Single Variable Data

5 Key excerpts on "Single Variable Data"

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  Introduction to Probability and Statistics Metric Edition

  • William Mendenhall, Robert Beaver, Barbara Beaver, , William Mendenhall, Robert Beaver, Barbara Beaver(Authors)
  • 2019(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. CHAPTER 1 Describing Data with Graphs 8 1.1 Variables and Data In Chapters 1 and 2, we will present some basic techniques in descriptive statistics —the branch of statistics concerned with describing sets of measurements, both samples and populations . Once you have collected a set of measurements, how can you display this set in a clear, understandable, and readable form? First, you must be able to define what is meant by measurements or “data” and to categorize the types of data that you are likely to encounter in real life. We begin by introducing some definitions. DEFINITION A variable is a characteristic that changes or varies over time and/or for different individuals or objects under consideration. DEFINITION A population is the set of all measurements of interest to the investigator. DEFINITION A sample is a subset of measurements selected from the population of interest. For example, body temperature is a variable that changes over time within a single indi-vidual; it also varies from person to person. Religious affiliation, ethnic origin, income, height, age, and number of offspring are all variables—characteristics that vary depending on the individual chosen. In the Introduction, we defined an experimental unit or an element of the sample as the object on which a measurement is taken. This is the same as saying that an experimental unit is the object on which a variable is measured. When a variable is actually measured on a set of experimental units, a set of measurements or data result. DEFINITION An experimental unit is the individual or object on which a variable is measured. A single measurement or data value results when a variable is actually measured on an experimental unit. If a measurement is obtained for every experimental unit in the entire collection, the resulting data set constitutes the population of interest.

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  eBook - ePub

  Applied Predictive Analytics

  Principles and Techniques for the Professional Data Analyst

  • Dean Abbott(Author)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  Categorical variables have a limited number of values whose intent is to label the variable rather than measure it. Examples of categorical variables include State, Title Code, SKU, and target variables such as Responder. Categorical variables are sometimes called nominal variables or discrete variables. Flag variables or binary variables are often used as names for categorical variables having only two values, like Gender (M,F), responses to questions (Yes and No), and dummy variables (1 and 0).

  Single Variable Summaries

  After data has been loaded into any software tool for analysis, it is readily apparent that for all but the smallest data sets, there is far too much data to be able to make sense of it through visual inspection of the values.

  At the core of the Data Understanding stage is using summary statistics and data visualization to gain insight into what data you have available for modeling. Is the data any good? Is it clean? Is it representative of what it is supposed to measure? Is it populated? Is it distributed as you expect? Will it be useful for building predictive models? These are all questions that should be answered before you begin to build predictive models.

  The simplest way to gain insight into variables is to assess them one at a time through the calculation of summary statistics, including the mean, standard deviation, skewness, and kurtosis.

  Mean

  The mean of a distribution is its average, simply the sum of all values for the variable divided by the count of how many values the variable has. It is sometimes represented by the Greek symbol mu, μ.

  The mean value is often understood to represent the middle of the distribution or a typical value. This is true when variables match a normal or uniform distribution, but often this is not the case.

  Standard Deviation

  The Standard Deviation measures the spread of the distribution; a larger standard deviation means the distribution of values for the variable has a greater range. Most often in predictive modeling, the standard deviation is considered in the context of normal distributions. The Greek symbol sigma,

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  eBook - ePub

  Understanding Statistics

  • Bruce J. Chalmer(Author)
  • 2020(Publication Date)
  • CRC Press

    (Publisher)

  3

  Describing Data for a Single Variable

  3.1    There are many ways of summarizing a set of data.

  In Chapter 1 we discussed the distinctions between a sample and a population and between a statistic and a parameter. We considered the ideas of systematic bias and randomization and mentioned some techniques used in selecting random samples. In Chapter 2 we discussed how we can use information from samples (statistics) to draw inferences about population characteristics (parameters). We saw how knowledge of the sampling distribution of our statistic allows us to say how certain we are about our conclusions.

