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Statistics For Data Science | Data Science Tutorial | Simplilearn

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Statistics is a branch of applied mathematics, that is the study and manipulation of data, including ways to gather, review, analyze, and draw conclusions. This is why statistics still holds a very important place in todayâ€™s data science and business intelligence world. In this statistics tutorial, you will learn all about statistics, including, percentile in statistics, what is normal distribution, Central Limit Theorem, and probability density function. Start learning the statistics tutorial now.
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âžˇď¸Ź About Caltech Post Graduate Program In Data Science
This Post Graduation in Data Science leverages the superiority of Caltech's academic eminence. The Data Science program covers critical Data Science topics like Python programming, R programming, Machine Learning, Deep Learning, and Data Visualization tools through an interactive learning model with live sessions by global practitioners and practical labs.
âś… Key Features
- Simplilearn's JobAssist helps you get noticed by top hiring companies
- Caltech PG program in Data Science completion certificate
- Earn up to 14 CEUs from Caltech CTME
- Masterclasses delivered by distinguished Caltech faculty and IBM experts
- Caltech CTME Circle membership
- Online convocation by Caltech CTME Program Director
- IBM certificates for IBM courses
- Access to hackathons and Ask Me Anything sessions from IBM
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- Seamless access to integrated labs
- Capstone projects in 3 domains
- Simplilearnâ€™s Career Assistance to help you get noticed by top hiring companies
- 8X higher interaction in live online classes by industry experts
âś… Skills Covered
- Exploratory Data Analysis
- Descriptive Statistics
- Inferential Statistics
- Model Building and Fine Tuning
- Supervised and Unsupervised Learning
- Ensemble Learning
- Deep Learning
- Data Visualization
đź”Ą Learn More: https://www.simplilearn.com/big-data-and-analytics/senior-data-scientist-masters-program-training?utm_campaign=StatisticsForDataScience&utm_medium=Description&utm_source=youtube
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{'title': 'Statistics For Data Science | Data Science Tutorial | Simplilearn', 'heatmap': [{'end': 136.671, 'start': 82.443, 'weight': 0.825}, {'end': 243.754, 'start': 192.911, 'weight': 0.786}, {'end': 317.661, 'start': 250.611, 'weight': 0.796}, {'end': 659.789, 'start': 642.521, 'weight': 0.72}, {'end': 889.338, 'start': 869.282, 'weight': 0.723}, {'end': 938.84, 'start': 921.581, 'weight': 0.924}, {'end': 987.494, 'start': 945.563, 'weight': 0.774}, {'end': 1030.734, 'start': 994.198, 'weight': 0.915}], 'summary': 'Covers the importance of statistics in data analysis, introduction to sas data analysis tools, descriptive and inferential statistics, different statistical concepts, hypothesis testing with sas, and statistical tests including parametric and non-parametric tests, providing comprehensive insights into statistical analysis for data science.', 'chapters': [{'end': 317.661, 'segs': [{'end': 37.604, 'src': 'embed', 'start': 9.406, 'weight': 0, 'content': [{'end': 12.848, 'text': "Let's begin this lesson by defining the term statistics.", 'start': 9.406, 'duration': 3.442}, {'end': 20.673, 'text': 'Statistics is a mathematical science pertaining to the collection, presentation, analysis, and interpretation of data.', 'start': 13.529, 'duration': 7.144}, {'end': 28.298, 'text': "It's widely used to understand the complex problems of the real world and simplify them to make well-informed decisions.", 'start': 21.454, 'duration': 6.844}, {'end': 37.604, 'text': 'Several statistical principles, functions, and algorithms can be used to analyze primary data, build a statistical model, and predict the outcomes.', 'start': 29.098, 'duration': 8.506}], 'summary': 'Statistics is a mathematical science used to analyze and interpret data for informed decision-making.', 'duration': 28.198, 'max_score': 9.406, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Lv0xcdeXaGU/pics/Lv0xcdeXaGU9406.jpg'}, {'end': 107.578, 'src': 'embed', 'start': 82.443, 'weight': 1, 'content': [{'end': 88.386, 'text': 'There are two major categories of statistics, descriptive statistics and inferential statistics.', 'start': 82.443, 'duration': 5.943}, {'end': 94.709, 'text': 'Descriptive statistics helps organize data and focuses on the main characteristics of the data.', 'start': 89.386, 'duration': 5.323}, {'end': 98.692, 'text': 'It provides a summary of the data numerically or graphically.', 'start': 95.45, 'duration': 3.242}, {'end': 107.578, 'text': 'Numerical measures such as average, mode, standard deviation or SD, and correlation are used to describe the features of a dataset.', 'start': 99.453, 'duration': 8.125}], 'summary': 'Descriptive and inferential statistics organize data using numerical measures and graphics.', 'duration': 25.135, 'max_score': 82.443, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Lv0xcdeXaGU/pics/Lv0xcdeXaGU82443.jpg'}, {'end': 159.923, 'src': 'heatmap', 'start': 82.443, 'weight': 2, 'content': [{'end': 88.386, 'text': 'There are two major categories of statistics, descriptive statistics and inferential statistics.', 'start': 82.443, 'duration': 5.943}, {'end': 94.709, 'text': 'Descriptive statistics helps organize data and focuses on the main characteristics of the data.', 'start': 89.386, 'duration': 5.323}, {'end': 98.692, 'text': 'It provides a summary of the data numerically or graphically.', 'start': 95.45, 'duration': 3.242}, {'end': 107.578, 'text': 'Numerical measures such as average, mode, standard deviation or SD, and correlation are used to describe the features of a dataset.', 'start': 99.453, 'duration': 8.125}, {'end': 111.627, 'text': 'Suppose you want to study the height of students in a classroom.', 'start': 108.703, 'duration': 2.924}, {'end': 119.599, 'text': 'In the descriptive statistics, you would record the height of every person in the classroom and then find out the maximum height,', 'start': 112.388, 'duration': 7.211}, {'end': 122.102, 'text': 'minimum height and average height of the population.', 'start': 119.599, 'duration': 2.503}, {'end': 129.568, 'text': 'Inferential statistics generalizes the larger dataset and applies probability theory to draw a conclusion.', 'start': 123.208, 'duration': 