Data analysis

File:Data visualization process v1.png

Data analysis is a process for obtaining raw data, and subsequently converting it into information useful for decision-making by users. 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."{{Cite journal |url=http://projecteuclid.org/download/pdf_1/euclid.aoms/1177704711 |title=John Tukey-The Future of Data Analysis-July 1961 |journal=The Annals of Mathematical Statistics |date=March 1962 |volume=33 |issue=1 |pages=1–67 |doi=10.1214/aoms/1177704711 |access-date=2015-01-01 |archive-date=2020-01-26 |archive-url=https://web.archive.org/web/20200126232007/https://projecteuclid.org/download/pdf_1/euclid.aoms/1177704711 |url-status=live |last1=Tukey |first1=John W. }}

There are several phases, and they are iterative, in that feedback from later phases may result in additional work in earlier phases.{{cite book

| author2-last = O'Neil | author2-first= Cathy | author2-link= Cathy O'Neil

| author1-last = Schutt | author1-first= Rachel

| year = 2013

| title = Doing Data Science | publisher = O'Reilly Media

| isbn = 978-1-449-35865-5}}

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).{{Citation|title=USE OF THE DATA|date=2015-02-06|url=http://dx.doi.org/10.1002/9781118986370.ch18|work=Handbook of Petroleum Product Analysis|pages=296–303|place=Hoboken, NJ|publisher=John Wiley & Sons, Inc|doi=10.1002/9781118986370.ch18|isbn=978-1-118-98637-0|access-date=2021-05-29}} 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).

Data may be collected from a variety of sources.{{Cite journal|title=Table 1: Data type and sources of data collected for this research.|journal=PeerJ|date=7 May 2021|volume=9|pages=e11387|doi=10.7717/peerj.11387/table-1|last1=Olusola|first1=Johnson Adedeji|last2=Shote|first2=Adebola Adekunle|last3=Ouigmane|first3=Abdellah|last4=Isaifan|first4=Rima J. |doi-access=free }} 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.{{Citation|last=MacPherson|first=Derek|title=Information Technology Analysts' Perspectives|date=2019-10-16|url=http://dx.doi.org/10.4324/9780429437564-12|work=Data Strategy in Colleges and Universities|pages=168–183|publisher=Routledge|doi=10.4324/9780429437564-12|isbn=978-0-429-43756-4|s2cid=211738958|access-date=2021-05-29}} 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.

File:Relationship of data, information and intelligence.png used to convert raw information into actionable intelligence or knowledge are conceptually similar to the phases in data analysis.]]

Data integration is a precursor to data analysis: Data, when initially obtained, must be processed or organized for analysis. For instance, this 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.

{{Main|Data cleansing}}

Once processed and organized, the data may be incomplete, contain duplicates, or contain errors.{{Cite journal |last=Bohannon|first=John|date=2016-02-24|title=Many surveys, about one in five, may contain fraudulent data |journal=Science|doi=10.1126/science.aaf4104|issn=0036-8075|doi-access=free}} The need for data cleaning will arise from problems in the way that the data is entered and stored.{{Cite journal |last1=Hancock |first1=R.G.V. |last2=Carter |first2=Tristan |date=February 2010|title=How reliable are our published archaeometric analyses? Effects of analytical techniques through time on the elemental analysis of obsidians |url=http://dx.doi.org/10.1016/j.jas.2009.10.004|journal=Journal of Archaeological Science|volume=37|issue=2|pages=243–250|doi=10.1016/j.jas.2009.10.004|bibcode=2010JArSc..37..243H |issn=0305-4403}} 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.{{cite web|title=Data Cleaning |url=http://research.microsoft.com/en-us/projects/datacleaning/ |publisher=Microsoft Research |access-date=26 October 2013|archive-date=29 October 2013 |archive-url=https://web.archive.org/web/20131029200356/http://research.microsoft.com/en-us/projects/datacleaning/|url-status=live}}{{cite journal |last=Hellerstein |first=Joseph |title=Quantitative Data Cleaning for Large Databases|journal=EECS Computer Science Division|date=27 February 2008|page=3|url=http://db.cs.berkeley.edu/jmh/papers/cleaning-unece.pdf|access-date=26 October 2013|archive-date=13 October 2013|archive-url=https://web.archive.org/web/20131013011223/http://db.cs.berkeley.edu/jmh/papers/cleaning-unece.pdf|url-status=live}}

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.{{Cite web |url=http://www.perceptualedge.com/articles/b-eye/quantitative_data.pdf |title=Perceptual Edge-Jonathan Koomey-Best practices for understanding quantitative data-February 14, 2006 |access-date=November 12, 2014 |archive-date=October 5, 2014 |archive-url=https://web.archive.org/web/20141005075112/http://www.perceptualedge.com/articles/b-eye/quantitative_data.pdf |url-status=live }} 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.{{Cite journal |last1=Peleg |first1=Roni |last2=Avdalimov |first2=Angelika |last3=Freud |first3=Tamar|date=2011-03-23|title=Providing cell phone numbers and email addresses to Patients: the physician's perspective|journal=BMC Research Notes|volume=4|issue=1|page=76|doi=10.1186/1756-0500-4-76|pmid=21426591|issn=1756-0500|pmc=3076270 |doi-access=free }} 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. Text data spell checkers can be used to lessen the amount of mistyped words. However, it is harder to tell if the words are contextually (i.e., semantically and idiomatically) correct.

