Scatter plot, a randomly plotted points of data with coordinates of independent and dependent  variables. There are developed techniques how to determine whether the set of points are some how correlated ( with some degree of correlation values on [-1,1]) and determining the regression line as a function of the independent variable. The analysis that is needed to be done on the points and any other point within the grid  is done through the linear function of best fit or regression line. And I assume the line is only used for values within  the minimum and the maximum values of the independent variable in the data set, which makes it a constricted method some how.   

A scatter plot is used at initial steps in order to see if there exist a kind of known correlation or even functional form. If we have a 'cloud' then we will probably not have a linear relation. If we 'see' a well known pattern (a line, a parabola,...) then we could try to fit it using the relevant model. Another issue is the concept of variance change, since it is not constant over time and we can observe this fact from a scatterplot easily.

If I understand correctly, Data analysis is a bottom up method at reaching some decisions, without presupposing a probability or Statistical model. In this respect scatterplots fall in data analysis method. Gauss least squares method can give a law. Thus starting from data we add up with a theoretical law! This is the essence of "Data Analysis".

Scatter plot, a randomly plotted points of data with coordinates of independent and dependent  variables. There are developed techniques how to determine whether the set of points are some how correlated ( with some degree of correlation values on [-1,1]) and determining the regression line as a function of the independent variable. The analysis that is needed to be done on the points and any other point within the grid  is done through the linear function of best fit or regression line. And I assume the line is only used for values within  the minimum and the maximum values of the independent variable in the data set, which makes it a constricted method some how.   

i prefer to read the asnwers of this question.

Scatter plot is used to display the degree of association of two variables. Normally, data plotting is the first stage in data analysis concerning relationship of two or more variables before proceeding to a higher degree of statistical testing/analysis. But of course, variable selection has to be supported with assumptions and theories about their association. In some cases also, scatter plot is used to identify outliers in the dataset. Some friends of mine in the field used it to identify which data (outliers) will be excluded in the analysis.

FYI: Example scatterplots used for editing outliers are found in the seminar slides which may be located through the first link attached to the question above.

One 3-D scatterplot done by my coauthor, Joel Douglas, was fun to play with, for purposes of analyzing the estimator for totals (in addition to the variance of the prediction error, and also performance history) for establishment survey sampling. You could turn this scatterplot to see current gross generation of electricity on the y-axis vs either past gross generation or against generating capacity, or intermediate views. ('Current' gross electric generation was from a monthly quasi-cutoff sample, and the two regressors were from previously collected census data.) The 3-D graph was informative. The red dots were estimated or 'imputed' points, depending upon your viewpoint about cutoff sampling.

Attached is that 3-D scatterplot, which I also used as an illustration at the end of "Quasi-Cutoff Sampling and Inference," on my profile, originally on InterStat in 2007.  The scatterplots as shown through the first attached link to the question above might be easier to see. - Click on figure below to enlarge.  

Thank you all for the responses thus far.   The use of scatterplots for continuous data for purposes of data exploration in support of model selection, and data editing, is noted, but I think scatterplots are still possibly underappreciated and underused in general, so I think more attention to this topic would be helpful. Your responses are appreciated. Additional responses, particularly with examples, would be useful. Unusual applications would also be interesting.    

I'll ask Joel Douglas, but it may take a few days to get an answer.  (He changed agencies last May, and is very busy, and I retired in August, and we have not had a great deal of contact since then.)    

We use scatterplot to analyze if there are relationships between some physical properties. This is a preliminary analysis, but very useful one. Scatterplot analysis is one of the step of Exploratory Data Analysis.

This is a very useful link

http://www.itl.nist.gov/div898/handbook/index.htm

mariano.

Mariano - I remember that NIST handbook (by a contractor, SEMATECH, I think), being generally quite good, though I had a problem with their logic concerning weighted least squares, as noted in section 3 of the attached. - Thank you. - Jim 

Article Properties of Weighted Least Squares Regression for Cutoff S...

Sayed - 

i received this text message from Joel in answer to your question regarding the 3D graphic software: 

"SAS 'proc g3d'...these also had the SAS graph JAVA plugin embedded in them to make them interactve"

I thought it was an independent piece of software, but apparently it was a SAS PROC. 

Cheers - 

jim 

PS - Perhaps R has something similar??? 

Reviewing ... it appears to me that Rudy Lange's very clear response has been underrated.  I suggest giving it a look.  

