Computational physics
Mechanics Electromagnetics Multiphysics Particle physics Thermodynamics Simulation
Potentials Morse/Long-range potential Lennard-Jones potential Yukawa potential Morse potential
Fluid dynamics Finite difference Finite volume Finite element Boundary element Lattice Boltzmann Riemann solver Dissipative particle dynamics Smoothed particle hydrodynamics Turbulence models
Monte Carlo methods Integration Gibbs sampling Metropolis algorithm
Particle N-body Particle-in-cell Molecular dynamics
Scientists Godunov Ulam von Neumann Galerkin Lorenz Wilson Alder Richtmyer

Data analysis is the process of inspecting, cleansing, transforming, and modeling data with the goal of discovering useful information, informing conclusions, and supporting decision-making.^[1] Data analysis has multiple facets and approaches, encompassing diverse techniques under a variety of names, and is used in different business, science, and social science domains.^[2] In today's business world, data analysis plays a role in making decisions more scientific and helping businesses operate more effectively.^[3]

Data mining is a particular data analysis technique that focuses on statistical modeling and knowledge discovery for predictive rather than purely descriptive purposes, while business intelligence covers data analysis that relies heavily on aggregation, focusing mainly on business information.^[4] In statistical applications, data analysis can be divided into descriptive statistics, exploratory data analysis (EDA), and confirmatory data analysis (CDA).^[5] EDA focuses on discovering new features in the data while CDA focuses on confirming or falsifying existing hypotheses.^[6][7] Predictive analytics focuses on the application of statistical models for predictive forecasting or classification, while text analytics applies statistical, linguistic, and structural techniques to extract and classify information from textual sources, a species of unstructured data. All of the above are varieties of data analysis.^[8]

Data integration is a precursor to data analysis, and data analysis is closely linked to data visualization and data dissemination.^[9]

Analysis refers to dividing a whole into its separate components for individual examination.^[10] Data analysis is a process for obtaining raw data, and subsequently converting it into information useful for decision-making by users.^[1] Data is collected and analyzed to answer questions, test hypotheses, or disprove theories.^[11]

Statistician John Tukey, defined data analysis in 1961, as:

"Procedures for analyzing data, techniques for interpreting the results of such procedures, ways of planning the gathering of data to make its analysis easier, more precise or more accurate, and all the machinery and results of (mathematical) statistics which apply to analyzing data."^[12]

There are several phases that can be distinguished, described below. The phases are iterative, in that feedback from later phases may result in additional work in earlier phases.^[13] The CRISP framework, used in data mining, has similar steps.

The data is necessary as inputs to the analysis, which is specified based upon the requirements of those directing the analytics (or customers, who will use the finished product of the analysis).^[14][15] The general type of entity upon which the data will be collected is referred to as an experimental unit (e.g., a person or population of people). Specific variables regarding a population (e.g., age and income) may be specified and obtained. Data may be numerical or categorical (i.e., a text label for numbers).^[13]

Data is collected from a variety of sources.^[16][17] A list of data sources are available for study & research. The requirements may be communicated by analysts to custodians of the data; such as, Information Technology personnel within an organization.^[18] Data collection or data gathering is the process of gathering and measuring information on targeted variables in an established system, which then enables one to answer relevant questions and evaluate outcomes. The data may also be collected from sensors in the environment, including traffic cameras, satellites, recording devices, etc. It may also be obtained through interviews, downloads from online sources, or reading documentation.^[13]

Data, when initially obtained, must be processed or organized for analysis.^[19][20] For instance, these may involve placing data into rows and columns in a table format (known as structured data) for further analysis, often through the use of spreadsheet(excel) or statistical software.^[13]

