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Brilliant explanation provided by all respondents. Best answer was provided by Johanan Collins. His answer also highlights the difference between the 3 variations of Rootogram.

Similar to histogram , Rootogram is a graphical tool to visually depict the distribution of the variable data. One of the limitation of histogram is that it is difficult to can be difficult because small frequencies are dominated by the larger frequencies and it is hard to understand the pattern of  histogram bars when compared with the distribution. Comparison can become easier by ‘hanging’ the observed results from the theoretical curve, and  that way the discrepancies are seen by comparison with the horizontal axis rather than a sloping curve. Here vertical axis is scaled to the square-root of the frequencies so as to draw attention to discrepancies in the tails of the distribution.   Reference: https://datavizproject.com/data-type/rootogram/

A rootogram is a data visualization technique to summarize a distribution of a variable. It has the frequencies in the Y axis and the response variable on the X axis. The frequencies are square root or relative frequencies. Rootogram can be for absolute count, relative rootogram converts counts into proportions, cumulative rootogram and cumulative relative rootogram. Its variation to the histogram, bars are plotted for observed frequencies and a curve for the fitted frequencies all on  square-root scale. Overlaying the distribution curve tell us how an actual histogram differs from a distribution estimate.  Mathematician John Tukey noted that the difference of comparing the distribution of data with a theoretical distribution from an ordinary histogram can be difficult because small frequencies are dominated by the larger frequencies so it difficult to understand the pattern of differences between the histogram bars and the curve. Advantages: The data visualization becomes much better if we use hanging bars- from the fitted curve, or a "suspended" histogram of deviations can be drawn. ‘hanging’ the observed results from the theoretical curve is drawn, so that the discrepancies are seen by comparison with the straight reference line at zero (horizontal axis) rather than a sloping curve.   Image courtesy: andrewpwheeler.com    

We all know about our good old histograms - having bar charts with continuous numeric axes. For e.g., here is a simple histogram of transaction wise freight variation:   Truck Freight Distribution Histogram   X- axis is represented as freight cost bucket and y axis is represented as no. of transactions. Now to understand the overall distribution pattern i.e., we will overlay the histogram with a normal distribution curve on the top.   Truck Freight Distribution Histogram   Now if you look above for the overlaying of the distribution curve and the histogram, it is obvious that line graphs that is overlapping the bar chart is not be flat, hence difficult to approximate the horizontal midpoint of the bar is:   And thus, to solve this visualization challenge and to display data in such a way that interesting features will become apparent Tukey proposed this Rootogram also known as Tukey’s Hanging Rootogram.   Truck Freight Distribution Histogram   Now you can see difference become much easy to estimate, since the bars are hanging from the curve and using X-axis as flat line for comparison.   One more critical point related to Rootogram, is that it plots the square roots of the number of observations observed in different ranges of a quantitative variable. Here the requirement of using square roots is to equalize the variance of the deviations between the curve and the bars, which otherwise would increase with increasing frequency.

Q 441. While doing ANOVA or DOE, a researcher usually prefers a balanced design. However, in real world, we might get an unbalanced design. Illustrate the difference between the two using an example. What are the methods to deal with unbalanced designs?   Note for website visitors - Two questions are asked every week on this platform. One on Tuesday and the other on Friday. All questions so far can be seen here - https://www.benchmarksixsigma.com/forum/lean-six-sigma-business-excellence-questions/ Please visit the forum home page at https://www.benchmarksixsigma.com/forum/ to respond to the latest question open till the next Tuesday/ Friday evening 5 PM as per Indian Standard Time. Questions launched on Tuesdays are open till Friday and questions launched on Friday are open till Tuesday.  When you respond to this question, your answer will not be visible till it is reviewed. Only non-plagiarised (plagiarism below 5-10%) responses will be approved. If you have doubts about plagiarism, please check your answer with a plagiarism checker tool like https://smallseotools.com/plagiarism-checker/ before submitting.  The best answer is always shown at the top among responses and the author finds honorable mention in our Business Excellence dictionary at https://www.benchmarksixsigma.com/forum/business-excellence-dictionary-glossary/ along with the related term

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

Very interesting answers to an equally interesting topic    There are two winners for this question - Johanan Collins and Sai Kotari. Johanan for the shear vastness of information provided with respect to  ANOVA and Sai for highlighting various related concepts like within and between variations plus confidence intervals.   Well done!