Scatter plots is usually the first step to solve a problem in data analysis; the following link entitled:  " What is a Scatter Plot Saying? " discusses this issue. Brief Overview:

This unit is designed to solidify students’ understanding of creating, analyzing, and

interpreting scatter plots. It starts out with the basic ideas of collecting data and making a scatter plot. It then focuses on analyzing the scatter plot to find relationships and interpret the data. This unit introduces major ideas of linear and non-linear relationships, the y=x  line, the line of best fit, and positive and negative relationships (association).   

https://www.nsa.gov/academia/_files/collected_learning/middle_school/interdisciplinary/scatter_plot.pdf

Here is an example scatterplot for editing data, taken from the first link under the question, which is particularly effective if either y or x are in the wrong units (as long as both are not basically in error by the same factor).   -  Click on the figure below to enlarge.  

For a study on the relevance of units and x,y-axes transformations you can also look my work here:

https://www.researchgate.net/publication/268977798

Data Reliable computations of knee point for a curve and introduc...

Interesting application, Demetris, especially, in my opinion, since you include this section: 

"4. The problem of noisy data" 

Dear James.

It is interesting that there exist some very high leverage values. Before I see the data I had this in mind. I think for some data scatter plots are crucial because of leverages.

In this dataset high leverage values can have dramatic effects. Besides, high leverage values, in most cases, lead to heteroscedastic models which needs WLS method. WLS method, itself, can define prediction effect of a datum. So i think, we should omit data which have less than 5 percent real effect of all values. On the other hand, as a datum goes farther from the sample mean, both leverage and heteroscedasticity will increase. 

QQ plot shows that these high leverage data can even change the model type, and here, we really do not have a linear model.

Therefore, plotting the data depends on their nature.

scatter plot inferences:

1) recognition of pattern type

2) a particular trend between variables can be studied. i.e. increasing / decreasing or constant

3) correlation between two variables can be studied

4) market behavior can be forecast fro previous data plots.

    Just as James said, scatterplots can be used to explore whether a relationship may be considered linear or nonlinear, positive or negative. Thus, it is very important for mathematical modeling. In 1997, when I studied taper equation, I used scatterplot to explore the variation of power parameter b in the taper equation presented by Ormerod (1971):

d/D=[(H-h)/(H-1.3)]^b

and found that b can be expressed as a 1/2 power parabola as:

b=c0+c1*z^1/4+c2*z^1/2

which is better than the more complicated 7 power parabola:

b=c0+c1*z+c2*z^2+c3*z^3+c4*z^4+c5*z^5+c6*z^6+c7*z^7 presented by other people. See the attached file.

    Recently, I have being engaged in biomass modeling and scatterplots were used very often (see my recent publications, such as the 2nd attached file). For example, the proportion of stem biomass to aboveground biomass increases with growing diameter of stem, while the proportion of crown biomass to aboveground biomass decreases. These relations can be showed obviously by scatterplots. In addition, as Demetris said, we can understand the characteristc of variance from the scatterplot, and determine whether it is heteroscedasticity or not.

The concept of data analysis is to extract information from a noisy source, the better the quality of xtraction, the better the ranking of the relevant algorithm. Regression uses the fact that E(Y|X)=bhat0+bhat1*X because E(error)=0, if the later does not hold, then bye!

In 1982, when I was Chairman of a Group evaluating he impact of sulphur and nitrogen oxide emissions on the acidification  of poorly buffered waters in the UK, we were faced with a mass of pH measurements made routinely at monthly intervals over the previous 20 years.  The data  was not used for anything and, until we requested it, had lain, covered with real and metaphorical cobwebs, in River Authority Archives.  The error was typically about +/- 1.5 pH unit at mean values of 6-7 for several hundred sites.  We knew the data was useless but we had to go through the motions of seeing if we could detect any trends. 

The analysis was carried out by the best statistician I know who works on this kind of problem, who plotted CUSUMs (CUmulative SUMs of the deviations from the global mean) of the data for each site.  [For any regular data set with an underlying fall the sum of the deviations in the early data will be increasingly above the mean and will then decline until they reach the mean.  If there is an underlying rise, the first part of the plot will be below the line and the last part will rise until it meets the line.]   This elegant method of extracting meaning from poor quality, noisy time-series data will show not only when a trend exists but also when a  trend changes.  It eliminates the noise in the data because the cumulative error is much larger than individual errors.