6.36}, {'end': 136.671, 'text': 'It allows you to infer population parameters based on the sample statistics and to model relationships within the data.', 'start': 130.31, 'duration': 6.361}, {'end': 143.953, 'text': 'Modeling allows you to develop mathematical equations which describe the interrelationships between two or more variables.', 'start': 137.511, 'duration': 6.442}, {'end': 148.495, 'text': 'Consider the same example of calculating the height of students in the classroom.', 'start': 144.733, 'duration': 3.762}, {'end': 153.258, 'text': 'In inferential statistics, you would categorize height as tall,', 'start': 149.076, 'duration': 4.182}, {'end': 159.923, 'text': 'medium and small and then take only a small sample from the population to study the height of students in the classroom.', 'start': 153.258, 'duration': 6.665}], 'summary': 'Descriptive statistics organize data; inferential statistics generalize and apply probability theory to draw conclusions.', 'duration': 77.48, 'max_score': 82.443, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Lv0xcdeXaGU/pics/Lv0xcdeXaGU82443.jpg'}, {'end': 249.87, 'src': 'heatmap', 'start': 172.776, 'weight': 3, 'content': [{'end': 177.219, 'text': 'There are various statistical terms that one should be aware of while dealing with statistics.', 'start': 172.776, 'duration': 4.443}, {'end': 186.306, 'text': 'Population, sample, variable, quantitative variable, qualitative variable, discrete variable, continuous variable.', 'start': 178.06, 'duration': 8.246}, {'end': 191.17, 'text': 'A population is the group from which data is to be collected.', 'start': 188.028, 'duration': 3.142}, {'end': 194.993, 'text': 'A sample is a subset of a population.', 'start': 192.911, 'duration': 2.082}, {'end': 204.973, 'text': 'A variable is a feature that is characteristic of any member of the population differing in quality or quantity from another member.', 'start': 198.007, 'duration': 6.966}, {'end': 209.778, 'text': 'A variable differing in quantity is called a quantitative variable.', 'start': 206.154, 'duration': 3.624}, {'end': 213.461, 'text': 'For example, the weight of a person, number of people in a car.', 'start': 210.278, 'duration': 3.183}, {'end': 219.787, 'text': 'A variable differing in quality is called a qualitative variable or attribute.', 'start': 215.523, 'duration': 4.264}, {'end': 224.251, 'text': 'For example, color, the degree of damage of a car in an accident.', 'start': 220.307, 'duration': 3.944}, {'end': 233.804, 'text': 'A discrete variable is one which no value can be assumed between the two given values, for example, the number of children in a family.', 'start': 225.615, 'duration': 8.189}, {'end': 243.754, 'text': 'A continuous variable is one in which any value can be assumed between the two given values, for example, the time taken for a 100-meter run.', 'start': 235.265, 'duration': 8.489}, {'end': 249.87, 'text': 'Typically, there are four types of statistical measures used to describe the data.', 'start': 245.649, 'duration': 4.221}], 'summary': 'Understanding statistical terms: population, sample, variable, and types of statistical measures.', 'duration': 77.094, 'max_score': 172.776, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Lv0xcdeXaGU/pics/Lv0xcdeXaGU172776.jpg'}, {'end': 330.367, 'src': 'heatmap', 'start': 245.649, 'weight': 4, 'content': [{'end': 249.87, 'text': 'Typically, there are four types of statistical measures used to describe the data.', 'start': 245.649, 'duration': 4.221}, {'end': 257.512, 'text': 'They are measures of frequency, measures of central tendency, measures of spread, measures of position.', 'start': 250.611, 'duration': 6.901}, {'end': 259.654, 'text': "Let's learn each in detail.", 'start': 258.392, 'duration': 1.262}, {'end': 266.216, 'text': 'Frequency of the data indicates the number of times a particular data value occurs in the given data set.', 'start': 260.793, 'duration': 5.423}, {'end': 269.737, 'text': 'The measures of frequency are number and percentage.', 'start': 267.096, 'duration': 2.641}, {'end': 277.39, 'text': 'Central tendency indicates whether the data values tend to accumulate in the middle of the distribution or toward the end.', 'start': 271.246, 'duration': 6.144}, {'end': 282.313, 'text': 'The measures of central tendency are mean, median, and mode.', 'start': 278.33, 'duration': 3.983}, {'end': 289.417, 'text': 'Spread describes how similar or varied the set of observed values are for a particular variable.', 'start': 284.094, 'duration': 5.323}, {'end': 294.3, 'text': 'The measures of spread are standard deviation, variance, and quartiles.', 'start': 290.197, 'duration': 4.103}, {'end': 298.062, 'text': 'The measure of spread are also called measures of dispersion.', 'start': 295.18, 'duration': 2.882}, {'end': 304.249, 'text': 'Position identifies the exact location of a particular data value in the given data set.', 'start': 299.465, 'duration': 4.784}, {'end': 309.314, 'text': 'The measures of position are percentiles, quartiles, and standard scores.', 'start': 305.07, 'duration': 4.244}, {'end': 316.04, 'text': 'Statistical Analysis System, or SAS, provides a list of procedures to perform descriptive statistics.', 'start': 309.935, 'duration': 6.105}, {'end': 317.661, 'text': 'They are as follows.', 'start': 316.801, 'duration': 0.86}, {'end': 330.367, 'text': 'PROC PRINT, PROC CONTENTS, PROC MEANS, PROC FREQUENCY, PROC UNIVARIATE, PROC GCHART, PROC BOXPLOT, PROC GPLOT.', 'start': 318.522, 'duration': 11.845}], 'summary': 'Descriptive statistics cover frequency, central tendency, spread, and position using measures like mean, median, and standard deviation. sas offers procedures for statistical analysis.', 'duration': 84.718, 'max_score': 245.649, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Lv0xcdeXaGU/pics/Lv0xcdeXaGU245649.jpg'}], 'start': 9.406, 'title': 'Importance of statistics in data analysis', 'summary': 'Covers the significance of statistics in simplifying real-world problems, making informed decisions, identifying patterns, and trends in data. it also discusses descriptive and inferential statistics and introduces four types of statistical measures.', 'chapters': [{'end': 81.423, 'start': 9.406, 'title': 'Understanding statistics for analysis', 'summary': 'Discusses the importance of statistics in analyzing data, including its role in simplifying complex real-world problems and making well-informed decisions. it highlights the use of statistical analysis in identifying patterns and trends, and the significance of statistical analysis over non-statistical analysis in providing clearer insights for businesses.', 'duration': 72.017, 'highlights': ['Statistical analysis is the science of collecting, exploring, and presenting large amounts of data to identify the patterns and trends, providing clearer insights for businesses.', 'Statistics is a mathematical science used to understand complex real-world problems and simplify them to make well-informed decisions.', 'Several statistical principles, functions, and algorithms can be used to analyze primary data, build a statistical model, and predict the outcomes.']