Mathematics
Frequency, Frequency Tables and Levels of Measurement
Frequency refers to the number of times a particular value occurs in a dataset. Frequency tables organize this information by listing the values and their corresponding frequencies. Levels of measurement categorize data as nominal, ordinal, interval, or ratio, based on the nature of the values being measured.
Written by Perlego with AI-assistance
Related key terms
1 of 5
12 Key excerpts on "Frequency, Frequency Tables and Levels of Measurement"
eBook - PDF- M Smit R Jonker(Author)
- 2017(Publication Date)
- Macmillan(Publisher)
Some of the numbers in the data set may be repeated twice or more often. The frequency of a number (or the variable to which that number has been assigned) is the number of times the number is repeated in the data set. When the frequencies of variables in a data set are listed in a table, the table is known as a frequency distribution table and the list is referred to as a frequency distribution . 112 Module 12 Example 12.2: Frequency distribution By counting frequencies, we can make a frequency distribution table. We will use the example of number of goals scored in a soccer tournament. Victor Nkosi’s team scored the following number of goals in a recent tournament: 2, 3, 1, 2, 1, 3, 2, 3, 4, 5, 4, 2, 2, 3 We place the numbers in order then add up: l How often 1 occurs (2 times). l How often 2 occurs (5 times). l How often 3 occurs (4 times), and so on. This is shown in the frequency table below: SCORES: 2, 3, 1, 2, 1, 3, 2, 3, 4, 5, 4, 2, 2, 3 Score Frequency 1 2 2 5 3 4 4 2 5 1 From the table, we can make certain observations, such as: l Scoring 2 goals happened most often. l The team managed to score 5 goals only once. Discrete data is generated by counting. We count how many times a particular value is repeated and the number of times a number is repeated is known as its frequency. In a frequency distribution table for discrete data, the range of the data is so small that the class of data is limited. Example 12.3: Frequency distribution for discrete data A Mathematics test counted out of 10 marks. There are 26 learners in the class. Calculate how many of the learners obtained 1 out 10, 2 out 10, and so on. The frequency table will only look at learners with marks ranging from 1 to 10. This means the class of the table is limited (discrete) to 1–10. The range of the class is limited. The class intervals are 1. range: the difference between the largest and smallest values of a set of data class: a group of data that is related by some user defined property; e.g.
eBook - PDF- Alison Jane Pickard(Author)
- 2017(Publication Date)
- Facet Publishing(Publisher)
Examples of interval measures are age, population size, number of years spent in an occupation and distance between one location and another. Quantitative analysis 285 Frequency distribution One of the first stages in analysing your data is to calculate and present the frequency distribution of your dataset. Some say that frequency distributions are the ‘bedrock’ of subsequent analysis of your data, as the distributions relate to the number of responses you get to each of the options available to your respondents in each question you ask. It is easy to assume that this is too boring; you may be eager to perform more complex calculations on your data and start to look for deeper meaning behind your evidence. This is all well and good but remember that before you do anything you should give your reader an overview of the data you are working with. Provide them with the information they need to make sense of the results you present from the analytical procedures that follow. Frequency distributions are often seen as data processing, sorting your data and saying very little about any relationship between the variables. This is true but presenting your frequency distributions provides your reader with the basic facts. You need to understand what they mean for your data and how they could influence any subsequent analysis. The elements of a frequency distribution table differ depending on the level of measurement used. See Table 24.2 for all possible elements of a frequency table but remember they will not all be used, depending on your level of measurement.
eBook - PDF- Jingmei Jiang(Author)
- 2022(Publication Date)
- Wiley(Publisher)
2 Descriptive Statistics 12 2.1 Frequency Tables and Graphs The frequency distribution table and the frequency distribution diagram are starting points for summarizing data and provide a basis for observing the characteristics of the distribution. They are made by grouping observations and obtaining the frequency distribution by counting the number of observations in each group. 2.1.1 Frequency Distribution of Numerical Data Most of numerical data can be regarded as continuous, theoretically taking on an infi-nite number of values. Thus, one is essentially always dealing with a frequency distri-bution tabulated by group (i.e., the data are grouped into several non-overlapping intervals). Therefore, the number of observations falling into each interval, which we call the frequency, can be determined. We use these numbers to construct the fre-quency distribution table and the frequency distribution diagram. In the following steps, we use the data shown in Table 2.1 to illustrate the proce-dures for organizing a frequency table. Table 2.1 Raw data on the height (cm) of 153 10-year-old girls.