Once the datasets are cleaned, they can then begin to be analyzed using exploratory data analysis. The process of data exploration may result in additional data cleaning or additional requests for data; thus, the initialization of the iterative phases mentioned above.{{Cite journal|date=December 1999|title=FTC requests additional data|url=http://dx.doi.org/10.1016/s1359-6128(99)90509-8|journal=Pump Industry Analyst|volume=1999|issue=48|page=12|doi=10.1016/s1359-6128(99)90509-8|issn=1359-6128}} Descriptive statistics, such as the average, median, and standard deviation, are often used to broadly characterize the data.{{Cite journal|date=2017|title=Exploring your Data with Data Visualization & Descriptive Statistics: Common Descriptive Statistics for Quantitative Data|url=http://dx.doi.org/10.4135/9781529732795|doi=10.4135/9781529732795}}{{Cite book |last=Murray|first=Daniel G.|title=Tableau your data! : fast and easy visual analysis with Tableau Software|date=2013|publisher=J. Wiley & Sons |isbn=978-1-118-61204-0|oclc=873810654}} Data visualization is also used, in which the analyst is able to examine the data in a graphical format in order to obtain additional insights about messages within the data.

Mathematical formulas or models (also known as algorithms), may be applied to the data in order to identify relationships among the variables; for example, checking for correlation and by determining whether or not there is the presence of causality. 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).{{Cite journal|title=Figure 2. Variable importance by permutation, averaged over 25 models.|journal=eLife|date=28 February 2017|volume=6|pages=e22053|doi=10.7554/elife.22053.004|last1=Evans|first1=Michelle V.|last2=Dallas|first2=Tad A.|last3=Han|first3=Barbara A.|last4=Murdock|first4=Courtney C.|last5=Drake|first5=John M. |editor1=Brady, Oliver |doi-access=free }}

Inferential statistics utilizes techniques that measure the relationships between particular variables.{{Cite journal|title=Table 3: Descriptive (mean ± SD), inferential (95% CI) and qualitative statistics (ES) of all variables between self-selected and predetermined conditions |journal=PeerJ|date=12 November 2020|volume=8|pages=e10361|doi=10.7717/peerj.10361/table-3 |last1=Watson|first1=Kevin|last2=Halperin|first2=Israel|last3=Aguilera-Castells|first3=Joan|last4=Iacono|first4=Antonio Dello |doi-access=free }} 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), i.e. is Y a function of X? This can 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.{{Cite journal|last=Nwabueze|first=JC|date=2008-05-21|title=Performances of estimators of linear model with auto-correlated error terms when the independent variable is normal |url=http://dx.doi.org/10.4314/jonamp.v9i1.40071 |journal=Journal of the Nigerian Association of Mathematical Physics |volume=9 |issue=1|doi=10.4314/jonamp.v9i1.40071|issn=1116-4336}}

A data product is a computer application that takes data inputs and generates outputs, feeding them back into the environment.{{Cite journal|last=Conway|first=Steve|date=2012-07-04|title=A Cautionary Note on Data Inputs and Visual Outputs in Social Network Analysis|url=http://dx.doi.org/10.1111/j.1467-8551.2012.00835.x|journal=British Journal of Management |volume=25|issue=1|pages=102–117|doi=10.1111/j.1467-8551.2012.00835.x|hdl=2381/36068|s2cid=154347514|issn=1045-3172}} 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.{{Citation|title=Customer Purchases and Other Repeated Events|date=2016-01-29|url=http://dx.doi.org/10.1002/9781119183419.ch8|work=Data Analysis Using SQL and Excel®|pages=367–420|place=Indianapolis, Indiana|publisher=John Wiley & Sons, Inc.|doi=10.1002/9781119183419.ch8|isbn=978-1-119-18341-9|access-date=2021-05-31}}

File:Social Network Analysis Visualization.png is used to help understand the results after data is analyzed.{{Cite journal| volume = 10| issue = 3| last = Grandjean| first = Martin| title = La connaissance est un réseau| journal = Les Cahiers du Numérique| date = 2014| pages = 37–54| url = http://www.martingrandjean.ch/wp-content/uploads/2015/02/Grandjean-2014-Connaissance-reseau.pdf| doi = 10.3166/lcn.10.3.37-54| access-date = 2015-05-05| archive-date = 2015-09-27| archive-url = https://web.archive.org/web/20150927170721/http://www.martingrandjean.ch/wp-content/uploads/2015/02/Grandjean-2014-Connaissance-reseau.pdf| url-status = live}}]]

{{Main|Data and information visualization}}

Once data is analyzed, it may be reported in many formats to the users of the analysis to support their requirements.{{Citation|title=Data requirements for semiconductor die. Exchange data formats and data dictionary|url=http://dx.doi.org/10.3403/02271298|publisher=BSI British Standards|doi=10.3403/02271298|access-date=2021-05-31}} The users may have feedback, which results in additional analysis.