}, {'end': 317.661, 'start': 82.443, 'title': 'Understanding descriptive and inferential statistics', 'summary': 'Discusses the two major categories of statistics - descriptive statistics and inferential statistics, providing examples and defining various statistical terms while also introducing the four types of statistical measures used to describe the data.', 'duration': 235.218, 'highlights': ['Descriptive statistics focuses on the main characteristics of the data and provides a summary numerically or graphically. Descriptive statistics helps organize data and provides a summary through numerical measures such as average, mode, standard deviation, and correlation.', 'Inferential statistics applies probability theory to draw conclusions and allows the inference of population parameters based on sample statistics. Inferential statistics generalizes the larger dataset, applies probability theory, and allows the inference of population parameters based on sample statistics.', 'Various statistical terms such as population, sample, variable, quantitative variable, qualitative variable, discrete variable, and continuous variable are defined. The chapter defines various statistical terms, including population, sample, and different types of variables such as quantitative, qualitative, discrete, and continuous.', 'The four types of statistical measures used to describe the data are measures of frequency, measures of central tendency, measures of spread, and measures of position. The chapter introduces the four types of statistical measures used to describe the data, including frequency, central tendency, spread, and position.', 'The measures of frequency include number and percentage, while the measures of central tendency include mean, median, and mode. The chapter explains that measures of frequency include number and percentage, while measures of central tendency include mean, median, and mode.']}], 'duration': 308.255, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Lv0xcdeXaGU/pics/Lv0xcdeXaGU9406.jpg', 'highlights': ['Statistics simplifies real-world problems, providing clearer insights for businesses.', 'Descriptive statistics organizes data and provides a summary through numerical measures.', 'Inferential statistics generalizes the larger dataset and allows the inference of population parameters.', 'The chapter defines various statistical terms, including population, sample, and different types of variables.', 'The chapter introduces the four types of statistical measures used to describe the data.']}, {'end': 480.927, 'segs': [{'end': 374.902, 'src': 'embed', 'start': 318.522, 'weight': 0, 'content': [{'end': 330.367, 'text': 'PROC PRINT, PROC CONTENTS, PROC MEANS, PROC FREQUENCY, PROC UNIVARIATE, PROC GCHART, PROC BOXPLOT, PROC GPLOT.', 'start': 318.522, 'duration': 11.845}, {'end': 332.068, 'text': 'PROC PRINT.', 'start': 331.447, 'duration': 0.621}, {'end': 335.129, 'text': 'It prints all the variables in a SAS dataset.', 'start': 332.728, 'duration': 2.401}, {'end': 337.43, 'text': 'PROC CONTENTS.', 'start': 336.549, 'duration': 0.881}, {'end': 339.931, 'text': 'It describes the structure of a dataset.', 'start': 338.07, 'duration': 1.861}, {'end': 342.094, 'text': 'PROC MEANS.', 'start': 341.493, 'duration': 0.601}, {'end': 350.941, 'text': 'It provides data summarization tools to compute descriptive statistics for variables across all observations and within the groups of observations.', 'start': 342.814, 'duration': 8.127}, {'end': 353.723, 'text': 'PROC FREQUENCY.', 'start': 352.903, 'duration': 0.82}, {'end': 358.748, 'text': 'It produces one-way to n-way frequency and cross-tabulation tables.', 'start': 354.464, 'duration': 4.284}, {'end': 362.591, 'text': 'Frequencies can also be an output of a SAS dataset.', 'start': 359.468, 'duration': 3.123}, {'end': 374.902, 'text': 'ProcUnivariate. It goes beyond what ProcMeans does and is useful in conducting some basic statistical analyses, and includes high-resolution graphical features.', 'start': 365.151, 'duration': 9.751}], 'summary': 'Sas procedures include print, contents, means, frequency, univariate for data summarization and descriptive statistics.', 'duration': 56.38, 'max_score': 318.522, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Lv0xcdeXaGU/pics/Lv0xcdeXaGU318522.jpg'}, {'end': 451.104, 'src': 'embed', 'start': 423.061, 'weight': 2, 'content': [{'end': 430.843, 'text': 'overlay plots in which multiple sets of data points are displayed on one set of axes, plots against the second, vertical axis,', 'start': 423.061, 'duration': 7.782}, {'end': 433.024, 'text': 'bubble plots and logarithmic plots.', 'start': 430.843, 'duration': 2.181}, {'end': 440.206, 'text': "In this demo, you'll learn how to use descriptive statistics to analyze the mean from the electronic data set.", 'start': 434.224, 'duration': 5.982}, {'end': 444.187, 'text': "Let's import the electronic data set into the SAS console.", 'start': 441.066, 'duration': 3.121}, {'end': 451.104, 'text': 'In the left plane, right-click the Electronic.xlsx dataset and click Import Data.', 'start': 445.12, 'duration': 5.984}], 'summary': 'Learn to overlay plots, analyze mean from electronic data set with descriptive statistics in sas.', 'duration': 28.043, 'max_score': 423.061, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Lv0xcdeXaGU/pics/Lv0xcdeXaGU423061.jpg'}], 'start': 318.522, 'title': 'Sas data analysis tools', 'summary': 'Introduces various sas procedures including proc means, proc frequency, and proc gplot, offering data summarization, frequency tables, and graphical representations for statistical analysis on sas datasets.', 'chapters': [{'end': 480.927, 'start': 318.522, 'title': 'Sas data analysis tools', 'summary': 'Introduces various sas procedures such as proc means, proc frequency, and proc gplot, which provide data summarization, frequency tables, and graphical representations for statistical analysis on sas datasets.', 'duration': 162.405, 'highlights': ['PROC MEANS provides data summarization tools to compute descriptive statistics for variables across all observations and within the groups of observations. It provides data summarization tools to compute descriptive statistics for variables across all observations and within the groups of observations.', 'PROC FREQUENCY produces one-way to n-way frequency and cross-tabulation tables, and can also be an output of a SAS dataset. It produces one-way to n-way frequency and cross-tabulation tables. Frequencies can also be an output of a SAS dataset.', 'PROC GPLOT creates two-dimensional graphs including simple scatter plots, overlay plots, plots against the second vertical axis, bubble plots, and logarithmic plots. It creates two-dimensional graphs including simple scatter plots, overlay plots, plots against the second vertical axis, bubble plots, and logarithmic plots.']