Once processed and organized, the data may be incomplete, contain duplicates, or contain errors.^[21][22] The need for data cleaning will arise from problems in the way that the datum are entered and stored.^[21] Data cleaning is the process of preventing and correcting these errors. Common tasks include record matching, identifying inaccuracy of data, overall quality of existing data, deduplication, and column segmentation.^[23] Such data problems can also be identified through a variety of analytical techniques. For example; with financial information, the totals for particular variables may be compared against separately published numbers that are believed to be reliable.^[24][25] Unusual amounts, above or below predetermined thresholds, may also be reviewed. There are several types of data cleaning, that are dependent upon the type of data in the set; this could be phone numbers, email addresses, employers, or other values.^[26][27] Quantitative data methods for outlier detection, can be used to get rid of data that appears to have a higher likelihood of being input incorrectly.^[28] Textual data spell checkers can be used to lessen the amount of mistyped words. However, it is harder to tell if the words themselves are correct.^[29]

Once the datasets are cleaned, they can then be analyzed. Analysts may apply a variety of techniques, referred to as exploratory data analysis, to begin understanding the messages contained within the obtained data.^[30] The process of data exploration may result in additional data cleaning or additional requests for data; thus, the initialization of the iterative phases mentioned in the lead paragraph of this section.^[31] Descriptive statistics, such as, the average or median, can be generated to aid in understanding the data.^[32][33] Data visualization is also a technique used, in which the analyst is able to examine the data in a graphical format in order to obtain additional insights, regarding the messages within the data.^[13]

Mathematical formulas or models (also known as algorithms), may be applied to the data in order to identify relationships among the variables; for example, using correlation or causation.^[34][35] In general terms, models may be developed to evaluate a specific variable based on other variable(s) contained within the dataset, with some residual error depending on the implemented model's accuracy (e.g., Data = Model + Error).^[36][11]

Inferential statistics includes utilizing techniques that measure the relationships between particular variables.^[37] For example, regression analysis may be used to model whether a change in advertising (independent variable X), provides an explanation for the variation in sales (dependent variable Y).^[38] In mathematical terms, Y (sales) is a function of X (advertising).^[39] It may be described as (Y = aX + b + error), where the model is designed such that (a) and (b) minimize the error when the model predicts Y for a given range of values of X.^[40] Analysts may also attempt to build models that are descriptive of the data, in an aim to simplify analysis and communicate results.^[11]

A data product is a computer application that takes data inputs and generates outputs, feeding them back into the environment.^[41] It may be based on a model or algorithm. For instance, an application that analyzes data about customer purchase history, and uses the results to recommend other purchases the customer might enjoy.^[42][13]

Once data is analyzed, it may be reported in many formats to the users of the analysis to support their requirements.^[44] The users may have feedback, which results in additional analysis. As such, much of the analytical cycle is iterative.^[13]

When determining how to communicate the results, the analyst may consider implementing a variety of data visualization techniques to help communicate the message more clearly and efficiently to the audience.^[45] Data visualization uses information displays (graphics such as, tables and charts) to help communicate key messages contained in the data.^[46] Tables are a valuable tool by enabling the ability of a user to query and focus on specific numbers; while charts (e.g., bar charts or line charts), may help explain the quantitative messages contained in the data.^[47]

Stephen Few described eight types of quantitative messages that users may attempt to understand or communicate from a set of data and the associated graphs used to help communicate the message.^[48] Customers specifying requirements and analysts performing the data analysis may consider these messages during the course of the process.^[49]