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. Data visualization uses information displays (graphics such as, tables and charts) to help communicate key messages contained in the data. 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.{{Cite book|date=2021|title=Visualizing Data About UK Museums: Bar Charts, Line Charts and Heat Maps |url=http://dx.doi.org/10.4135/9781529768749 |doi=10.4135/9781529768749 |isbn=9781529768749 |s2cid=240967380}}

{{Main|Data and information visualization}}

File:Total Revenues and Outlays as Percent GDP 2013.png

File:U.S. Phillips Curve 2000 to 2013.png

Stephen Few described eight types of quantitative messages that users may attempt to communicate from a set of data, including the associated graphs.{{Cite web |url=http://www.perceptualedge.com/articles/ie/the_right_graph.pdf |title=Stephen Few-Perceptual Edge-Selecting the Right Graph for Your Message-2004 |access-date=2014-10-29 |archive-date=2014-10-05 |archive-url=https://web.archive.org/web/20141005080924/http://www.perceptualedge.com/articles/ie/the_right_graph.pdf |url-status=live }}{{Cite web |url=http://www.perceptualedge.com/articles/misc/Graph_Selection_Matrix.pdf |title=Stephen Few-Perceptual Edge-Graph Selection Matrix |access-date=2014-10-29 |archive-date=2014-10-05 |archive-url=https://web.archive.org/web/20141005080945/http://www.perceptualedge.com/articles/misc/Graph_Selection_Matrix.pdf |url-status=live }}

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.
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. A bar chart may be used to show the comparison across the salespersons.{{cite book | doi=10.1007/1-4020-0612-8_1063 | chapter=X-Bar Chart | title=Encyclopedia of Production and Manufacturing Management | date=2000 | page=841 | isbn=978-0-7923-8630-8 | last1=Swamidass | first1=P. M. }}
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.{{Cite journal |title=Chart C5.3. Percentage of 15-19 year-olds not in education, by labour market status (2012)|url=http://dx.doi.org/10.1787/888933119055|access-date=2021-06-03|doi=10.1787/888933119055}}
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.{{Cite journal|title=Chart 7: Households: final consumption expenditure versus actual individual consumption|url=http://dx.doi.org/10.1787/665527077310|access-date=2021-06-03|doi=10.1787/665527077310}}
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.
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.{{Cite journal|title=Table 2: Graph comparison between Scatter plot, Violin + Scatter plot, Heatmap and ViSiElse graph.|journal=PeerJ|date=3 February 2020|volume=8|pages=e8341|doi=10.7717/peerj.8341/table-2|last1=Garnier|first1=Elodie M.|last2=Fouret|first2=Nastasia|last3=Descoins|first3=Médéric |doi-access=free }}
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.{{Cite journal|date=2009|title=Product comparison chart: Wearables |url=http://dx.doi.org/10.1037/e539162010-006|access-date=2021-06-03|website=PsycEXTRA Dataset|doi=10.1037/e539162010-006}}
8. Geographic or geo-spatial: 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 typically used.

{{See also|Problem solving}}

Author Jonathan Koomey has recommended a series of best practices for understanding quantitative data. 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.

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

File:US_Employment_Statistics_-_March_2015.png used for data analysis]]

McKinsey and Company named a technique for breaking down a quantitative problem into its component parts called the MECE principle. MECE means "Mutually Exclusive and Collectively Exhaustive".{{Citation|title=Consultants Employed by McKinsey & Company|date=2008-07-30|url=http://dx.doi.org/10.4324/9781315701974-15|work=Organizational Behavior 5|pages=77–82|publisher=Routledge|doi=10.4324/9781315701974-15|isbn=978-1-315-70197-4|access-date=2021-06-03}} 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. For example, profit by definition can be broken down into total revenue and total cost.{{Cite journal|last=Carey|first=Malachy|date=November 1981|title=On Mutually Exclusive and Collectively Exhaustive Properties of Demand Functions |url=http://dx.doi.org/10.2307/2553697 |journal=Economica |volume=48|issue=192|pages=407–415|doi=10.2307/2553697|jstor=2553697|issn=0013-0427}}

Analysts may use robust statistical measurements to solve certain analytical problems. 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 hypothesis is true or false.{{Cite journal|last=Heckman|date=1978|title=Simple Statistical Models for Discrete Panel Data Developed and Applied to Test the Hypothesis of True State Dependence against the Hypothesis of Spurious State Dependence |url=http://dx.doi.org/10.2307/20075292|journal=Annales de l'inséé|issue=30/31|pages=227–269 |doi=10.2307/20075292 |jstor=20075292 |issn=0019-0209}} For example, the hypothesis might be that "Unemployment has no effect on inflation", which relates to an economics concept called the Phillips Curve.{{Citation|last=Munday|first=Stephen C. R. |title=Unemployment, Inflation and the Phillips Curve|date=1996|url=http://dx.doi.org/10.1007/978-1-349-24986-2_11|work=Current Developments in Economics|pages=186–218|place=London|publisher=Macmillan Education UK|doi=10.1007/978-1-349-24986-2_11|isbn=978-0-333-64444-7|access-date=2021-06-03}} 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.{{Cite journal |last=Louangrath|first=Paul I.|date=2013|title=Alpha and Beta Tests for Type I and Type II Inferential Errors Determination in Hypothesis Testing |url=http://dx.doi.org/10.2139/ssrn.2332756|journal=SSRN Electronic Journal |doi=10.2139/ssrn.2332756|issn=1556-5068}}