}], 'duration': 162.405, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Lv0xcdeXaGU/pics/Lv0xcdeXaGU318522.jpg', 'highlights': ['PROC MEANS provides data summarization tools to compute descriptive statistics for variables across all observations and within the groups of observations.', 'PROC FREQUENCY produces one-way to n-way frequency and cross-tabulation tables, and can also be an output of a SAS dataset.', 'PROC GPLOT creates two-dimensional graphs including simple scatter plots, overlay plots, plots against the second vertical axis, bubble plots, and logarithmic plots.']}, {'end': 614.43, 'segs': [{'end': 528.997, 'src': 'embed', 'start': 483.489, 'weight': 0, 'content': [{'end': 485.651, 'text': 'The output obtained is shown on the screen.', 'start': 483.489, 'duration': 2.162}, {'end': 495.739, 'text': 'Note that the number of observations mean standard deviation and maximum and minimum values of the electronic data set are obtained.', 'start': 488.073, 'duration': 7.666}, {'end': 503.673, 'text': 'This concludes the demo on how to use descriptive statistics to analyze the mean from the electronic data set.', 'start': 497.668, 'duration': 6.005}, {'end': 507.216, 'text': 'So far, you have learned about descriptive statistics.', 'start': 504.314, 'duration': 2.902}, {'end': 510.018, 'text': "Let's now learn about inferential statistics.", 'start': 507.897, 'duration': 2.121}, {'end': 521.751, 'text': 'Hypothesis testing is an inferential statistical technique to determine whether there is enough evidence in a data sample to infer that a certain condition holds true for the entire population.', 'start': 511.141, 'duration': 10.61}, {'end': 528.997, 'text': 'To understand the characteristics of the general population, we take a random sample and analyze the properties of the sample.', 'start': 522.53, 'duration': 6.467}], 'summary': 'Demonstrated descriptive statistics with electronic data set, now moving on to inferential statistics and hypothesis testing.', 'duration': 45.508, 'max_score': 483.489, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Lv0xcdeXaGU/pics/Lv0xcdeXaGU483489.jpg'}, {'end': 593.905, 'src': 'embed', 'start': 553.663, 'weight': 1, 'content': [{'end': 557.084, 'text': 'Hypothesis testing is formulated in terms of two hypotheses.', 'start': 553.663, 'duration': 3.421}, {'end': 561.165, 'text': 'Null hypothesis, which is referred to as H null.', 'start': 557.984, 'duration': 3.181}, {'end': 564.446, 'text': 'Alternative hypothesis, which is referred to as H1.', 'start': 562.025, 'duration': 2.421}, {'end': 570.928, 'text': 'The null hypothesis is assumed to be true unless there is strong evidence to the contrary.', 'start': 566.206, 'duration': 4.722}, {'end': 576.551, 'text': 'The alternative hypothesis is assumed to be true when the null hypothesis is proven false.', 'start': 571.829, 'duration': 4.722}, {'end': 582.473, 'text': "Let's understand the null hypothesis and alternative hypothesis using a general example.", 'start': 577.691, 'duration': 4.782}, {'end': 591.718, 'text': 'Null hypothesis attempts to show that no variation exists between variables and alternative hypothesis is any hypothesis other than the null.', 'start': 583.614, 'duration': 8.104}, {'end': 593.905, 'text': 'For example, say,', 'start': 592.605, 'duration': 1.3}], 'summary': 'Hypothesis testing involves null and alternative hypotheses to test for variation between variables.', 'duration': 40.242, 'max_score': 553.663, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Lv0xcdeXaGU/pics/Lv0xcdeXaGU553663.jpg'}], 'start': 483.489, 'title': 'Descriptive and inferential statistics', 'summary': 'Covers the application of descriptive statistics to analyze electronic data sets and provides an introduction to inferential statistics, emphasizing the significance of null and alternative hypotheses.', 'chapters': [{'end': 614.43, 'start': 483.489, 'title': 'Descriptive and inferential statistics', 'summary': 'Covers the application of descriptive statistics to analyze electronic data sets, and provides an introduction to inferential statistics and hypothesis testing, emphasizing the significance of null and alternative hypotheses in making inferences about population parameters.', 'duration': 130.941, 'highlights': ['Hypothesis testing is used to determine if there is enough evidence in a data sample to make inferences about the entire population, and involves choosing between null and alternative hypotheses. Hypothesis testing is a technique to determine evidence in a data sample for making inferences about the population, and it involves choosing between null and alternative hypotheses.', 'The chapter explains the significance of null and alternative hypotheses in hypothesis testing, where the null hypothesis is assumed true unless there is strong evidence to the contrary. The significance of null and alternative hypotheses in hypothesis testing is explained, where the null hypothesis is assumed true unless strong evidence is found.', "An example is provided to illustrate the concepts of null and alternative hypotheses, emphasizing that the null hypothesis attempts to show no variation between variables, while the alternative hypothesis represents any hypothesis other than the null. An example is provided to illustrate the concepts of null and alternative hypotheses, emphasizing the null hypothesis's attempt to show no variation between variables, and the alternative hypothesis representing any other hypothesis.", 'The demo on using descriptive statistics to analyze the mean from an electronic data set is covered, including obtaining the number of observations, mean, standard deviation, and maximum and minimum values of the data set. The demo on using descriptive statistics to analyze the mean from an electronic data set is covered, including obtaining the number of observations, mean, standard deviation, maximum, and minimum values.']