1. Time-series: A single variable is captured over a period of time, such as the unemployment rate over a 10-year period. A line chart may be used to demonstrate the trend.^[50]
2. Ranking: Categorical subdivisions are ranked in ascending or descending order, such as a ranking of sales performance (the measure) by salespersons (the category, with each salesperson a categorical subdivision) during a single period.^[51] A bar chart may be used to show the comparison across the salespersons.^[52]
3. Part-to-whole: Categorical subdivisions are measured as a ratio to the whole (i.e., a percentage out of 100%). A pie chart or bar chart can show the comparison of ratios, such as the market share represented by competitors in a market.^[53]
4. Deviation: Categorical subdivisions are compared against a reference, such as a comparison of actual vs. budget expenses for several departments of a business for a given time period. A bar chart can show the comparison of the actual versus the reference amount.^[54]
5. Frequency distribution: Shows the number of observations of a particular variable for a given interval, such as the number of years in which the stock market return is between intervals such as 0–10%, 11–20%, etc. A histogram, a type of bar chart, may be used for this analysis.^[55]
6. Correlation: Comparison between observations represented by two variables (X,Y) to determine if they tend to move in the same or opposite directions. For example, plotting unemployment (X) and inflation (Y) for a sample of months. A scatter plot is typically used for this message.^[56]
7. Nominal comparison: Comparing categorical subdivisions in no particular order, such as the sales volume by product code. A bar chart may be used for this comparison.^[57]
8. Geographic or geospatial: Comparison of a variable across a map or layout, such as the unemployment rate by state or the number of persons on the various floors of a building. A cartogram is a typical graphic used.^[58][59]

Author Jonathan Koomey has recommended a series of best practices for understanding quantitative data.^[60] These include:

• Check raw data for anomalies prior to performing an analysis;
• Re-perform important calculations, such as verifying columns of data that are formula driven;
• Confirm main totals are the sum of subtotals;
• Check relationships between numbers that should be related in a predictable way, such as ratios over time;
• Normalize numbers to make comparisons easier, such as analyzing amounts per person or relative to GDP or as an index value relative to a base year;
• Break problems into component parts by analyzing factors that led to the results, such as DuPont analysis of return on equity.^[25]

For the variables under examination, analysts typically obtain descriptive statistics for them, such as the mean (average), median, and standard deviation.^[61] They may also analyze the distribution of the key variables to see how the individual values cluster around the mean.^[62]

The consultants at McKinsey and Company named a technique for breaking a quantitative problem down into its component parts called the MECE principle.^[63] Each layer can be broken down into its components; each of the sub-components must be mutually exclusive of each other and collectively add up to the layer above them.^[64] The relationship is referred to as "Mutually Exclusive and Collectively Exhaustive" or MECE. For example, profit by definition can be broken down into total revenue and total cost.^[65] In turn, total revenue can be analyzed by its components, such as the revenue of divisions A, B, and C (which are mutually exclusive of each other) and should add to the total revenue (collectively exhaustive).^[66]

Analysts may use robust statistical measurements to solve certain analytical problems.^[67] Hypothesis testing is used when a particular hypothesis about the true state of affairs is made by the analyst and data is gathered to determine whether that state of affairs is true or false.^[68][69] For example, the hypothesis might be that "Unemployment has no effect on inflation", which relates to an economics concept called the Phillips Curve.^[70] Hypothesis testing involves considering the likelihood of Type I and type II errors, which relate to whether the data supports accepting or rejecting the hypothesis.^[71][72]

Regression analysis may be used when the analyst is trying to determine the extent to which independent variable X affects dependent variable Y (e.g., "To what extent do changes in the unemployment rate (X) affect the inflation rate (Y)?").^[73] This is an attempt to model or fit an equation line or curve to the data, such that Y is a function of X.^[74][75]

Necessary condition analysis (NCA) may be used when the analyst is trying to determine the extent to which independent variable X allows variable Y (e.g., "To what extent is a certain unemployment rate (X) necessary for a certain inflation rate (Y)?").^[73] Whereas (multiple) regression analysis uses additive logic where each X-variable can produce the outcome and the X's can compensate for each other (they are sufficient but not necessary),^[76] necessary condition analysis (NCA) uses necessity logic, where one or more X-variables allow the outcome to exist, but may not produce it (they are necessary but not sufficient). Each single necessary condition must be present and compensation is not possible.^[77]

Users may have particular data points of interest within a data set, as opposed to the general messaging outlined above. Such low-level user analytic activities are presented in the following table. The taxonomy can also be organized by three poles of activities: retrieving values, finding data points, and arranging data points.^{[78][79][80][81]}

Barriers to effective analysis may exist among the analysts performing the data analysis or among the audience. Distinguishing fact from opinion, cognitive biases, and innumeracy are all challenges to sound data analysis.^[82]