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)?").{{Cite journal|last=Yanamandra|first=Venkataramana|date=September 2015|title=Exchange rate changes and inflation in India: What is the extent of exchange rate pass-through to imports?|url=http://dx.doi.org/10.1016/j.eap.2015.07.004 |journal=Economic Analysis and Policy |volume=47 |pages=57–68 |doi=10.1016/j.eap.2015.07.004|issn=0313-5926}}

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)?"). 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),{{Cite web |url=https://doi.org/10.1049%2Fiet-tv.48.859 |last=Feinmann|first=Jane|title=How Can Engineers and Journalists Help Each Other?|access-date=2021-06-03|doi=10.1049/iet-tv.48.859|url-access=subscription|type=Video|publisher=The Institute of Engineering & Technology}} 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.{{Cite journal|last=Dul|first=Jan|date=2015|title=Necessary Condition Analysis (NCA): Logic and Methodology of 'Necessary But Not Sufficient' Causality|url=http://dx.doi.org/10.2139/ssrn.2588480|journal=SSRN Electronic Journal|doi=10.2139/ssrn.2588480|hdl=1765/77890|s2cid=219380122|issn=1556-5068}}

File:User-activities.png

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.Robert Amar, James Eagan, and John Stasko (2005) [http://www.cc.gatech.edu/~stasko/papers/infovis05.pdf "Low-Level Components of Analytic Activity in Information Visualization"] {{Webarchive|url=https://web.archive.org/web/20150213074349/http://www.cc.gatech.edu/~stasko/papers/infovis05.pdf |date=2015-02-13 }}William Newman (1994) [http://www.mdnpress.com/wmn/pdfs/chi94-pro-formas-2.pdf "A Preliminary Analysis of the Products of HCI Research, Using Pro Forma Abstracts"] {{Webarchive|url=https://web.archive.org/web/20160303212019/http://www.mdnpress.com/wmn/pdfs/chi94-pro-formas-2.pdf |date=2016-03-03 }}Mary Shaw (2002) [https://www.cs.cmu.edu/~Compose/ftp/shaw-fin-etaps.pdf "What Makes Good Research in Software Engineering?"] {{Webarchive|url=https://web.archive.org/web/20181105042928/http://www.cs.cmu.edu/~Compose/ftp/shaw-fin-etaps.pdf |date=2018-11-05 }}

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.{{Cite journal|date=July 1989|title=Connectivity tool transfers data among database and statistical products|url=http://dx.doi.org/10.1016/0167-9473(89)90021-2|journal=Computational Statistics & Data Analysis|volume=8|issue=2|pages=224|doi=10.1016/0167-9473(89)90021-2|issn=0167-9473}}

{{quote box|quote=You are entitled to your own opinion, but you are not entitled to your own facts.|source=Daniel Patrick Moynihan|width = 250px}}

Effective analysis requires obtaining relevant facts to answer questions, support a conclusion or formal opinion, or test hypotheses.{{Citation|title=Information relevant to your job|date=2007-07-11 |url=http://dx.doi.org/10.4324/9780080544304-16 |work=Obtaining Information for Effective Management|pages=48–54 |publisher=Routledge|doi=10.4324/9780080544304-16|doi-broken-date=17 December 2024 |isbn=978-0-08-054430-4|access-date=2021-06-03}} Facts by definition are irrefutable, meaning that any person involved in the analysis should be able to agree upon them. 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".{{Cite journal|last=Gordon|first=Roger|date=March 1990|title=Do Publicly Traded Corporations Act in the Public Interest?|url=http://dx.doi.org/10.3386/w3303|location=Cambridge, MA|doi=10.3386/w3303 |journal=National Bureau of Economic Research Working Papers}} This requires extensive analysis of factual data and evidence to support their opinion.

