Theory as the arrival rate, whereas the Exponential Distribution is used in the Service times.
Managing Demand. Study of the demand through the Poisson Distribution can help in developing complementary services, managing the reservation system and overbookings such as in hotels and airlines, segmenting demand, offering price incentives, or promoting off-peak demand.
Managing Supply. Similarly, the study of the supply can help in sharing capacity, take decisions on cross-training of staff, using part-time employees, increasing customer participation, scheduling work shifts, or creating adjustable capacity.
References
https://en.wikipedia.org/wiki/Poisson_distribution

Deductive reasoning is a scientific method that uses deduction to check or test theories and hypothesis.
 
Deductive reasoning consists of three steps, premise one, premise two and an inference. For Example, Premise one is all men are mortal, premise two is Sampson is a man. Thus, we infer that Sampson is a mortal.
 
Deductive Reasoning is the foundation of research and academic research in specific. In academic research we first make claims, gather data, and then test the claims or to be specific the hypothesis.
 
Inductive Reasoning makes broad claims about an observation. Data is gathered and then conclusions are drawn making it opposite to deductive reasoning.
 
The main difference between the two forms of reasoning is that deductive makes claims based on theory while inductive makes claims based on observations.
 
The main way that the two are used in the Lean six sigma world is inductive reasoning is used in Gemba walks and “management by walking around”, where the practitioner is observing the processes and then making decisions. While the entire DMAIC process under six sigma would be grouped under deductive reasoning as the practitioner theorizes a result, gathers data and tests if their results hold true.

ECRS or Eliminate, Combine, Rearrange and Simplify is a Lean technique. ECRS is used to reduce or eliminate wasteful steps from any business processes, be they a process in manufacturing or a process in the office. ECRS is a method that can be used for fast improvements.
           
Eliminate: In a process which currently takes a certain number of steps, if we were to eliminate a step it would shorten the process. For funny example would be Tom Smykowski in the 1999 Cult comedy Office space, Tom’s job was to take “the specifications from the customer and bring them down to the software engineers”. The consultants eliminated his job as they had the customers directly send the specifications to the software engineers. A funny example from a cult classic but it does the job of explaining elimination.
 
Combine: When elimination is not possible, the next step is to see if any of the steps in the process can be combined. A classic example would be the use of macros in Microsoft Excel. By creating a macro of multiple steps when working on a excel file you are combining all those steps together. The analyst does not have to do those steps again as the macro does it for him.
 
Rearrange: When elimination or combining is not possible, rearrangement of steps is done to make the process safer, easier, or faster. A classic example of rearrangement of steps is Henry Ford’s assembly line for the production of cars. One of the main benefits was the reduction in the assembly of a car from half a day to under an hour.
 
Simplify: When faced with a complicated process, simplifying steps is undertaken. This done to ensure that the process is easy to understand by the operators. Visual aids are an easy low-cost method of simplification.  
 
Sources: https://www.creativesafetysupply.com/glossary/ecrs/#:~:text=ECRS%20stands%20for%20Eliminate%2C%20Combine,processes%20or%20even%20office%20procedures.
https://www.lean.org/LeanPost/Posting.cfm?LeanPostId=1245

ECRS or Eliminate, Combine, Rearrange and Simplify is a Lean technique. ECRS is used to reduce or eliminate wasteful steps from any business processes, be they a process in manufacturing or a process in the office. ECRS is a method that can be used for fast improvements.
           
Eliminate: In a process which currently takes a certain number of steps, if we were to eliminate a step it would shorten the process. For funny example would be Tom Smykowski in the 1999 Cult comedy Office space, Tom’s job was to take “the specifications from the customer and bring them down to the software engineers”. The consultants eliminated his job as they had the customers directly send the specifications to the software engineers. A funny example from a cult classic but it does the job of explaining elimination.
 
Combine: When elimination is not possible, the next step is to see if any of the steps in the process can be combined. A classic example would be the use of macros in Microsoft Excel. By creating a macro of multiple steps when working on a excel file you are combining all those steps together. The analyst does not have to do those steps again as the macro does it for him.
 
Rearrange: When elimination or combining is not possible, rearrangement of steps is done to make the process safer, easier, or faster. A classic example of rearrangement of steps is Henry Ford’s assembly line for the production of cars. One of the main benefits was the reduction in the assembly of a car from half a day to under an hour.
 
Simplify: When faced with a complicated process, simplifying steps is undertaken. This done to ensure that the process is easy to understand by the operators. Visual aids are an easy low-cost method of simplification.  
 
Sources: https://www.creativesafetysupply.com/glossary/ecrs/#:~:text=ECRS%20stands%20for%20Eliminate%2C%20Combine,processes%20or%20even%20office%20procedures.
https://www.lean.org/LeanPost/Posting.cfm?LeanPostId=1245