Our data?  We christened the CUSUM results Manhattan Diagrams because of their resemblance to the NY skyline!

The link is the 2nd file for reference.

http://link.springer.com/article/10.1007/s00468-014-1112-0

I have worked on the mirror problem.  You can plot the germination of bacterial spores against time and obtain a smooth curve, starting with a shoulder and then an acceleration before the rate dies away.  It was thought that the shoulder represented an initialization period but two Americans showed that the population is heterogeneous  - some spores germinate quickly and some more slowly. The distribution is asymmetric, (no spore can germinate before t=0 but some will never germinate) and fits a Weibull distribution in which the shape parameter, r=2.  A simple model shows that the shoulder derives from the heterogeneity of the spore population and not the biochemistry. 

A picture is worth a thousand tables.  A scatterplot is useful only if it helps you to decide what kind off analysis will give you more information.

Writing some software to explore the Mandelbrot set I wanted a quick way to test whether the trajectory of a given pair of coordinates in the Argand diagram of the complex plain would disappear to infinity and which would remain finite after iterations according to the standard formula  (Zn+! = [Zn + C]2).   Naturally I used Excel to make scatter plots of successive values.  The results were better than I expected but  I had to join the dots to show the trajectory,  The attached file shows three pairs of coordinates, very close together but two are outside and the third, between them, is probably inside the set.

I had to plot the trajectories in successive batches of 20-25 iterations to avoid overcrowding.

I have more if anyone is interested.

The heteroscedasticity is quite pronounced, and even though heteroscedasticity is an integral part of such graphs, it is not always so visually obvious. Another thing is that one graph looks like linear regression through the origin, while the other looks like nonlinear regression with heteroscedasticity in the opposite 'direction' as the other

In analytical chemistry we often have to go beyond simple scatterplots, because the number of variables can be huge. Typically, we have a spectrum or a chromatogram of each sample, each having hundreds or thousands of points. Frequently we compound the data deluge by obtaining one or more kinds of spectrum at each point on the chromatogram.

"Chemometric" techniques are used when no single variable is specific for a component of the mixture being examined.

There are two main classes of problem to be solved: classification (e.g. distinguishing genuine vs counterfeit chemicals by the pattern of trace impurities), and calibration (finding quantitative signals that are specific for one of the substances in a mixture).

Chemometric scatter plots ("cluster analysis") require linear algebraic methods such as principal component analysis to produce a small set of statistically independent variables giving maximal discrimination.

It's all very complicated, and intelligent exploration may be required to find the most appropriate method. However, a rapid web seach reveals that the market for chemometrics software suitable for mathematically-challenged scientists is still thriving.

Cluster analysis scatterplots using R:  

I never used R, but it seems a useful tool - available free from the efforts of other researchers, but of course, without warrantee. In doing an Internet search, I found a handbook chapter on this topic, and in glancing at it, I saw a familiar type of scatterplot display. It is found through the attached link, and the figure I mention is labeled as follows:

"Figure 18.6 Scatterplot matrix of planets data showing a three-cluster solution from Mclust."  

This webpage is at the first link below - from authors Brian S. Everitt and Torsten Hothorn.  

The second link is the cran.r-project website overview.     

I would have displayed Figure 18.6 here, but even though R is free, I was not certain of the applicability of copyright laws. 

If you have any displays of your own to exhibit, that would be of interest to this thread, I am certain.   

http://cran.r-project.org/web/packages/HSAUR2/vignettes/Ch_cluster_analysis.pdf

http://cran.r-project.org/

Pertinent examples of scatterplot type graphs are boxcox plots to find power of a transformation, diff plots in engineering, influence plots of the residuals, interaction plots, normal plots, quality control charts, and Pareto charts when working with nonreplicated data. Data can be processed faster visually in a graph than it can be processed in a table of numbers. Do a google search on these examples; I think you will be amazed as to how it can help.    Good Luck.  MM

Be careful. There are plenty of nice linear correlations that are spurious (sales of German cars and US suicides; US unemployment rate and snowfall in inches in Amherst MA; etc.) and there may be correlations where the data don't match up well (global warming and C02(?)). Data are funky.

As Peter said, on occasion a spurious relationship may be found. That is why you should start with theory and use data to check it, not just expect data to discover all of your relationships, though exploratory data analysis (EDA) might give you some ideas.  But one may need to check a relationship from various points of view.  Statistics can show correlation, not causality.