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.{{Cite thesis|title=Confirmation bias in witness interviewing: Can interviewers ignore their preconceptions?|url=http://dx.doi.org/10.25148/etd.fi14071109|publisher=Florida International University|first=Jillian R|last=Rivard|year=2014 |doi=10.25148/etd.fi14071109}} In addition, individuals may discredit information that does not support their views.{{Citation|last=Papineau|first=David|title=Does the Sociology of Science Discredit Science?|date=1988|url=http://dx.doi.org/10.1007/978-94-009-2877-0_2|work=Relativism and Realism in Science|pages=37–57|place=Dordrecht|publisher=Springer Netherlands|doi=10.1007/978-94-009-2877-0_2|isbn=978-94-010-7795-8|access-date=2021-06-03}}

Analysts may be trained specifically to be aware of these biases and how to overcome them.{{Cite book|date=2005|editor-last=Bromme|editor-first=Rainer|editor2-last=Hesse|editor2-first=Friedrich W.|editor3-last=Spada|editor3-first=Hans|title=Barriers and Biases in Computer-Mediated Knowledge Communication |url=http://dx.doi.org/10.1007/b105100|doi=10.1007/b105100|isbn=978-0-387-24317-7}} 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.{{Cite book |last=Heuer |first=Richards |editor1-first=Richards J|editor1-last=Heuer|date=2019-06-10|title=Quantitative Approaches to Political Intelligence|url=http://dx.doi.org/10.4324/9780429303647|doi=10.4324/9780429303647|isbn=9780429303647|s2cid=145675822}} He emphasized procedures to help surface and debate alternative points of view.{{cite web |url=https://www.cia.gov/static/9a5f1162fd0932c29bfed1c030edf4ae/Pyschology-of-Intelligence-Analysis.pdf |title=Introduction|publisher=Central Intelligence Agency|access-date=2021-10-25|archive-date=2021-10-25|archive-url=https://web.archive.org/web/20211025160526/https://www.cia.gov/static/9a5f1162fd0932c29bfed1c030edf4ae/Pyschology-of-Intelligence-Analysis.pdf|url-status=live}}

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.{{Cite web |title=Figure 6.7. Differences in literacy scores across OECD countries generally mirror those in numeracy|url=http://dx.doi.org/10.1787/888934081549|access-date=2021-06-03|doi=10.1787/888934081549}} Persons communicating the data may also be attempting to mislead or misinform, deliberately using bad numerical techniques.{{Cite web |url=http://www.bloombergview.com/articles/2014-10-28/bad-math-that-passes-for-insight |title=Bad Math that Passes for Insight |last=Ritholz |first=Barry |work=Bloomberg View |access-date=2014-10-29 |archive-date=2014-10-29 |archive-url=https://web.archive.org/web/20141029014527/http://www.bloombergview.com/articles/2014-10-28/bad-math-that-passes-for-insight |url-status=dead }}

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.{{Cite journal |last1=Gusnaini |first1=Nuriska |last2=Andesto |first2=Rony |last3=Ermawati|date=2020-12-15|title=The Effect of Regional Government Size, Legislative Size, Number of Population, and Intergovernmental Revenue on The Financial Statements Disclosure |url=http://dx.doi.org/10.24018/ejbmr.2020.5.6.651 |journal=European Journal of Business and Management Research |volume=5 |issue=6 |doi=10.24018/ejbmr.2020.5.6.651 |s2cid=231675715|issn=2507-1076}} This numerical technique is referred to as normalization 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.{{cite book |last1=Taura |first1=Toshiharu |last2=Nagai |first2=Yukari |title=Design Creativity 2010 |date=2011 |publisher=Springer-Verlag London |location=London |isbn=978-0-85729-223-0 |pages=165-171 |chapter=Comparing Nominal Groups to Real Teams}}

Analysts may also analyze data under different assumptions or scenarios. 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.{{Cite journal|last=Gross|first=William H.|date=July 1979|title=Coupon Valuation and Interest Rate Cycles |journal=Financial Analysts Journal|volume=35|issue=4|pages=68–71 |doi=10.2469/faj.v35.n4.68|issn=0015-198X}} 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.{{Cite web |title=25. General government total outlays|url=http://dx.doi.org/10.1787/888932348795|access-date=2021-06-03|doi=10.1787/888932348795}}

{{Main|Analytics}}

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.{{cite book|last1=Davenport|last2=Harris|first1=Thomas|first2=Jeanne|year=2007|title=Competing on Analytics|publisher = O'Reilly|isbn = 978-1-4221-0332-6}}

In education, most educators have access to a data system for the purpose of analyzing student data.Aarons, D. (2009). [https://www.proquest.com/docview/202710770 Report finds states on course to build pupil-data systems.] Education Week, 29(13), 6. 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.Rankin, J. (2013, March 28). [https://sas.elluminate.com/site/external/recording/playback/link/table/dropin?sid=2008350&suid=D.4DF60C7117D5A77FE3AED546909ED2 How data Systems & reports can either fight or propagate the data analysis error epidemic, and how educator leaders can help.] {{Webarchive|url=https://web.archive.org/web/20190326201414/https://sas.elluminate.com/site/external/recording/playback/link/table/dropin?sid=2008350&suid=D.4DF60C7117D5A77FE3AED546909ED2 |date=2019-03-26 }} Presentation conducted from Technology Information Center for Administrative Leadership (TICAL) School Leadership Summit.