Effective analysis requires obtaining relevant facts to answer questions, support a conclusion or formal opinion, or test hypotheses.^[83][84] Facts by definition are irrefutable, meaning that any person involved in the analysis should be able to agree upon them.^[85] For example, in August 2010, the Congressional Budget Office (CBO) estimated that extending the Bush tax cuts of 2001 and 2003 for the 2011–2020 time period would add approximately $3.3 trillion to the national debt.^[86] Everyone should be able to agree that indeed this is what CBO reported; they can all examine the report. This makes it a fact. Whether persons agree or disagree with the CBO is their own opinion.^[87]

As another example, the auditor of a public company must arrive at a formal opinion on whether financial statements of publicly traded corporations are "fairly stated, in all material respects".^[88] This requires extensive analysis of factual data and evidence to support their opinion. When making the leap from facts to opinions, there is always the possibility that the opinion is erroneous.^[89]

There are a variety of cognitive biases that can adversely affect analysis. For example, confirmation bias is the tendency to search for or interpret information in a way that confirms one's preconceptions.^[90] In addition, individuals may discredit information that does not support their views.^[91]

Analysts may be trained specifically to be aware of these biases and how to overcome them.^[92] In his book Psychology of Intelligence Analysis, retired CIA analyst Richards Heuer wrote that analysts should clearly delineate their assumptions and chains of inference and specify the degree and source of the uncertainty involved in the conclusions.^[93] He emphasized procedures to help surface and debate alternative points of view.^[94]

Effective analysts are generally adept with a variety of numerical techniques. However, audiences may not have such literacy with numbers or numeracy; they are said to be innumerate.^[95] Persons communicating the data may also be attempting to mislead or misinform, deliberately using bad numerical techniques.^[96]

For example, whether a number is rising or falling may not be the key factor. More important may be the number relative to another number, such as the size of government revenue or spending relative to the size of the economy (GDP) or the amount of cost relative to revenue in corporate financial statements.^[97] This numerical technique is referred to as normalization^[25] or common-sizing. There are many such techniques employed by analysts, whether adjusting for inflation (i.e., comparing real vs. nominal data) or considering population increases, demographics, etc.^[98] Analysts apply a variety of techniques to address the various quantitative messages described in the section above.^[99]

Analysts may also analyze data under different assumptions or scenario. For example, when analysts perform financial statement analysis, they will often recast the financial statements under different assumptions to help arrive at an estimate of future cash flow, which they then discount to present value based on some interest rate, to determine the valuation of the company or its stock.^[100][101] Similarly, the CBO analyzes the effects of various policy options on the government's revenue, outlays and deficits, creating alternative future scenarios for key measures.^[102]

A data analytics approach can be used in order to predict energy consumption in buildings.^[103] The different steps of the data analysis process are carried out in order to realise smart buildings, where the building management and control operations including heating, ventilation, air conditioning, lighting and security are realised automatically by miming the needs of the building users and optimising resources like energy and time.^[104]

Analytics is the "extensive use of data, statistical and quantitative analysis, explanatory and predictive models, and fact-based management to drive decisions and actions." It is a subset of business intelligence, which is a set of technologies and processes that uses data to understand and analyze business performance to drive decision-making .^[105]

In education, most educators have access to a data system for the purpose of analyzing student data.^[106] These data systems present data to educators in an over-the-counter data format (embedding labels, supplemental documentation, and a help system and making key package/display and content decisions) to improve the accuracy of educators' data analyses.^[107]

This section contains rather technical explanations that may assist practitioners but are beyond the typical scope of a Wikipedia article.^[108]

The most important distinction between the initial data analysis phase and the main analysis phase, is that during initial data analysis one refrains from any analysis that is aimed at answering the original research question.^[109] The initial data analysis phase is guided by the following four questions:^[110]

The quality of the data should be checked as early as possible. Data quality can be assessed in several ways, using different types of analysis: frequency counts, descriptive statistics (mean, standard deviation, median), normality (skewness, kurtosis, frequency histograms), normal imputation is needed.^[111]

• Analysis of extreme observations: outlying observations in the data are analyzed to see if they seem to disturb the distribution.^[112]
• Comparison and correction of differences in coding schemes: variables are compared with coding schemes of variables external to the data set, and possibly corrected if coding schemes are not comparable.^[113]
• Test for common-method variance.