In many situations , we have to compare central tendencies(generally mean is compared) of multiple samples. For 2 sample, t test is widely used. We can use Z test for large sample size. ANOVA (Analysis of variance) is used in cases when we have to compare more than 2 means. Though we can use multiple t test but it will be time consuming and less efficient. ANOVA is a parametric test and requires various assumption for the populations samples to be compared. This includes assumption of Normality, Independence and approximately equal variance. ANOVA is not used in the Nominal data . Below are the steps to perform the ANOVA: Calculate the mean of all the samples Define the null and alternate hypothesis (e.g.: Ho= All samples have equal mean, Ha= All Sample means are not equal) Perform calculation to get the Sum of squares and mean squares based on degrees of freedom for within and between samples differences. Calculate the F statistic. Look up statistical Table , Compare the F statistic with the tabular values and conclude on the results.   ANOVA is a parametric test and hence can not be used in all situations. We can use Kruskal Wallis test which is the non parametric alternative to the One Way ANOVA. It compares the sum of ranks in the samples instead of mean.  

Analysis of variance (ANOVA) is used to assess the differences between means of 2 or more groups. It is a statistical hypothesis test to determine whether the means of at least two populations are different. Conditions for using ANOVA are: ·         a continuous dependent variable (Y) ad and a discrete independent variable (X) ·         should be a normal distribution. ·         Samples must be independent. ·         Population variances must be equal ·         Groups must have equal sample sizes.   Hypothesis: Ho: μ1 = μ2 = μ3 =… μn Ha: μ1 ≠ μn   A significant P value implies a low probability that the mean values for all groups are equal; it only tests for an overall difference between groups. Once the overall significance is arrived at, then we can use multiple comparison procedures for individual group comparisons.   The right AVOVA test to perform can be decided basis the number of independent variables that are included in the ANOVA test A.      One-way means the analysis of variance has one independent variable. Example: AHT of staff at different experience levels like <6 months and > 6 months B.      Two-way means the test has two independent variables (which can have multiple levels), Example, For Jam sales/week, independent variables of brand of Jams and how many calories it has. Another example: AHT of staff at different experience levels like 3-6 months, 6-12 and >12 months for Different experience background like Insurance, Banking, etc. o   The results from a Two- Way ANOVA will calculate a main effect and an interaction effect. o   The main effect is similar to a One -Way ANOVA: each factor’s effect is considered separately and with the interaction effect, all factors are considered at the same time. C.      MANOVA (multivariate analysis of variant) is another form of ANOVA for several dependent variables. For example, AHT for Task 1 for different Tenure groups/Experience background and AHT for Task 2 as well. Possibly to assess if the combination has any adverse impact on speed of Task 2. It tests multiple dependent variables at the same time by testing: a.      changes to the independent variables have statistically significant effects on dependent variables b.      interactions among dependent variables c.       interactions among independent variables For data that is not Normally distributed, we can use a non-parametric, analogue of one-way ANOVA (called Kruskal-Wallis ANOVA)

Q 440. Rootogram is a modified version of the good old histogram. What are the advantages of working with a rootogram? Explain using an example.   Note for website visitors - Two questions are asked every week on this platform. One on Tuesday and the other on Friday. All questions so far can be seen here - https://www.benchmarksixsigma.com/forum/lean-six-sigma-business-excellence-questions/ Please visit the forum home page at https://www.benchmarksixsigma.com/forum/ to respond to the latest question open till the next Tuesday/ Friday evening 5 PM as per Indian Standard Time. Questions launched on Tuesdays are open till Friday and questions launched on Friday are open till Tuesday.  When you respond to this question, your answer will not be visible till it is reviewed. Only non-plagiarised (plagiarism below 5-10%) responses will be approved. If you have doubts about plagiarism, please check your answer with a plagiarism checker tool like https://smallseotools.com/plagiarism-checker/ before submitting.  The best answer is always shown at the top among responses and the author finds honorable mention in our Business Excellence dictionary at https://www.benchmarksixsigma.com/forum/business-excellence-dictionary-glossary/ along with the related term