Continuous and Discrete Data

12 Key excerpts on "Continuous and Discrete Data"

• 

  eBook - PDF

  Statistics All-in-One For Dummies

  • Deborah J. Rumsey(Author)
  • 2022(Publication Date)
  • For Dummies

    (Publisher)

  So, in this section, you get a taste of statistical jargon; I send you to the appropriate chapters elsewhere in the book to get details. Data Data are the actual pieces of information that you collect through your study. For example, say that you ask five of your friends how many pets they own, and they give you the following data: 0, 2, 1, 4, 18. (The fifth friend counted each of their aquarium fish as a separate pet.) Not all data are numbers; you also record the gender of each of your friends, giving you the following data: male, male, female, male, female. Most data fall into one of two groups: numerical or categorical. (I present the main ideas about these variables here; see Chapters 4 and 5 for more details.) » Numerical data: These data have meaning as a measurement, such as a person’s height, weight, IQ, or blood pressure; or they’re a count, such as the number of stock shares a person owns, how many teeth a dog has, or how many pages you can read of your favorite book before you fall asleep. (Statisticians also call numerical data quantitative data.) Numerical data can be further divided into two types: discrete and continuous. • Discrete data represent items that can be counted; they take on possible values that can be listed out. The list of possible values may be fixed (also called finite); or it may go from 0, 1, 2, on to infinity (making it countably infinite). For example, the number of heads in 100 coin flips takes on values from 0 through 100 (finite case), but the number of flips needed to get 100 heads takes on values from 100 (the fastest scenario) on up to infinity. Its possible values are listed as 100, 101, 102, 103, (representing the countably infinite case). • Continuous data represent measurements; their possible values cannot be counted and can only be described using intervals on the real number line.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Statistics with Common Sense

  • David Kault(Author)
  • 2003(Publication Date)
  • Greenwood

    (Publisher)

  Descriptive Statistics 11 cutoff in size between one measurement and the next. Since the size of the different measurements are not necessarily cut off from each other by any fixed amount, we regard the measurements as continuous data. Discrete Data In the case of discrete data, the measurement has to fall on separated (or discrete) values. Discrete measurements almost always arise from counting. An example is the number of accidents the clients of an insurance company have in one year. For any client the measurement can be 0 or 1 or 2 or 3 and so on. These numbers are discrete in that they are separated by whole units from each other. It would make no sense to have the number 1.3 as the number of accidents a client had. A second example of a discrete measurement is the number of children in a family, since children also come in whole numbers: Having 1 child or 2 children in a family makes sense, but no family contains 1.3 children. Categorical and Ordinal Data Data can also be categorical. This means that our measurement consists of simply classifying the object into one of several categories. An example is classifying people by their religion. Here our measurement is simply the clas- sification of each individual as Christian, Buddhist, Moslem, and so on. A second example of a categorical measurement is to record the species of plants in a field. When there is only two categories possible, categorical data is some- times called dichotomous data. Examples of pairs of dichotomous categories are yes-no, better-worse, and alive-dead. There is another variant of categorical measurement. Categorical measure- ments are referred to as ordinal if the categories can be sensibly ordered. The different religions can't be ordered (except perhaps to some religious bigot it would not make sense to put Hinduism above or below Buddhism), so reli- gion is not ordinal data.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Statistics for the Social Sciences