}], 'duration': 130.941, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Lv0xcdeXaGU/pics/Lv0xcdeXaGU483489.jpg', 'highlights': ['Hypothesis testing is used to determine if there is enough evidence in a data sample to make inferences about the entire population, and involves choosing between null and alternative hypotheses.', 'The chapter explains the significance of null and alternative hypotheses in hypothesis testing, where the null hypothesis is assumed true unless there is strong evidence to the contrary.', 'An example is provided to illustrate the concepts of null and alternative hypotheses, emphasizing that the null hypothesis attempts to show no variation between variables, while the alternative hypothesis represents any hypothesis other than the null.', 'The demo on using descriptive statistics to analyze the mean from an electronic data set is covered, including obtaining the number of observations, mean, standard deviation, and maximum and minimum values of the data set.']}, {'end': 834.126, 'segs': [{'end': 701.168, 'src': 'heatmap', 'start': 642.521, 'weight': 0, 'content': [{'end': 647.884, 'text': "Nominal variables are ones which have two or more categories, and it's impossible to order the values.", 'start': 642.521, 'duration': 5.363}, {'end': 651.885, 'text': 'Examples of nominal variables include gender and blood group.', 'start': 648.664, 'duration': 3.221}, {'end': 655.787, 'text': 'Ordinal variables have values ordered logically.', 'start': 653.146, 'duration': 2.641}, {'end': 659.789, 'text': 'However, the relative distance between two data values is not clear.', 'start': 656.327, 'duration': 3.462}, {'end': 669.693, 'text': 'Examples of ordinal variables include considering the size of a coffee cup large, medium and small, and considering the ratings of a product bad,', 'start': 660.569, 'duration': 9.124}, {'end': 670.594, 'text': 'good and best.', 'start': 669.693, 'duration': 0.901}, {'end': 678.591, 'text': 'Interval variables are similar to ordinal variables except that the values are measured in a way where their differences are meaningful.', 'start': 671.88, 'duration': 6.711}, {'end': 684.931, 'text': 'With an interval scale, equal differences between scale values do have equal quantitative meaning.', 'start': 679.485, 'duration': 5.446}, {'end': 690.757, 'text': 'For this reason, an interval scale provides more quantitative information than the ordinal scale.', 'start': 685.672, 'duration': 5.085}, {'end': 694.021, 'text': 'The interval scale does not have a true zero point.', 'start': 691.598, 'duration': 2.423}, {'end': 701.168, 'text': 'A true zero point means that a value of zero on the scale represents zero quantity of the construct being assessed.', 'start': 694.681, 'duration': 6.487}], 'summary': 'Nominal variables have unordered categories, ordinal variables have logical order, and interval variables have meaningful differences.', 'duration': 85.998, 'max_score': 642.521, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Lv0xcdeXaGU/pics/Lv0xcdeXaGU642521.jpg'}, {'end': 834.126, 'src': 'embed', 'start': 724.403, 'weight': 3, 'content': [{'end': 729.528, 'text': 'For example, the system of inches used with a common ruler is an example of a ratio scale.', 'start': 724.403, 'duration': 5.125}, {'end': 735.873, 'text': 'There is a true zero point because zero inches does in fact indicate a complete absence of length.', 'start': 730.508, 'duration': 5.365}, {'end': 741.924, 'text': "In this demo, you'll learn how to perform the hypothesis testing using SAS.", 'start': 737.461, 'duration': 4.463}, {'end': 749.089, 'text': "In this example, let's check against the length of certain observations from a random sample.", 'start': 744.066, 'duration': 5.023}, {'end': 753.592, 'text': 'The keyword data identifies the input dataset.', 'start': 750.59, 'duration': 3.002}, {'end': 761.298, 'text': 'The input statement is used to declare the aging variable and cards to read data into SAS.', 'start': 755.834, 'duration': 5.464}, {'end': 777.619, 'text': "Let's perform a t-test to check the null hypothesis.", 'start': 774.597, 'duration': 3.022}, {'end': 790.269, 'text': "Let's assume that the null hypothesis to be that the mean days to deliver a product is 6 days.", 'start': 784.465, 'duration': 5.804}, {'end': 796.434, 'text': 'So, null hypothesis equals 6.', 'start': 793.311, 'duration': 3.123}, {'end': 806.647, 'text': 'Alpha value is the probability of making an error, which is 5% standard, and hence alpha equals 0.05.', 'start': 796.434, 'duration': 10.213}, {'end': 810.35, 'text': 'The variable statement names the variable to be used in the analysis.', 'start': 806.647, 'duration': 3.703}, {'end': 823.539, 'text': 'The output is shown on the screen.', 'start': 821.617, 'duration': 1.922}, {'end': 831.284, 'text': 'Note that the p-value is greater than the alpha value, which is 0.05.', 'start': 826.04, 'duration': 5.244}, {'end': 834.126, 'text': 'Therefore, we fail to reject the null hypothesis.', 'start': 831.284, 'duration': 2.842}], 'summary': 'Using sas, a t-test was performed to check the null hypothesis that the mean days to deliver a product is 6 days, with a 5% alpha level. the p-value was greater than the alpha value, resulting in the failure to reject the null hypothesis.', 'duration': 109.723, 'max_score': 724.403, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Lv0xcdeXaGU/pics/Lv0xcdeXaGU724403.jpg'}], 'start': 615.17, 'title': 'Different statistical concepts', 'summary': 'Covers the significance of recognizing variable types (nominal, ordinal, interval, ratio), differences between interval and ratio scales, and performing sas t-test to test the null hypothesis with a 5% alpha value.', 'chapters': [{'end': 684.931, 'start': 615.17, 'title': 'Types of variables in statistics', 'summary': 'Discusses the significance of recognizing the nature of variables before performing statistical tests and classifies variables into four types: nominal, ordinal, interval, and ratio.', 'duration': 69.761, 'highlights': ['The chapter emphasizes the significance of recognizing the nature of variables before performing statistical tests.', "Nominal variables, such as gender and blood group, are ones which have two or more categories and it's impossible to order the values.", 'Examples of ordinal variables include considering the size of a coffee cup large, medium, and small, and considering the ratings of a product bad, good, and best.', 'Interval variables are similar to ordinal variables except that the values are measured in a way where their differences are meaningful.']