[Aside: Sometimes a relationship will be missing a step.  A dermatologist once told me that since I had such dark hair, he was surprised I was having skin cancer problems because those with dark hair usually did not have that problem. What he was missing is that even though I had dark hair, my pale skin is the problem.  Most people with darker hair have darker skin, and that is what protects against skin cancer. The darker hair was doing me no good at all.]  

We are looking at correlation, not causality, and an odd, spurious relationship may be encountered on the rare occasion.  But scatterplots are still a terrific tool.  Like everything else, however, you need to look at the 'big picture.'  You need to consider how everything fits together. 

Essentially accidental relationships are not spurious when gene linkage is involved. Jim's unfortunate situation could be an example.

One that comes to mind, writing from memory, relates to tomato breeding. A gene that causes the fruit to ripen progressively from one end of a truss to the other was bred out. This conferred an obvious economic and commercial advantage though it would be a disadvantage in the wild and for those who grow their own.

The big disadvantage was that the new tomatoes had no taste; this was because a gene for the relevant tasty metabolite just happens to be on the same chromosome and close to the one for progressive ripening. The odds on a crossover occurring between two genes (thus separating them) decrease the closer they are together. Conversely, the chances of two loci recombining are greater the farther they are apart.

There are two links I attached to this question at the outset. One contains some slides showing scatterplots useful for editing of continuous data from establishment surveys, and the second shows how to create confidence intervals about predicted y-values for a simple regression through the origin.

Does anyone have a comment, and/or similar graphics to share?  

Many use of scatter plot...What type of transformer need for the data...

Currently I and few of my students are rigorously researching all the EDA topics. We try to look at all angles of the knowledge of EDA. Boxplots, Histogram etc. Wish us luck!

Theory and the nature of the environment of data collection are very significant to obtain natural trend as defined by the dependent and independent variables. This determines the nature of correlation....whether it is realistic or ideal for reliability and replication. This is to avoid the appraisal by Peter (above). Remember, other factors must acknowledged through scientific norm.

Be careful when looking for any relationship.... distinguish between independent and dependents variables are very important. Multicollinearity problem should be carefully address. Thanks for the reminder. Use that scatter plot with care.

You guys are scaring me a bit.

As an economist who works with time series data all the time, data that are influenced by myriad factors most of which we can't see or quantify can be distorted. Sometimes nothing shows, but it's actually there when proper variables are included. Sometimes something shows, but it's a ruse because of James' third (outside) factors.

Many variables rise over time in macroeconomics. They all appear to correlate. But they're not necessarily correlated. A time series scatter plot can really give the wrong impression.

Take a simple demand curve. If we're trying to empirically find one, we get data for prices and quantities all at different times, but because supply factors are in there too, all we get are supply and demand equilibrium prices at different times at best. The scatter plot reflects both. We don't even know if there were disequilibrium positions recorded too. You need to account for both supply and demand impacts to get the "actual" demand. Even then, the assumption is that the demand curve elasticities didn't change through time. Yikes.

As I mentioned before, data are funky. Be careful.

Actually, Peter, I have not dealt with time series very much, so your insight on that is interesting to me.  What I really had in mind was inference from sampling from finite populations.  

I think it is good to keep in mind that we all have different background experiences and areas of application.  I have learned a thing or two from exchanges like this with people of different areas of interest.  I think that can make ResearchGate useful.  (By the way, I have gone to statistics conferences and tried to go to at least one presentation in an area unfamiliar to me.  Besides differences in emphasis, it also seems that sometimes parallel development of similar ideas occurs in different subject matter disciplines, using different terminology, or sometimes the same terms are used for different things.)  

Anyway, thank you for your input.  

Actually the plane coordinate of y and x axises..is not interested anymore in looking for any information for the data as the scatter plot only reveal bivariate case...3D of scatter plot will give extra tri-dimension of the data. Trend might be detected or maybe some outliers or maybe some clustering exist... Very limited usage.

Would you please send your data to me?

Masoud - 

For finite population data such as used in https://www.researchgate.net/publication/263927238_Cutoff_Sampling_and_Estimation_for_Establishment_Surveys above, you might inquire at https://www.eia.gov

(In the abstract note that it says that "The 3-D scatterplots (Joel Douglas) would be rotated to 2-D views for analyses, but are shown in partially rotated form here, for purposes of qualitative illustration.") 

Cheers - Jim 

Conference Paper Cutoff Sampling and Estimation for Establishment Surveys

nice information thank you

Good submissions