This section contains rather technical explanations that may assist practitioners but are beyond the typical scope of a Wikipedia article.{{Citation|last=Brödermann|first=Eckart J.|title=Article 2.2.1 (Scope of the Section) |date=2018 |url=http://dx.doi.org/10.5771/9783845276564-525|work=Commercial Law|pages=525|publisher=Nomos Verlagsgesellschaft mbH & Co. KG|doi=10.5771/9783845276564-525|isbn=978-3-8452-7656-4|access-date=2021-06-03}}

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. The initial data analysis phase is guided by the following four questions:{{sfn|Adèr|2008a|p=337}}

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.{{Cite journal|title=Descriptive statistics indicating the mean, standard deviation and frequency of missing values for each condition (N = number of participants), and for the dependent variables (DV)|journal=PeerJ|date = 19 December 2013|volume = 1|pages = e231|doi=10.7717/peerj.231/table-1|last1 = Kjell|first1 = Oscar N. E.|last2 = Thompson|first2 = Sam | doi-access=free }}

• Analysis of extreme observations: outlying observations in the data are analyzed to see if they seem to disturb the distribution.{{Citation|title=Practice for Dealing With Outlying Observations|url=http://dx.doi.org/10.1520/e0178-16a|publisher=ASTM International|doi=10.1520/e0178-16a|access-date=2021-06-03}}
• 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.{{Citation|title=Alternative Coding Schemes for Dummy Variables|url=http://dx.doi.org/10.4135/9781412985628.n5|work=Regression with Dummy Variables|year=1993|pages=64–75|location=Newbury Park, CA|publisher=SAGE Publications, Inc.|doi=10.4135/9781412985628.n5|isbn=978-0-8039-5128-0|access-date=2021-06-03}}
• 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.{{sfn|Adèr|2008a|pp=338-341}}

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.{{Cite book|first=Isadore|last=Newman|title=Qualitative-quantitative research methodology : exploring the interactive continuum|date=1998|publisher=Southern Illinois University Press|isbn=0-585-17889-5|oclc=44962443}} 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.{{Cite journal|last1=Terwilliger|first1=James S.|last2=Lele|first2=Kaustubh|title=Some Relationships Among Internal Consistency, Reproducibility, and Homogeneity|date=June 1979|url=http://dx.doi.org/10.1111/j.1745-3984.1979.tb00091.x|journal=Journal of Educational Measurement|volume=16|issue=2|pages=101–108|doi=10.1111/j.1745-3984.1979.tb00091.x|issn=0022-0655}} 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{{sfn|Adèr|2008a|pp=341-342}}

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.{{sfn|Adèr|2008a|p=344}}

Possible transformations of variables are:Tabachnick & Fidell, 2007, p. 87-88.

• 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. 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 the sample.{{Citation|title=Random sampling and randomization procedures |url=http://dx.doi.org/10.3403/30137438 |publisher=BSI British Standards|doi=10.3403/30137438|access-date=2021-06-03}}
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).{{sfn|Adèr|2008a|pp=344-345}}

In any report or article, the structure of the sample must be accurately described. It is especially important to exactly determine the size of the subgroup when subgroup analyses will be performed during the main analysis phase.{{Cite journal|title=Figure 4: Centroid size regression analyses for the main sample |journal=PeerJ|date=18 January 2016 |volume=4 |pages=e1589 |doi=10.7717/peerj.1589/fig-4|last1=Foth|first1=Christian|last2=Hedrick|first2=Brandon P.|last3=Ezcurra|first3=Martin D. |doi-access=free }}
The characteristics of the data sample can be assessed by looking at:

• Basic statistics of important variables
• Scatter plots
• Correlations and associations
• Cross-tabulations{{sfn|Adèr|2008a|p=345}}

During the final stage, the findings of the initial data analysis are documented, and necessary, preferable, and possible corrective actions are taken. Also, the original plan for the main data analyses can and should be specified in more detail or rewritten. 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?{{sfn|Adèr|2008a|pp=345-346}}

Several analyses can be used during the initial data analysis phase:{{sfn|Adèr|2008a|pp=346-347}}

• 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:{{sfn|Adèr|2008a|pp=349-353}}

• 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.Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013

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.{{sfn|Adèr|2008b|p=363}}

In the main analysis phase, either an exploratory or confirmatory approach can be adopted. Usually the approach is decided before data is collected.{{Citation|title=Exploratory Data Analysis |date=2017-10-13 |url=http://dx.doi.org/10.1002/9781119126805.ch4 |work=Python® for R Users|pages=119–138|place=Hoboken, NJ, USA |publisher=John Wiley & Sons, Inc.|doi=10.1002/9781119126805.ch4|hdl=11380/971504|isbn=978-1-119-12680-5|access-date=2021-06-03}} 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.{{Citation|title=Engaging in Exploratory Data Analysis, Visualization, and Hypothesis Testing – Exploratory Data Analysis, Geovisualization, and Data|date=2015-07-28|url=http://dx.doi.org/10.1201/b18808-8|work=Spatial Analysis|pages=106–139|publisher=CRC Press|doi=10.1201/b18808-8|isbn=978-0-429-06936-9|s2cid=133412598 |access-date=2021-06-03}} In a confirmatory analysis, clear hypotheses about the data are tested.{{Citation|title=Hypotheses About Categories|url=http://dx.doi.org/10.4135/9781446287873.n14|work=Starting Statistics: A Short, Clear Guide|year=2010|pages=138–151|location=London|publisher=SAGE Publications Ltd |doi=10.4135/9781446287873.n14 |isbn=978-1-84920-098-1|access-date=2021-06-03}}