}, {'end': 749.089, 'start': 685.672, 'title': 'Types of measurement scales', 'summary': 'Explains the differences between interval and ratio scales, highlighting the presence of a true zero point in ratio scales, with examples of fahrenheit and inches scales and their applications in temperature and length measurements.', 'duration': 63.417, 'highlights': ['Ratio scales have a true zero point, as demonstrated by the example of the system of inches used with a common ruler, where zero inches indicates a complete absence of length.', 'Interval scales provide more quantitative information than ordinal scales, with examples such as the Fahrenheit scale used to measure temperature and distance between two compartments in a train.', 'The chapter also briefly mentions performing hypothesis testing using SAS and checking the length of certain observations from a random sample in an example.']}, {'end': 834.126, 'start': 750.59, 'title': 'Sas t-test analysis', 'summary': 'Covers performing a t-test in sas to test the null hypothesis that the mean days to deliver a product is 6 days, with a 5% alpha value, and concludes that the null hypothesis is not rejected as the p-value is greater than alpha.', 'duration': 83.536, 'highlights': ['The p-value is greater than the alpha value of 0.05, leading to the failure to reject the null hypothesis. The p-value is compared with the alpha value of 0.05, resulting in the conclusion of failing to reject the null hypothesis.', 'The null hypothesis states that the mean days to deliver a product is 6 days. The null hypothesis is set as the mean days to deliver a product being 6 days.', 'The alpha value for the test is 0.05, representing the probability of error. The alpha value for the test is 0.05, indicating the probability of error in the test.']}], 'duration': 218.956, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Lv0xcdeXaGU/pics/Lv0xcdeXaGU615170.jpg', 'highlights': ['Nominal variables have two or more categories and cannot be ordered.', 'Recognizing the nature of variables is significant before performing statistical tests.', 'Interval variables have meaningful differences between values.', 'Ratio scales have a true zero point, demonstrated by the example of the system of inches.', 'Interval scales provide more quantitative information than ordinal scales.', 'The p-value is compared with the alpha value of 0.05, resulting in the conclusion of failing to reject the null hypothesis.', 'The null hypothesis states that the mean days to deliver a product is 6 days.', 'The alpha value for the test is 0.05, representing the probability of error.']}, {'end': 1000.541, 'segs': [{'end': 868.402, 'src': 'embed', 'start': 837.003, 'weight': 0, 'content': [{'end': 841.446, 'text': 'This concludes the demo on how to perform the hypothesis testing using SAS.', 'start': 837.003, 'duration': 4.443}, {'end': 845.688, 'text': "Let's now learn about hypothesis testing procedures.", 'start': 843.047, 'duration': 2.641}, {'end': 848.85, 'text': 'There are two types of hypothesis testing procedures.', 'start': 846.289, 'duration': 2.561}, {'end': 852.553, 'text': 'They are parametric tests and nonparametric tests.', 'start': 849.551, 'duration': 3.002}, {'end': 861.939, 'text': 'In statistical inference or hypothesis testing, the traditional tests, such as t-test and ANOVA, are called parametric tests.', 'start': 853.773, 'duration': 8.166}, {'end': 868.402, 'text': 'They depend on the specification of a probability distribution except for a set of free parameters.', 'start': 862.719, 'duration': 5.683}], 'summary': 'Demo on hypothesis testing in sas: covers parametric and nonparametric tests.', 'duration': 31.399, 'max_score': 837.003, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Lv0xcdeXaGU/pics/Lv0xcdeXaGU837003.jpg'}, {'end': 894.202, 'src': 'heatmap', 'start': 869.282, 'weight': 0.723, 'content': [{'end': 877.567, 'text': 'In simple words, you can say that if the population information is known completely by its parameter, then it is called a parametric test.', 'start': 869.282, 'duration': 8.285}, {'end': 886.034, 'text': 'if the population or parameter information is not known and you are still required to test the hypothesis of the population,', 'start': 878.547, 'duration': 7.487}, {'end': 889.338, 'text': "then it's called a non-parametric test.", 'start': 886.034, 'duration': 3.304}, {'end': 894.202, 'text': 'non-parametric tests do not require any strict distributional assumptions.', 'start': 889.338, 'duration': 4.864}], 'summary': 'Parametric tests use known population information, while non-parametric tests do not require strict distributional assumptions.', 'duration': 24.92, 'max_score': 869.282, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Lv0xcdeXaGU/pics/Lv0xcdeXaGU869282.jpg'}, {'end': 944.283, 'src': 'heatmap', 'start': 916.578, 'weight': 1, 'content': [{'end': 920.6, 'text': 'To test if the mean is significantly different than a hypothesized value.', 'start': 916.578, 'duration': 4.022}, {'end': 925.363, 'text': 'To test if the mean for two independent groups is significantly different.', 'start': 921.581, 'duration': 3.782}, {'end': 931.106, 'text': 'To test if the mean for two dependent or paired groups is significantly different.', 'start': 926.463, 'duration': 4.643}, {'end': 938.84, 'text': "For example, let's say you have to find out which region spends the highest amount of money on shopping.", 'start': 933.117, 'duration': 5.723}, {'end': 944.283, 'text': "It's impractical to ask everyone in the different regions about their shopping expenditure.", 'start': 939.74, 'duration': 4.543}], 'summary': 'Testing mean differences for independent and dependent groups in practical research scenarios.', 'duration': 27.705, 'max_score': 916.578, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Lv0xcdeXaGU/pics/Lv0xcdeXaGU916578.jpg'}, {'end': 987.494, 'src': 'heatmap', 'start': 945.563, 'weight': 0.774, 'content': [{'end': 951.746, 'text': 'In this case, you can calculate the highest shopping expenditure by collecting sample observations from each region.', 