eBook - PDF- Frederick Gravetter, Larry Wallnau(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
Thus, a frequency distribution presents a picture of how the individual scores are dis-tributed on the measurement scale—hence the name frequency distribution. ■ ■ Frequency Distribution Tables The simplest frequency distribution table presents the measurement scale by listing the different measurement categories ( X values) in a column from highest to lowest. Beside each X value, we indicate the frequency, or the number of times that particular measure-ment occurred in the data. It is customary to use an X as the column heading for the scores and an f as the column heading for the frequencies. An example of a frequency distribution table follows. The following set of N = 20 scores was obtained from a 10-point statistics quiz. We will organize these scores by constructing a frequency distribution table. Scores: 8, 9, 8, 7, 10, 9, 6, 4, 9, 8, 7, 8, 10, 9, 8, 6, 9, 7, 8, 8 LEARNING OBJECTIVES D E FINITIO N It is customary to list categories from high-est to lowest, but this is an arbitrary arrange-ment. Many computer programs list categories from lowest to highest. E X A M P L E 2.1 SECTION 2.1 | Frequency Distributions and Frequency Distribution Tables 35 Copyright 2017 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. 1. The highest score is X = 10, and the lowest score is X = 4. Therefore, the first column of the table lists the categories that make up the scale of measurement ( X values) from 10 down to 4. Notice that all of the possible values are listed in the table. For example, no one had a score of X = 5, but this value is included.
eBook - PDF- Alan Monroe(Author)
- 2018(Publication Date)
- Routledge(Publisher)
6 Statistics: An Introduction Once the observations of the variables in a hypothesis have been made and assembled into a data set, the next step in the research process is to analyze those data in order to draw conclusions about the hypothesis. However, the bits of data are often numerous indeed. This is particularly true in the social sciences, where we may have survey results on dozens of questions from hundreds or even thou-sands of respondents. To look over such a vast array of data to “see” what is there would be a very difficult task. In order to evaluate our data and determine what patterns are present, we need statistics. There are many statistical measures. Chapters 8, 9, and 10 will show you how to compute several of them. This chapter presents an overview, beginning with some basic information that is neces-sary to be able to use any statistical measures correctly. Levels of Measurement The term level of measurement refers to the classifications or units that result when a variable has been operationally defined. There are three levels of measurement with which you need to be famil-iar: nominal, ordinal, and interval data. Nominal Variables The “lowest” level of measurement, that is, the least precise, is the nominal level. A nominal variable simply places each case into one 83 84 Statistics: An Introduction of several unordered categories. Examples would include an indi-vidual’s racial/ethnic status (African American, white, Hispanic, Asian, Native American, or other), religious preference (Protestant, Catholic, Jewish, none, other), and vote for president (Clinton, Dole, Perot, other, not voting). Note that it would make no sense to describe such variables in quantitative terms. To speak of “more race,” “less religion,” or “more voting” from data on these mea-sures would be silly. Nominal variables contain information on “what kind,” not “how much.” Ordinal Variables As the name implies, ordinal variables rank cases in relation to each other.
eBook - PDFBiostatistics
Basic Concepts and Methodology for the Health Sciences, 10th Edition International Student Version
- Wayne W. Daniel, Chad L. Cross(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
When we do this for our example, we have Table 2.3.1. A table such as Table 2.3.1 is called a frequency distribution. This table shows the way in which the values of the variable are distributed among the specified class intervals. By consulting it, we can determine the frequency of occurrence of values within any one of the class intervals shown. Relative Frequencies It may be useful at times to know the proportion, rather than the number, of values falling within a particular class interval. We obtain this information by dividing the number of values in the particular class interval by the total number of values. If, in our example, we wish to know the proportion of values between 50 and 59, inclusive, we divide 70 by 189, obtaining .3704. Thus we say that 70 out of 189, or 70/189ths, or .3704, of the values are between 50 and 59. Multiplying .3704 by 100 gives us the percentage of values between 50 and 59. We can say, then, that 37.04 percent of the subjects are between 50 and 59 years of age. We may refer to the proportion of values falling within a class interval as the relative frequency of occurrence of values in that interval. In Section 3.2 we shall see that a relative frequency may be interpreted also as the probability of occurrence within the given interval. This probability of occurrence is also called the experimental probability or the empirical probability . TABLE 2.3.1 Frequency Distribution of Ages of 189 Subjects Shown in Tables 1.4.1 and 2.2.1 Class Interval Frequency 30–39 11 40–49 46 50–59 70 60–69 45 70–79 16 80–89 1 Total 189 24 CHAPTER 2 STRATEGIES FOR UNDERSTANDING THE MEANINGS OF DATA In determining the frequency of values falling within two or more class intervals, we obtain the sum of the number of values falling within the class intervals of interest. Similarly, if we want to know the relative frequency of occurrence of values falling within two or more class intervals, we add the respective relative frequencies.