The choice of analyses to assess the data quality during the initial data analysis phase depends on the analyses that will be conducted in the main analysis phase.^[114]

The quality of the measurement instruments should only be checked during the initial data analysis phase when this is not the focus or research question of the study.^[115][116] One should check whether structure of measurement instruments corresponds to structure reported in the literature.

There are two ways to assess measurement quality:

• Confirmatory factor analysis
• Analysis of homogeneity (internal consistency), which gives an indication of the reliability of a measurement instrument.^[117] During this analysis, one inspects the variances of the items and the scales, the Cronbach's α of the scales, and the change in the Cronbach's alpha when an item would be deleted from a scale^[118]

After assessing the quality of the data and of the measurements, one might decide to impute missing data, or to perform initial transformations of one or more variables, although this can also be done during the main analysis phase.^[119]
Possible transformations of variables are:^[120]

• Square root transformation (if the distribution differs moderately from normal)
• Log-transformation (if the distribution differs substantially from normal)
• Inverse transformation (if the distribution differs severely from normal)
• Make categorical (ordinal / dichotomous) (if the distribution differs severely from normal, and no transformations help)

One should check the success of the randomization procedure, for instance by checking whether background and substantive variables are equally distributed within and across groups.^[121]
If the study did not need or use a randomization procedure, one should check the success of the non-random sampling, for instance by checking whether all subgroups of the population of interest are represented in sample.^[122]
Other possible data distortions that should be checked are:

• dropout (this should be identified during the initial data analysis phase)
• Item non-response (whether this is random or not should be assessed during the initial data analysis phase)
• Treatment quality (using manipulation checks).^[123]

In any report or article, the structure of the sample must be accurately described.^[124][125] It is especially important to exactly determine the structure of the sample (and specifically the size of the subgroups) when subgroup analyses will be performed during the main analysis phase.^[126]
The characteristics of the data sample can be assessed by looking at:

• Basic statistics of important variables
• Scatter plots
• Correlations and associations
• Cross-tabulations^[127]

During the final stage, the findings of the initial data analysis are documented, and necessary, preferable, and possible corrective actions are taken.^[128]
Also, the original plan for the main data analyses can and should be specified in more detail or rewritten.^[129] In order to do this, several decisions about the main data analyses can and should be made:

• In the case of non-normals: should one transform variables; make variables categorical (ordinal/dichotomous); adapt the analysis method?
• In the case of missing data: should one neglect or impute the missing data; which imputation technique should be used?
• In the case of outliers: should one use robust analysis techniques?
• In case items do not fit the scale: should one adapt the measurement instrument by omitting items, or rather ensure comparability with other (uses of the) measurement instrument(s)?
• In the case of (too) small subgroups: should one drop the hypothesis about inter-group differences, or use small sample techniques, like exact tests or bootstrapping?
• In case the randomization procedure seems to be defective: can and should one calculate propensity scores and include them as covariates in the main analyses?^[130]

Several analyses can be used during the initial data analysis phase:^[131]

• Univariate statistics (single variable)
• Bivariate associations (correlations)
• Graphical techniques (scatter plots)

It is important to take the measurement levels of the variables into account for the analyses, as special statistical techniques are available for each level:^[132]

• Nominal and ordinal variables
  • Frequency counts (numbers and percentages)
  • Associations
    • circumambulations (crosstabulations)
    • hierarchical loglinear analysis (restricted to a maximum of 8 variables)
    • loglinear analysis (to identify relevant/important variables and possible confounders)
  • Exact tests or bootstrapping (in case subgroups are small)
  • Computation of new variables
• Continuous variables
  • Distribution
    • Statistics (M, SD, variance, skewness, kurtosis)
    • Stem-and-leaf displays
    • Box plots