'start': 945.563, 'duration': 6.183}, {'end': 958.87, 'text': 'With the help of the t-test, you can check if the difference between the regions are significant or a statistical fluke.', 'start': 952.667, 'duration': 6.203}, {'end': 970.92, 'text': 'ANOVA. ANOVA is a generalized version of the t-test and used when the mean of the interval dependent variable is different to the categorical independent variable.', 'start': 960.928, 'duration': 9.992}, {'end': 976.467, 'text': 'When we want to check variance between two or more groups, we apply the ANOVA test.', 'start': 971.861, 'duration': 4.606}, {'end': 982.471, 'text': "For example, let's look at the same example of the t-test example.", 'start': 978.629, 'duration': 3.842}, {'end': 987.494, 'text': 'Now you want to check how much people in various regions spend every month on shopping.', 'start': 983.132, 'duration': 4.362}], 'summary': 'Use t-test and anova to analyze shopping expenditure across regions.', 'duration': 41.931, 'max_score': 945.563, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Lv0xcdeXaGU/pics/Lv0xcdeXaGU945563.jpg'}, {'end': 982.471, 'src': 'embed', 'start': 960.928, 'weight': 2, 'content': [{'end': 970.92, 'text': 'ANOVA. ANOVA is a generalized version of the t-test and used when the mean of the interval dependent variable is different to the categorical independent variable.', 'start': 960.928, 'duration': 9.992}, {'end': 976.467, 'text': 'When we want to check variance between two or more groups, we apply the ANOVA test.', 'start': 971.861, 'duration': 4.606}, {'end': 982.471, 'text': "For example, let's look at the same example of the t-test example.", 'start': 978.629, 'duration': 3.842}], 'summary': 'Anova is used to check variance between two or more groups.', 'duration': 21.543, 'max_score': 960.928, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Lv0xcdeXaGU/pics/Lv0xcdeXaGU960928.jpg'}], 'start': 837.003, 'title': 'Hypothesis testing with sas', 'summary': 'Explains hypothesis testing procedures, including parametric tests like t-test and anova, and non-parametric tests, and provides examples of their applications in determining significant differences in data sets.', 'chapters': [{'end': 1000.541, 'start': 837.003, 'title': 'Hypothesis testing with sas', 'summary': 'Explains hypothesis testing procedures, including parametric tests like t-test and anova, and non-parametric tests, and provides examples of their applications in determining significant differences in data sets.', 'duration': 163.538, 'highlights': ['The chapter introduces two types of hypothesis testing procedures: parametric tests and nonparametric tests, and their distinction based on the knowledge of population information and distributional assumptions.', 'The explanation of t-test includes its applications in determining significant differences in means for independent or dependent groups, with a practical example of comparing shopping expenditure in different regions.', 'The ANOVA test is described as a generalized version of the t-test, used to check variance between two or more groups, with a practical example of comparing shopping expenditure in various regions.', 'The chapter provides an overview of various parametric tests, including t-test, ANOVA, chi-squared, and linear regression, as part of hypothesis testing procedures.']}], 'duration': 163.538, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Lv0xcdeXaGU/pics/Lv0xcdeXaGU837003.jpg', 'highlights': ['The chapter provides an overview of various parametric tests, including t-test, ANOVA, chi-squared, and linear regression, as part of hypothesis testing procedures.', 'The explanation of t-test includes its applications in determining significant differences in means for independent or dependent groups, with a practical example of comparing shopping expenditure in different regions.', 'The ANOVA test is described as a generalized version of the t-test, used to check variance between two or more groups, with a practical example of comparing shopping expenditure in various regions.', 'The chapter introduces two types of hypothesis testing procedures: parametric tests and nonparametric tests, and their distinction based on the knowledge of population information and distributional assumptions.']}, {'end': 1211.273, 'segs': [{'end': 1030.734, 'src': 'embed', 'start': 1002.582, 'weight': 4, 'content': [{'end': 1012.128, 'text': 'CHI SQUARE Chi-square is a statistical test used to compare observed data with data you would expect to obtain according to a specific hypothesis.', 'start': 1002.582, 'duration': 9.546}, {'end': 1015.93, 'text': "Let's understand the chi-square test through an example.", 'start': 1013.208, 'duration': 2.722}, {'end': 1019.651, 'text': 'You have a data set of male shoppers and female shoppers.', 'start': 1016.81, 'duration': 2.841}, {'end': 1030.734, 'text': "Let's say you need to assess whether the probability of females purchasing items of $500 or more is significantly different from the probability of males purchasing items of $500 or more.", 'start': 1020.411, 'duration': 10.323}], 'summary': 'Chi-square test compares observed data with expected data based on hypothesis.', 'duration': 28.152, 'max_score': 1002.582, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Lv0xcdeXaGU/pics/Lv0xcdeXaGU1002582.jpg'}, {'end': 1071.186, 'src': 'embed', 'start': 1041.618, 'weight': 0, 'content': [{'end': 1047.226, 'text': 'Simple linear regression is used when one wants to test how well a variable predicts another variable.', 'start': 1041.618, 'duration': 5.608}, {'end': 1055.732, 'text': 'Multiple linear regression allows one to test how well multiple variables, or independent variables, predict a variable of interest.', 'start': 1048.146, 'duration': 7.586}, {'end': 1062.698, 'text': 'When using multiple linear regression, we additionally assume the predictor variables are independent.', 'start': 1056.453, 'duration': 6.245}, {'end': 1071.186, 'text': 'For example, finding relationship between any two variables, say sales and profit, is called simple linear regression.', 'start': 1064.18, 'duration': 7.006}], 'summary': 'Simple and multiple linear regression analyze relationships between variables.', 'duration': 29.568, 'max_score': 1041.618, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Lv0xcdeXaGU/pics/Lv0xcdeXaGU1041617.jpg'}, {'end': 1134.886, 'src': 'embed', 'start': 1089.023, 'weight': 1, 'content': [{'end': 1100.927, 'text': 'The Wilcoxon Signed Rank Test is a nonparametric statistical hypothesis