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. It is important to always adjust the significance level when testing multiple models with, for example, a Bonferroni correction.{{Cite journal|last1=Liquet|first1=Benoit |last2=Riou |first2=Jérémie|date=2013-06-08|title=Correction of the significance level when attempting multiple transformations of an explanatory variable in generalized linear models|journal=BMC Medical Research Methodology|volume=13|issue=1|page=75|doi=10.1186/1471-2288-13-75|pmid=23758852|pmc=3699399|issn=1471-2288 |doi-access=free }} Also, one should not follow up an exploratory analysis with a confirmatory analysis in the same dataset.{{Cite journal|last=Mcardle|first=John J.|date=2008|title=Some ethical issues in confirmatory versus exploratory analysis|url=http://dx.doi.org/10.1037/e503312008-001|access-date=2021-06-03|website=PsycEXTRA Dataset|doi=10.1037/e503312008-001}} An exploratory analysis is used to find ideas for a theory, but not to test that theory as well. 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. The confirmatory analysis therefore will not be more informative than the original exploratory analysis.{{sfn|Adèr|2008b|pp=361-362}}

It is important to obtain some indication about how generalizable the results are.{{sfn|Adèr|2008b|pp=361-371}} 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.

• 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.{{cite journal

| last1 = Benson | first1 = Noah C

| last2 = Winawer | first2 = Jonathan

| date = December 2018

| doi = 10.7554/elife.40224

| journal = eLife

| title = Bayesian analysis of retinotopic maps

| volume = 7| doi-access = free

| pmid = 30520736

| pmc = 6340702

}} Supplementary file 1. Cross-validation schema. {{doi|10.7554/elife.40224.014}} Cross-validation is generally inappropriate, though, if there are correlations within the data, e.g. with panel data.{{Citation|last=Hsiao|first=Cheng|title=Cross-Sectionally Dependent Panel Data|url=http://dx.doi.org/10.1017/cbo9781139839327.012|work=Analysis of Panel Data|year=2014|pages=327–368|place=Cambridge|publisher=Cambridge University Press|doi=10.1017/cbo9781139839327.012|isbn=978-1-139-83932-7|access-date=2021-06-03}} Hence other methods of validation sometimes need to be used. For more on this topic, see statistical model validation.{{Citation|last=Hjorth|first=J.S. Urban|title=Cross validation|date=2017-10-19|url=http://dx.doi.org/10.1201/9781315140056-3|work=Computer Intensive Statistical Methods|pages=24–56|publisher=Chapman and Hall/CRC|doi=10.1201/9781315140056-3|isbn=978-1-315-14005-6|access-date=2021-06-03}}

• 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.{{Cite journal |last1=Sheikholeslami|first1=Razi|last2=Razavi|first2=Saman|last3=Haghnegahdar|first3=Amin|date=2019-10-10|title=What should we do when a model crashes? Recommendations for global sensitivity analysis of Earth and environmental systems models|journal=Geoscientific Model Development|volume=12|issue=10|pages=4275–4296|doi=10.5194/gmd-12-4275-2019|bibcode=2019GMD....12.4275S|s2cid=204900339|issn=1991-9603 |doi-access=free }}

Free software for data analysis include:

• DevInfo – A database system endorsed by the United Nations Development Group for monitoring and analyzing human development.{{Cite book|chapter=Human development composite indices|title= Human Development Indices and Indicators 2018|pages=21–41|doi=10.18356/ce6f8e92-en|s2cid=240207510|author=United Nations Development Programme|date= 2018|publisher=United Nations }}
• 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.{{Citation|last1=Wiley|first1=Matt|title=Multivariate Data Visualization|date=2019|url=http://dx.doi.org/10.1007/978-1-4842-2872-2_2|work=Advanced R Statistical Programming and Data Models|pages=33–59|place=Berkeley, CA|publisher=Apress|isbn=978-1-4842-2871-5|access-date=2021-06-03|last2=Wiley|first2=Joshua F.|doi=10.1007/978-1-4842-2872-2_2|s2cid=86629516}}
• 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, 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 document analysis script content and workflow. Additionally, overall documentation is crucial, as well as providing reports that are understandable by both machines and humans, and ensuring accurate representation of the analysis workflow even as scripts evolve.{{Cite book |last=Mailund |first=Thomas |title=Beginning Data Science in R 4: Data Analysis, Visualization, and Modelling for the Data Scientist |year=2022 |isbn=978-148428155-0 |edition=2nd}}

Different companies and organizations hold data analysis contests to encourage researchers to utilize their data or to solve a particular question using data analysis. A few examples of well-known international data analysis contests are:

• Kaggle competitions; the Kaggle platform is owned and run by Google.{{cite news |title=The machine learning community takes on the Higgs |url=http://www.symmetrymagazine.org/article/july-2014/the-machine-learning-community-takes-on-the-higgs/ |access-date=14 January 2015|newspaper=Symmetry Magazine|date=July 15, 2014|archive-date=16 April 2021|archive-url=https://web.archive.org/web/20210416100455/https://www.symmetrymagazine.org/article/july-2014/the-machine-learning-community-takes-on-the-higgs|url-status=live}}
• LTPP data analysis contest{{cite web |date = May 26, 2016 |url = https://www.fhwa.dot.gov/research/tfhrc/programs/infrastructure/pavements/ltpp/ |title = Data.Gov:Long-Term Pavement Performance (LTPP) |access-date = November 10, 2017 |archive-date = November 1, 2017 |archive-url = https://web.archive.org/web/20171101191727/https://www.fhwa.dot.gov/research/tfhrc/programs/infrastructure/pavements/ltpp/ |url-status = live }} held by FHWA and ASCE.{{cite web |first = Jean |last = Nehme |date = September 29, 2016 |url = https://www.fhwa.dot.gov/research/tfhrc/programs/infrastructure/pavements/ltpp/2016_2017_asce_ltpp_contest_guidelines.cfm |title = LTPP International Data Analysis Contest |publisher = Federal Highway Administration |access-date = October 22, 2017 |archive-date = October 21, 2017 |archive-url = https://web.archive.org/web/20171021010012/https://www.fhwa.dot.gov/research/tfhrc/programs/infrastructure/pavements/ltpp/2016_2017_asce_ltpp_contest_guidelines.cfm |url-status = live }}

{{Div col|colwidth=20em}}

• Actuarial science
• Analytics
• Augmented Analytics
• Business intelligence
• Data presentation architecture
• Exploratory data analysis
• Machine learning
• Multiway data analysis
• Qualitative research
• Structured data analysis (statistics)
• Text mining
• Unstructured data
• List of datasets for machine-learning research

{{Div col end}}

{{Reflist}}

• {{cite book |author-link1=Herman J. Adèr |first1=Herman J. |last1=Adèr |editor-first1=Herman J. |editor-last1=Adèr |editor-link2=Gideon J. Mellenbergh |editor-first2=Gideon J. |editor-last2=Mellenbergh |editor-link3=David Hand (statistician) |editor-first3=David J |editor-last3=Hand |title=Advising on research methods : a consultant's companion |publisher=Johannes van Kessel Pub |location=Huizen, Netherlands |year=2008a |isbn=9789079418015 |oclc=905799857 |chapter=Chapter 14: Phases and initial steps in data analysis |pages=333–356 }}
• {{cite book |author-link1=Herman J. Adèr |first1=Herman J. |last1=Adèr |editor-first1=Herman J. |editor-last1=Adèr |editor-link2=Gideon J. Mellenbergh |editor-first2=Gideon J. |editor-last2=Mellenbergh |editor-link3=David Hand (statistician) |editor-first3=David J |editor-last3=Hand |title=Advising on research methods : a consultant's companion |publisher=Johannes van Kessel Pub |location=Huizen, Netherlands |year=2008b |isbn=9789079418015 |oclc=905799857 |chapter=Chapter 15: The main analysis phase |pages=357–386 }}
• Tabachnick, B.G. & Fidell, L.S. (2007). Chapter 4: Cleaning up your act. Screening data prior to analysis. In B.G. Tabachnick & L.S. Fidell (Eds.), Using Multivariate Statistics, Fifth Edition (pp. 60–116). Boston: Pearson Education, Inc. / Allyn and Bacon.

{{wikiversity}}

• Adèr, H.J. & Mellenbergh, G.J. (with contributions by D.J. Hand) (2008). Advising on Research Methods: A Consultant's Companion. Huizen, the Netherlands: Johannes van Kessel Publishing. {{ISBN|978-90-79418-01-5}}
• Chambers, John M.; Cleveland, William S.; Kleiner, Beat; Tukey, Paul A. (1983). Graphical Methods for Data Analysis, Wadsworth/Duxbury Press. {{ISBN|0-534-98052-X}}
• Fandango, Armando (2017). Python Data Analysis, 2nd Edition. Packt Publishers. {{ISBN|978-1787127487}}
• Juran, Joseph M.; Godfrey, A. Blanton (1999). Juran's Quality Handbook, 5th Edition. New York: McGraw Hill. {{ISBN|0-07-034003-X}}
• Lewis-Beck, Michael S. (1995). Data Analysis: an Introduction, Sage Publications Inc, {{ISBN|0-8039-5772-6}}
• NIST/SEMATECH (2008) [http://www.itl.nist.gov/div898/handbook/ Handbook of Statistical Methods]
• Pyzdek, T, (2003). Quality Engineering Handbook, {{ISBN|0-8247-4614-7}}
• Richard Veryard (1984). Pragmatic Data Analysis. Oxford : Blackwell Scientific Publications. {{ISBN|0-632-01311-7}}
• Tabachnick, B.G.; Fidell, L.S. (2007). Using Multivariate Statistics, 5th Edition. Boston: Pearson Education, Inc. / Allyn and Bacon, {{ISBN|978-0-205-45938-4}}

{{data}}

{{Authority control}}

Category:Data processing

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