  A General Linear Model Approach

  • Russell T. Warne(Author)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  Many types of data in the social sciences are continuous, such as intelligence test scores, which in a normal human population range from about 55 to 145 on most tests, with every whole number in between being a possible value for a person. Continuous data often permit scores that are expressed as fractions or decimals. All three temperature scales that I have discussed in this chapter (i.e., Fahrenheit, Celsius, and Kelvin) are continuous data, and with a sensitive enough thermometer it would be easy to gather temperature data measured at the half-degree or tenth-degree. The opposite of continuous data are discrete data, which are scores that have a limited range of possible values and do not form a constant, uninterrupted scale of scores. All nominal data are discrete, as are ordinal data that have a limited number of categories or large gaps between groups. A movie rating system where a critic gives every film a 1-, 2-, 3-, or 4-star rating would be discrete data because it only has four possible values. Most interval or ratio data, however, are continuous – not discrete – data. The point at which a variable has “too many” values to be discrete and is therefore continuous is often not entirely clear, and whether a particular variable consists of discrete or continuous data is sometimes a subjective judgment. To continue with the movie rating Other Ways to Classify Data 29 system example, the website Internet Movie Database (IMDb) asks users to rate films on a scale from 1 to 10. Whether ten categories are enough for the data to be continuous is a matter of argument, and opinions may vary from researcher to researcher. Check Yourself! • Why is it important to understand the four levels of data? • Why is there controversy about the level of data that rating scales produce? • What is the difference between discrete and continuous data? Summary Before conducting any statistical analyses, researchers must decide how to measure their variables.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Statistical Analysis in Forensic Science

  Evidential Value of Multivariate Physicochemical Data

  • Grzegorz Zadora, Agnieszka Martyna, Daniel Ramos, Colin Aitken(Authors)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  3 Continuous data 3.1 Introduction

  This chapter presents some general background information on continuous data that is necessary to fully understand the topics on which the book focuses in other chapters (see also Aitken and Taroni 2004; Curran 2011; Lucy 2005). Examinations performed by the application of various analytical methods (Chapter 1) return various kinds of information:

  • qualitative data, for example, information on compounds detected in fire debris samples based on a chromatogram, information obtained from the spectrum of an unknown sample, and morphological information such as the number and thicknesses of layers in a cross-section of car paints;
  • quantitative data, for example, the concentration of elements or value of the refractive index in a glass fragment, peak areas of a drug profile chromatogram or gasoline detected in fire debris, and the concentration of ethanol in blood samples.

  Another way to classify data is connected with the possible values they may generate. If the analysis produces data that can take any value within a specified range, they are considered to be continuous , as opposed to discrete data, which may take certain values. A Typical examples of continuous data are human height and the elemental composition of glass fragments. A typical example of discrete data is the number of children in a family, as it may take only integer values, and cannot take values such as 3.3. Most of the data that chemists have to deal with are of continuous type and so this book focuses on them.

  Physicochemical data analysis requires some data organisation. A recommended way to do so is to present the data generated during the analysis in the form of a rectangular table as in Figure 3.1 (see also the example of a database in the file glass_data.txt ).1 Such tables containing data may take the form of what is generally called a matrix, as in Figure 3.2(a) .

  Figure 3.1 Data organisation in the form of a matrix. Here the data refer to p variables (X p ) measured three times for each of m

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Handbook of Data Analysis

  • Melissa A Hardy, Alan Bryman, Melissa A Hardy, Alan Bryman(Authors)
  • 2009(Publication Date)
  • SAGE Publications Ltd

    (Publisher)

  But any summary measure is designed to succinctly portray a specific feature of the data, not to provide a detailed description. Therefore, it is impor-tant to choose measures suited to the ques-tion at hand and to the nature of the data. CLASSIFYING , COUNTING , AND MEASURING The tools that we use must be suited to the type of information we wish to analyze. In the same way that a carpenter learns that different types of saws with different types of blades are best suited to cutting different sorts of materials, so the analyst learns that the first task is to identify the nature of the information at hand. Initially, we can cate-gorize variables into two types: discrete and continuous. Discrete variables can take a finite number of values, whereas continu-ously measured variables can take on any value within a much more detailed range of possible values. In other words, the possible values for discrete variables are countable; the possible values for continuous measures 3 Summarizing Distributions MELISSA HARDY are not. Because of this difference, the sta-tistical approaches to describing distribu-tions of discrete and continuous measures utilize different branches of mathematics. Among discrete variables, we also have dif-ferent possible types. These types include nominal classifications, ordinal classifica-tions, and counts. Continuous variables may be measured on either interval or ratio scales, the difference being that ratio scales have an absolute zero point. To facilitate our discus-sion of distribution statistics, we will rely on an exemplary data set extracted from the National Longitudinal Survey of Youth (NLSY) that was initiated in 1979. Table 3.1 lists the variables we will use in examples. Most of the variables are from the 1994 wave; a few are taken from the initial wave in 1979. The sample consists of men and women who were initially interviewed when they were aged 14–22.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Applied Statistics For Agricultural Sciences