eBook - PDFStatistics Using IBM SPSS
An Integrative Approach
- Sharon Lawner Weinberg, Sarah Knapp Abramowitz(Authors)
- 2016(Publication Date)
- Cambridge University Press(Publisher)
WHEN VARIABLES ARE MEASURED AT THE NOMINAL LEVEL The variable REGION in the NELS data set is an example of a discrete variable with four possible values or categories: Northeast, North Central, South, and West. To display the number of student respondents who are from each of these regions, we may use frequency and percent tables as well as bar and pie graphs. FREQUENCY AND PERCENT DISTRIBUTION TABLES To create frequency and percent distribution tables, click Analyze on the main menu bar, Descriptive Statistics , and then Frequencies . From the list of variables that appears on the left side of the screen, click the variable, REGION, then click the arrow facing right to move REGION into the box of variables that will be analyzed. Click OK . WHEN VARIABLES ARE MEASURED AT THE NOMINAL LEVEL 21 The result, as shown in Table 2.1 , should appear in the Output Viewer on your screen. In Table 2.1 , the first column lists the four possible categories of this variable. The sec-ond column, labeled Frequency, presents the number of respondents from each region. From this column, we may note that the fewest number of respondents, 93, are from the West. Note that if we sum the frequency column we obtain 500, which represents the total number of respondents in our data set. This is the value that we should obtain if an appro-priate category is listed for each student and all students respond to one and only one cat-egory. If an appropriate category is listed for each student and all students respond to one and only one category, we say the categories are mutually exclusive and exhaustive . This is a desirable, although not essential, property of frequency distributions. Although the frequency of a category is informative, so is also the relative frequency of the category; that is, the frequency of the category relative to (divided by) the total number of individuals responding.
eBook - PDF- Lorena Madrigal(Author)
- 2012(Publication Date)
- Cambridge University Press(Publisher)
2 The first step in data analysis: summarizing and displaying data. Computing descriptive statistics In this chapter we review what should be the first steps in data analysis. A sure way to aggravate your adviser would be to bring her a table with your raw data (hundreds of observations) for her reading pleasure. Instead of making such a mistake, you would do well to summarize your data into an easy-to-read table, namely, a frequency dis- tribution, and to illustrate this table with a graph. Afterwards, you should compute simple descriptive statistics that summarize the sample. This chapter is divided into these two broad sections: (1) frequency distributions and graphs and (2) descriptive statistics. 2.1 Frequency distributions 2.1.1 Frequency distributions of discontinuous numeric and qualitative variables Frequency distributions are very useful as a first step to understand how data are dis- tributed, that is, what values are most frequent in a sample, and which ones are less frequent or even absent. The procedure for constructing a frequency distribution of a discontinuous numeric variable and a qualitative variable is very similar, so I dis- cuss the procedure for both variables together. Tables 2.1 and 2.2 show a frequency distribution of a discontinuous numeric variable (number of children produced) and a qualitative variable (religious membership). The latter one is rather uninteresting because there are only two categories (religious group one and two), so we discuss the former one. A frequency distribution should have as a first column a listing of the observations measured in the data set. You will notice that in group one we observed variates ranging from one through fourteen, but that variate thirteen was not recorded, so it is missing in the table. In contrast, the listing of observations for group two starts with the variate five and finishes with fifteen.
- Robert Pagano(Author)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
The Histogram The histogram is used to represent frequency distributions composed of interval or ratio data. It resembles the bar graph, but with the histogram, a bar is drawn for each class interval. The class intervals are plotted on the horizontal axis such that each class bar begins and terminates at the real limits of the interval. The height of the bar corresponds to the frequency of the class interval. Since the intervals are continuous, the vertical bars must touch each other rather than be spaced apart as is done with the bar graph. Figure 3.4 shows the statistics exam scores (Table 3.4, p. 52) displayed as a histogram. Note that it is customary to plot the midpoint of each class interval on the abscissa. The grouped scores have been presented again in the figure for your convenience. 600 500 400 300 200 100 0 Number of students Psychology Communications Undergraduate major Biological Sciences English Chemistry Philosophy figure 3.3 Bar graph: Students enrolled in various undergraduate majors in a college of arts and sciences. 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. 64 C H A P T E R 3 Frequency Distributions The Frequency Polygon The frequency polygon is also used to represent interval or ratio data. The horizontal axis is identical to that of the histogram. However, for this type of graph, instead of using bars, a point is plotted over the midpoint of each interval at a height corresponding to the frequency of the interval.