test used to compare two related samples or matched samples to assess whether or not their population mean ranks differ.', 'start': 1089.023, 'duration': 11.904}, {'end': 1108.031, 'text': 'In Wilcoxon Rank Sum Test, you can test the null hypothesis on the basis of the ranks of the observations.', 'start': 1101.767, 'duration': 6.264}, {'end': 1119.539, 'text': 'Kruskal-Wallis H Test Kruskal-Wallis H Test is a rank-based nonparametric test used to compare independent samples of equal or different sample sizes.', 'start': 1108.932, 'duration': 10.607}, {'end': 1125.763, 'text': 'In this test, you can test the null hypothesis on the basis of the ranks of the independent samples.', 'start': 1120.3, 'duration': 5.463}, {'end': 1129.125, 'text': 'The advantages of parametric tests are as follows.', 'start': 1126.584, 'duration': 2.541}, {'end': 1134.886, 'text': 'Provide information about the population in terms of parameters and confidence intervals.', 'start': 1130.083, 'duration': 4.803}], 'summary': 'Wilcoxon signed rank test compares related samples; kruskal-wallis h test compares independent samples.', 'duration': 45.863, 'max_score': 1089.023, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Lv0xcdeXaGU/pics/Lv0xcdeXaGU1089023.jpg'}], 'start': 1002.582, 'title': 'Statistical tests', 'summary': 'Covers chi-square test, linear regression, and nonparametric tests including wilcoxon rank sum test and kruskal-wallis h-test. it also explains wilcoxon signed rank test and emphasizes the advantages of parametric tests. additionally, it discusses the advantages and disadvantages of parametric and nonparametric tests.', 'chapters': [{'end': 1088.323, 'start': 1002.582, 'title': 'Understanding chi-square and linear regression', 'summary': 'Introduces the chi-square test for comparing observed and expected data, as well as the concepts of simple and multiple linear regression with examples. it also mentions nonparametric tests like wilcoxon rank sum test and kruskal-wallis h-test.', 'duration': 85.741, 'highlights': ['Chi-square is a statistical test used to compare observed data with data you would expect to obtain according to a specific hypothesis. Explains the purpose of Chi-square test for comparing observed and expected data.', 'Example of using Chi-square test to assess the probability of females purchasing items of $500 or more compared to males. Illustrates the application of Chi-square test with a specific example.', 'Introduction to simple and multiple linear regression, with the former used to test how well a variable predicts another variable and the latter to test how well multiple variables predict a variable of interest. Provides an overview of simple and multiple linear regression and their respective purposes.', 'Explanation of the Wilcoxon Rank Sum Test and its application as a nonparametric test. Introduces the Wilcoxon Rank Sum Test as a nonparametric test.']}, {'end': 1134.886, 'start': 1089.023, 'title': 'Rank-based nonparametric statistical tests', 'summary': 'Highlights the wilcoxon signed rank test and kruskal-wallis h test for comparing related and independent samples, emphasizing the advantages of parametric tests in providing population information.', 'duration': 45.863, 'highlights': ['The Wilcoxon Signed Rank Test and Kruskal-Wallis H Test are nonparametric statistical hypothesis tests used to compare related and independent samples, respectively.', 'The Wilcoxon Signed Rank Test assesses whether their population mean ranks differ.', 'The Kruskal-Wallis H Test compares independent samples of equal or different sample sizes based on their ranks.', 'Parametric tests provide information about the population in terms of parameters and confidence intervals.']}, {'end': 1211.273, 'start': 1136.307, 'title': 'Parametric and nonparametric tests', 'summary': 'Discusses the advantages and disadvantages of parametric and nonparametric tests, highlighting that nonparametric tests are simple and easy to understand, make fewer assumptions, and do not involve population parameters and sampling theory, but they are not as efficient as parametric tests and difficult to perform operations on large samples manually.', 'duration': 74.966, 'highlights': ['Nonparametric tests are simple and easy to understand, make fewer assumptions, and do not involve population parameters and sampling theory.', 'Parametric tests are easier to use in modeling, analyzing, and describing data with central tendencies and data transformations, and express the relationship between variables.', 'Parametric tests do not require data to be converted into rank order to test.', 'Nonparametric tests provide results similar to parametric procedures but are not as efficient and are difficult to perform operations on large samples manually.']}], 'duration': 208.691, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Lv0xcdeXaGU/pics/Lv0xcdeXaGU1002582.jpg', 'highlights': ['Introduction to simple and multiple linear regression, with the former used to test how well a variable predicts another variable and the latter to test how well multiple variables predict a variable of interest.', 'The Kruskal-Wallis H Test compares independent samples of equal or different sample sizes based on their ranks.', 'Explanation of the Wilcoxon Rank Sum Test and its application as a nonparametric test.', 'The Wilcoxon Signed Rank Test and Kruskal-Wallis H Test are nonparametric statistical hypothesis tests used to compare related and independent samples, respectively.', 'Chi-square is a statistical test used to compare observed data with data you would expect to obtain according to a specific hypothesis.']}], 'highlights': ['Statistics simplifies real-world problems, providing clearer insights for businesses.', 'Descriptive statistics organizes data and provides a summary through numerical measures.', 'Inferential statistics generalizes the larger dataset and allows the inference of population parameters.', 'Hypothesis testing is used to determine if there is enough evidence in a data sample to make inferences about the entire population, and involves choosing between null and alternative hypotheses.', 'The chapter provides an overview of various parametric tests, including t-test, ANOVA, chi-squared, and linear regression, as part of hypothesis testing procedures.', 'Introduction to simple and multiple linear regression, with the former used to test how well a variable predicts another variable and the latter to test how well multiple variables predict a variable of interest.']}