  • Venkatesan D.(Author)
  • 2020(Publication Date)
  • NEW INDIA PUBLISHING AGENCY (NIPA)

    (Publisher)

  For example (i) the age years of a person, (ii) the income in rupees of family, (iii) the weight in kilogram of student, (iv) the price in rupees of a commodity etc., are continuous variables. b. Discrete variable is a variable which is not capable of taking all possible values in a range and thus there are gaps between one value and the other. For example the numbers of births, deaths in a place, the number of houses in a city etc., are discrete variable. 2.3 Array The data before it is arranged and analysed is called raw data. It is ‘raw’ because it is unprocessed by statistical by statistical methods. The data array is one of the simplest ways to present data. It arranges the values from the lowest value to the highest in ascending order or from highest to the lowest in descending order. The following tables 2.2 and 2.3 illustrate the raw data and data array in ascending order relating to the average annual Inventory Turnover in weeks for 20 large construction firms. Table 2.2 : Raw data of Inventory Turnover in weeks for 20 construction firms 2.3 3.4 4.9 4.8 5.5 5.5 4.0 4.1 5.1 5.4 4.7 3.4 4.3 5.5 4.7 3.8 4.2 5.5 4.9 4.9 23 Classification, Tabulation and Graphical Representation Table 2.3 : Data array of Inventory Turnover in weeks for 20 construction firms 2.0 4.0 4.7 4.9 5.5 3.4 4.1 4.7 4.9 5.5 3.4 4.2 4.8 5.1 5.5 3.8 4.3 4.9 5.4 5.5 Data arrays offer several advantages over raw data: i. We can quickly notice the lowest and highest values in the data. ii. We can see whether any values appear more than once in the array. iii. We can easily divide the data into sections. iv. We can observed the distance between succeeding values in the data. Inspite of these advantages, sometimes a data array is not helpful. Since it lists every observation, it does not condense or compress the data. One way we can compress data is to use a frequency table of frequency distribution.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Quantitative Techniques in Business, Management and Finance

  A Case-Study Approach

  • Umeshkumar Dubey, D P Kothari, G K Awari(Authors)
  • 2016(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  For example, the employees of a company may be classi-fied according to their monthly salaries. Since quantitative data is characterised by differ-ent numerical values, the data represents the values of a variable. Quantitative data may be further classified into one or two types: discrete or continuous. The term discrete data refers to quantitative data that is limited to certain numerical values of a variable, for example the number of employees in an organisation or the number of machines in a factory. 2.5 Data Collection 2.5.1 Population This is the entire collection of entities that a manager is trying to study. 2.5.2 Sample This is a fraction of the population that represents the entire population in its characteris-tics proportionately. For example, when a magazine conducts an opinion poll among 1000 individuals from all over India, with a view to know the general opinion of Indians towards politicians, then all Indians is the target population and 1000 individuals represent the population and is referred to as the sample. In another example, to estimate the potential market for a new innovation, managers in a research department may study 1000 consumers in a particular territory. The managers make sure that this sample contains the consumers belonging to all cross sections (income, religion, education and locality) of the society. 2.5.3 Testing the Validity of Data This is testing the data for adequacy and reliability, as it influences the quality of the final decision. Managers can pose the following questions: 1. Data origin? 2. Is the source reliable? 22 Quantitative Techniques in Business, Management and Finance 3. Does the data support or contradict the previous decisions? 4.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Pharmaceutical Statistics