eBook - PDF- Peihua Qiu(Author)
- 2013(Publication Date)
- Chapman and Hall/CRC(Publisher)
See Section 2.7 for a related discussion. 2.6 Tabular and Graphical Methods for Describing Data 2.6.1 Frequency table, pie chart, and bar chart To discuss tabular and graphical methods for describing categorical data, let us con-sider the following example. Example 2.3 Before a presidential election, a TV network made a survey in the na-tion, and they randomly selected 1,000 legitimate voters to ask for their favorite candidates. Each voter was also asked to tell the TV network his or her party status. The data about the party status are summarized in Table 2.2. In Example 2.3, the data about party status is categorical with four categories. Table 2.2 lists all the categories of the observations and the corresponding counts, or frequencies. This table is called the frequency table . In the table, sometimes it is more convenient to list relative frequencies (or proportions) of the categories. The relative frequency of a category is defined by relative frequency = frequency sample size . For the data summarized in Table 2.2, the relative frequencies of the categories Democrat, Republican, Green Party, and Other are 0.437, 0.486, 0.053, and 0.024, respectively. In the table, if relative frequencies, instead of frequencies, are listed, then the table is often called the relative frequency table . Sometimes, in a frequency table, both frequencies and relative frequencies are listed for convenience of its ap-plications. There are two commonly used graphical methods to describe categorical data. One is the pie chart , by which a circle is divided into slices, each slice denotes a category, and the slice size is proportional to the relative frequency of the category. For the data in Table 2.2, the corresponding pie chart is shown in Figure 2.3(a).
- Ronald Asherson(Author)
- 2008(Publication Date)
- Elsevier Science(Publisher)
2 2 Elementary Descriptive Statistical Techniques 2.1 Summarizing Sets of Data Measured on a Ratio or Interval Scale In what follows we shall consider a variety of descriptive statistical techniques that may be employed to conveniently summarize, for the most part, a set of ratio- or interval-scale data. Unless otherwise stated, this data set will be assumed to represent the entire population. Two basic approaches will be advanced. First, we may summarize the data in a tabular (and, eventually, graphical) fashion. Second, rather than work with the entire mass of data in, say, tabular form, we may derive from it a set of concise quantitative summary characteristics (either computed or positional values arrived at largely from a set of formulas), which themselves describe the salient features of the data set. Once these summary measures have been obtained, they may be used to assess the current status of a particular population, measure differences between two or more populations, or consider changes in a given population over time. These approaches are contrasted in Figure 2.1. With respect to our second approach regarding various sets of quantitative/ positional summary figures, we note briefly that: (a) Measures of Central Location, broadly construed, are essentially averages that describe the location of the center of the data set. (b) Measures of Dispersion (typically taken about some point of central location) describe the spread or scatter of the observations along the horizontal axis. (c) Measures of Skewness indicate the degree of departure of the data distribution from symmetry and thus serve as measures of asymmetry. (d) Measures of Kurtosis indicate how flat or rounded the peak of the data distribution happens to be. 9 10 Chapter 2 Elementary Descriptive Statistical Techniques Descriptive Techniques for Summarizing Sets of Ratio-, Interval-Scale Data I. Tabular Methods (grouped or ungrouped data) 1. Absolute Frequency Distribution 2.
eBook - PDFPsychometric Methods
Theory into Practice
- Larry R. Price(Author)
- 2016(Publication Date)
- The Guilford Press(Publisher)
Graphs such as histograms and polygons are often used when two or more groups are compared on a set of scores such as our lan- guage development or vocabulary test. The choice between using a histogram and a polygon FIGURE 2.9. Relative frequency polygon for 100 individuals (from Table 2.4 data). Measurement and Statistical Concepts 29 depends on preference; however, the type and nature of the variable also serve as a guide for when to use one type of graph rather than another. For example, when a variable is discrete, score values can only take on whole numbers that can be measured exactly—and there are no intermediate values between the score points. Even though a variable may be continuous in theory, the process of measurement always reduces the scores on a variable to a discrete level (e.g., a discrete random variable; see the Appendix for a rigorous mathematical treatment of random variables and probability). In part, this is due to the accuracy and/or precision of the instrumentation used and the integrity of the data acquisition/collection method. Therefore, continuous scales are in fact discrete ones with varying degrees of precision or accuracy. Returning to Figure 2.1, for our test of general intelligence any of the scores may appear to be continuous but are actually discrete because a person can only obtain a numerical value based on the sum of his or her responses across the set of items on a test (e.g., it is not pos- sible for a person to obtain a score of 15.5 on a total test score). The frequency histogram can also be used with variables such as zip codes or family size (i.e., categorical variables with naturally occurring discrete structures). Alternatively, the nature of the frequency poly- gon technically suggests that there are intermediary score values (and therefore a continuous score scale) between the points and/or dots in the graph.
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