Nonlinear analysis is often necessary when the data is recorded from a nonlinear system. Nonlinear systems can exhibit complex dynamic effects including bifurcations, chaos, harmonics and subharmonics that cannot be analyzed using simple linear methods. Nonlinear data analysis is closely related to nonlinear system identification.^[133]

In the main analysis phase, analyses aimed at answering the research question are performed as well as any other relevant analysis needed to write the first draft of the research report.^[134]

In the main analysis phase, either an exploratory or confirmatory approach can be adopted. Usually the approach is decided before data is collected.^[135] In an exploratory analysis no clear hypothesis is stated before analysing the data, and the data is searched for models that describe the data well.^[136] In a confirmatory analysis clear hypotheses about the data are tested.^[137]

Exploratory data analysis should be interpreted carefully. When testing multiple models at once there is a high chance on finding at least one of them to be significant, but this can be due to a type 1 error.^[138] It is important to always adjust the significance level when testing multiple models with, for example, a Bonferroni correction.^[139] Also, one should not follow up an exploratory analysis with a confirmatory analysis in the same dataset.^[140] An exploratory analysis is used to find ideas for a theory, but not to test that theory as well.^[140] When a model is found exploratory in a dataset, then following up that analysis with a confirmatory analysis in the same dataset could simply mean that the results of the confirmatory analysis are due to the same type 1 error that resulted in the exploratory model in the first place.^[140] The confirmatory analysis therefore will not be more informative than the original exploratory analysis.^[141]

It is important to obtain some indication about how generalizable the results are.^[142] While this is often difficult to check, one can look at the stability of the results. Are the results reliable and reproducible? There are two main ways of doing that.^[143]

• Cross-validation. By splitting the data into multiple parts, we can check if an analysis (like a fitted model) based on one part of the data generalizes to another part of the data as well.^[144] Cross-validation is generally inappropriate, though, if there are correlations within the data, e.g. with panel data.^[145] Hence other methods of validation sometimes need to be used. For more on this topic, see statistical model validation.^[146]
• Sensitivity analysis. A procedure to study the behavior of a system or model when global parameters are (systematically) varied. One way to do that is via bootstrapping.^[147]

Notable free software for data analysis include:

• DevInfo – A database system endorsed by the United Nations Development Group for monitoring and analyzing human development.^[148]
• ELKI – Data mining framework in Java with data mining oriented visualization functions.
• KNIME – The Konstanz Information Miner, a user friendly and comprehensive data analytics framework.
• Orange – A visual programming tool featuring interactive data visualization and methods for statistical data analysis, data mining, and machine learning.
• Pandas – Python library for data analysis.
• PAW – FORTRAN/C data analysis framework developed at CERN.
• R – A programming language and software environment for statistical computing and graphics.^[149]
• ROOT – C++ data analysis framework developed at CERN.
• SciPy – Python library for scientific computing.
• Julia – A programming language well-suited for numerical analysis and computational science.

The typical data analysis workflow involves collecting data, running analyses through various scripts, creating visualizations, and writing reports. However, this workflow presents challenges, including a separation between analysis scripts and data, as well as a gap between analysis and documentation. Often, the correct order of running scripts is only described informally or resides in the data scientist's memory. The potential for losing this information creates issues for reproducibility. To address these challenges, it is essential to have analysis scripts written for automated, reproducible workflows. Additionally, dynamic documentation is crucial, providing reports that are understandable by both machines and humans, ensuring accurate representation of the analysis workflow even as scripts evolve.^[150]

Different companies or organizations hold data analysis contests to encourage researchers to utilize their data or to solve a particular question using data analysis.^[151][152] A few examples of well-known international data analysis contests are as follows:^[153]

• Kaggle competition, which is held by Kaggle.^[154]
• LTPP data analysis contest held by FHWA and ASCE.^[155][156]

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