  Practical and Clinical Applications, Fifth Edition

  • Sanford Bolton, Charles Bon(Authors)
  • 2009(Publication Date)
  • CRC Press

    (Publisher)

  2 CHAPTER 1 1.1.1 Continuous Variables Experimental data come in many forms. ∗ Probably the most commonly encountered variables are known as continuous variables. A continuous variable is one that can take on any value within some range or interval (i.e., within a specified lower and upper limit). The limiting factor for the total number of possible observations or results is the sensitivity of the measuring instrument. When weighing tablets or making blood pressure measurements, there are an infinite number of possible values that can be observed if the measurement could be made to an unlimited number of decimal places. However, if the balance, for example, is sensitive only to the nearest milligram, the data will appear as discrete values. For tablets targeted at 1 g and weighed to the nearest milligram, the tablet weights might range from 900 to 1100 mg, a total of 201 possible integral values (900, 901, 902, 903, . . . , 1098, 1099, 1100). For the same tablet weighed on a more sensitive balance, to the nearest 0.1 mg, values from 899.5 to 1100.4 might be possible, a total of 2010 possible values, and so on. Often, continuous variables cannot be easily measured but can be ranked in order of magnitude. In the assessment of pain in a clinical study of analgesics, a patient can have a continuum of pain. To measure pain on a continuous numerical scale would be difficult. On the other hand, a patient may be able to differentiate slight pain from moderate pain, moderate pain from severe pain, and so on. In analgesic studies, scores are commonly assigned to pain severity, such as no pain = 0, slight pain = 1, moderate pain = 2, and severe pain = 3. Although the scores cannot be thought of as an exact characterization of pain, the value 3 does represent more intense pain than the values 0, 1, or 2. The scoring system above is a representation of a continuous variable by discrete “scores” that can be rationally ordered or ranked from low to high.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Probability and Statistics with R

  • Maria Dolores Ugarte, Ana F. Militino, Alan T. Arnholt(Authors)
  • 2015(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  When a variable is qualitative, it is essentially defining groups or categories. When the categories have no ordering the variable is called nominal . For example, the variable gender can take on the values male and female or the variable music preference could have values such as classical , jazz , rock , or other . When the categories have a distinct ordering, the variable is called ordinal . Such a variable might be educational level with values elementary school , high school , college graduate , graduate , or professional school . Values on a scale can be either interval or ratio. Interval data have interpretable distances, while ratio data have a true zero. A variable that is quantitative (numeric) may be either discrete or continuous . A discrete variable is a numerical variable that can assume a finite, or at most a countably infinite, number of values. Such variables include the number of people arriving at a bank on Thursday, students in a class, or dogs in the pound. A continuous variable is a numerical variable that can assume an infinite number of values associated with the numbers on an interval of the real number line; for example, the height of a tree, the life of a light bulb, and the 97 98 Probability and Statistics with R , Second Edition weight of an apple are all continuous random variables. An important distinction between discrete and continuous variables is that discrete variables can take on the same value re-peatedly, while continuous variables have few or no repeated values. It is important to be able to distinguish between di ↵ erent types of variables because methods for viewing and summarizing data depend on variable type. More to the point, it will be imperative to dis-tinguish between qualitative (categorical) variables and quantitative (numerical) variables. The distribution of a variable specifies the values that the variable assumes and how often these values occur.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Statistics with JMP