  In our examples up to now, our parameter of interest has been either a proportion (indicating how prevalent some characteristic is in the population) or an average (indicating how large the scores on some variable tend to be in the population). But as we have noted previously, the concepts we have been discussing can be used to draw inferences about any type of parameter. In this chapter we begin our discussion of various types of parameters and the statistical methods used to draw inferences about them.

  There are many types of parameter, with each type describing a particular characteristic of the population. Sometimes we can summarize a great deal of information about a population in just a few parameters, at least approximately. This capability is what makes statistical analysis so useful. In this chapter we consider techniques for describing a population in terms of a single variable. Later chapters deal with situations involving several variables or several populations.

  3.2    Often we are interested in the entire population distribution.

  The population distribution

  The population distribution for some variable is simply the proportion of individuals with each possible score on the variable. As we have already seen, a histogram is a picture of a distribution. Everything we ever want to know about a variable in a population is contained in the population distribution.

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  Exploratory and Multivariate Data Analysis

  • Michel Jambu(Author)
  • 1991(Publication Date)
  • Academic Press

    (Publisher)

  Chapter 3

  1-D Statistical Data Analysis

  Michel Jambu

  1 Introduction

  A statistical series resulting from numerous observations needs to be summarized in a small set of numbers so that one can compare several series and understand them easily. A summary can be numerical or graphical:

  numerical: each numerical summary highlights a specific feature of a series. It is affected by the point of view of the statistician.

  graphical: graphics indicate more than numerical summaries, from which they are generally derived.

  Most of the time, graphical and numerical summaries are used simultaneously, and they are both studied for the following types of variables: quantitative, qualitative, multiple form qualitative, and chronological.

  2 1-D Analysis of a Quantitative Variable

  As an example, consider the series from the data set of prices of cars. (cf . Appendix 2 , §1). How can one summarize this series? A statistician can study it from three points of view:

  1.  

  What are the most representative values of the data set in terms of local concentration ?

  2.  

  What are the most representative values of the data set in terms of dispersion ?

  3.  

  What are the most representative values of the data set in terms of shape ?

  We study these three points of view successively.

  2.1 Measures of Central Tendency

  Our aim is to characterize a statistical series by a single number (a type-value), representing the order of magnitude of the whole set of numbers, so that we may compare two series by comparing their type-values. The type-value should satisfy the conditions given by Yule (1950) :

  1.  The type-value must be defined independently of the observer, and independently of the conditions under which the observations were taken.

  2.  The type-value must depend on all the values of the series. In particular, values considered as exceptional or irrelevant must be integrated when computing the type-value.

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  Educational Research

  A Contextual Approach

  • Ken Springer(Author)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  Single-Variable Tables Single-variable tables are most commonly used to represent frequencies. Frequency information may be reported in a conventional frequency table, or in a stem-and-leaf display. 346 Single-Variable Representation 347 FREQUENCY TABLES The number of observations for each level of a discrete (nominal or ordinal) variable is referred to as the frequency. Examples of frequency information for one discrete variable include the following: • The numbers of boys and girls in a particular elementary school • The numbers of first-, second-, and third-tier universities in a particular state • The number of charter schools that have been in existence for more than 10 years • The number of first-year students in four high schools who plan to take Advanced Placement (AP) classes As described in Chapter 10, all of the observations for a particular variable can be arranged into a frequency distribution, which depicts frequencies in the form of raw numbers. Frequency data may also be represented as proportions or percentages. (Proportions and percentages will be referred to interchangeably in this chapter, since you can readily convert one into the other without using a calculator.) Research reports typically include whichever form of information is most meaningful and readily comprehended in a particular context. Often, more than one type of frequency information will be helpful. Raw numbers are commonly reported when describing sample characteristics. If you are reading about a sample of students from different schools, it is helpful to know how many students from each school participated. At the same time, your evaluation of the quality of the sample will depend on knowing what percentage of the entire student body of each school participated in the study. Information about percentages is also helpful when describing the characteristics of large samples or populations, An example of a frequency table for a single nominal variable is given in Table 12.1.

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