  Graphs, Descriptive Statistics and Probability

  • Peter Goos, David Meintrup(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  2 Data and its representation A microphone in the sidewalk would provide an eavesdropper with a cacophony of clocks, seemingly random like the noise from a Geiger counter. But the right kind of per- son could abstract signal from noise and count the pedestrians, provide a male/female breakdown and a leg-length histogram … (from Cryptonomicon, Neal Stephenson, p. 147) Data is a set of measurements of one or more characteristics or variables of some elements of a population, or of a number of objects generated by a process. Different types of variables can be measured. 2.1 Types of data and measurement scales Variables are classified according to the measurement scale on which they are mea- sured. Categorical or qualitative variables are measured on a nominal scale or on an ordinal scale. Quantitative variables are either measured on an interval scale or on a ratio scale. 2.1.1 Categorical or qualitative variables 2.1.1.1 Nominal variables Elements of a sample or a population can be classified using a nominal variable: the value of the variable places an element in a certain class or category. Examples of such variables are • gender (male/female), • nationality (Belgian, German, and so on), Statistics with JMP: Graphs, Descriptive Statistics, and Probability, First Edition. Peter Goos and David Meintrup. © 2015 John Wiley & Sons, Ltd. Published 2015 by John Wiley & Sons, Ltd. Companion Website: wiley.com/go/goosandmeintrup DATA AND ITS REPRESENTATION 9 • religion (Catholic, Protestant, and so on), and • whether or not one owns a car (yes/no). Sometimes it can be useful to assign labels, code numbers, or code letters, to the different classes or categories. For example, a Belgian person may be assigned the code “1”, a Dutch person the code “2”, a French person the code “3”, and a German person the code “4”. It is important to note that these figures do not imply any order and/or quantity.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Statistical Quality Control

  Using MINITAB, R, JMP and Python

  • Bhisham C. Gupta(Author)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  3 Describing Quantitative and Qualitative Data 3.1 Introduction In all statistical applications, we collect sets of data. These sets could be small, containing just a few data points, or very large and messy, with millions of points or more. Not all data are of the same kind, and therefore not all data sets can be treated in the same manner. In this chapter, we intro- duce the classification of data into various categories, such as qualitative and quantitative, and apply appropriate graphical and numerical techniques to analyze them. Sometimes data sets contain data points that are erroneous and thus may not belong to the data set. These data points are commonly known as outliers. In this chapter, we apply a method that can detect such outliers so that they may be either corrected and included in the analysis or eliminated from the analysis. In certain applications, we are studying two characteristics of an individual or item, and we may be interested in finding any association between these characteristics; a measure of such association is introduced in this chapter. The concept of descriptive and inferential statistics is also introduced. Finally, we study some important probability distributions that are quite fre- quently used in the study of statistical quality control. 3.2 Classification of Various Types of Data In practice, it is common to collect a large amount of non-numerical and/or numerical data on a daily basis. For example, we may collect data concerning customer satisfaction comments of employees, perceptions of suppliers, etc. Or we may track the number of employees in various departments of a company, check weekly production volume in units produced or sales in dollars per unit of time, etc. However, not all of the data collected can be treated the same way, as there are different types of data.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Understanding Statistics in the Behavioral Sciences

  • Robert Pagano(Author)
  • 2020(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  M. Lord, “On the Statistical Treatment of Football Numbers,” American Psychologist, 8 (1953), 750–751; W. L. Hays, Statistics for the Social Sciences , 2nd ed., Holt, Rinehart and Winston, New York, 1973, pp. 87–90; S. Siegel, Nonparametric Statistics for the Behavioral Sciences , McGraw-Hill, New York, 1956, pp. 18–20; and S. S. Stevens, “Mathematics, Measurement, and Psy-chophysics,” in Handbook of Experimental Psychology , S. S. Stevens, ed., Wiley, New York, 1951, pp. 23–30. Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 36 C H A P T E R 2 Basic Mathematical and Measurement Concepts Weight, height, and time are examples of continuous variables. With each of these variables, there are potentially an infinite number of values between adjacent units. If we are measuring time and the smallest unit on the clock that we are using is 1 second, between 1 and 2 seconds there are an infinite number of possible values: 1.1 seconds, 1.01 seconds, 1.001 seconds, and so forth. The same argument could be made for weight and height. This is not the case with a discrete variable. “Number of children in a family” is an example of a discrete variable. Here the smallest unit is one child, and there are no possible values between one or two children, two or three children, and so on. The characteristic of a discrete variable is that the variable changes in fixed amounts, with no intermediate values possible.

  Sign up to read

  Learn more about book