Mathematics
Statistical Measures
Statistical measures are numerical values used to describe and summarize the features of a dataset. They include measures of central tendency (such as mean, median, and mode) and measures of dispersion (such as range, variance, and standard deviation). These measures provide insights into the distribution and characteristics of the data, aiding in the interpretation and analysis of statistical information.
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10 Key excerpts on "Statistical Measures"
eBook - PDFReasoning with Data
An Introduction to Traditional and Bayesian Statistics Using R
- Jeffrey M. Stanton(Author)
- 2017(Publication Date)
- The Guilford Press(Publisher)
I ’m the kind of person who thinks about the definitions of words and who tries to use the right word in the right situation. Knowing the correct terminology for important ideas makes it easier to think about and talk about things that really matter. As is the case with most fields, statistics has a specialized vocabu- lary that, when you learn it, helps you to think about and solve problems. In this chapter, I introduce some of the most essential statistical terminology and I explain how the concepts differ and are connected to one another. Once you have mastered this vocabulary, later parts of the book will be much easier to understand. DESCRIPTIVE STATISTICS Descriptive statistics accomplish just what you would think: they describe some data. (By the way, I typically use the word data as a plural noun because it usu- ally refers to a collection of pieces of information.) Usually, descriptive statistics describe some data by summarizing them as a single number—what we call a scalar value. Most of the time, when we calculate a descriptive statistic we do so over a vector of numbers—that is, a list of scalar values that all quantify one attribute such as height, weight, or alcohol content. To put it all together, a descriptive statistic summarizes a vector of data by calculating a scalar value that summarizes those data. Let’s look in detail at two kinds of descriptive sta- tistics: measures of central tendency and measures of dispersion. CHAPTER 1 Statistical Vocabulary 7 MEASURES OF CENTRAL TENDENCY If you have ever talked about or calculated the average of a set of numbers, you have used a “measure of central tendency.” Measures of central tendency sum- marize a vector of data by figuring out the location of the typical, the middle, or the most common point in the data. The three most common measures of central tendency are the mean, the median, and the mode.
- William P. Fox, Rodney X. Sturdivant(Authors)
- 2022(Publication Date)
- Chapman and Hall/CRC(Publisher)
3Statistical MeasuresDOI: 10.1201/9781003317906-3Objectives
- Learn how to use R to obtain and interpret descriptive statistics for the center (location), spread, and shape of the distribution of a data set.
- Learn how to use R to obtain displays and be able to interpret them.
- Know what the measures are and which ones are used for quantitative data and which ones are better for qualitative data.
3.1 Measures of Central Tendency or Location
3.1.1 Describing the Data
In addition to plots and tables, numerical descriptors are often used to summarize data. Three numerical descriptors, the mean, the median, and the mode, offer different ways to describe and compare data sets. These are generally known as the measures of location.3.1.2 The Mean
The mean is the arithmetic average, with which you are probably very familiar. For example, your academic average in a course is generally the arithmetic average of your graded work. The mean of a data set is found by summing all the data and dividing this sum by the number of data elements.The following data represent 10 scores earned by a student in a college mathematics course: 55, 75, 92, 83, 99, 62, 77, 89, 91, 72. We find the mean in the same way you would compute the student’s average. The mean can be found by summing. We first sum the 10 scores:55 + 75 + 92 + 83 + 99 + 62 + 77 + 89 + 91 + 72 = 795and then divide by the number of data elements (10), 795/10 = 79.5.To describe this process in general, we can represent each data element by a letter with a numerical subscript. Thus, for a class of n tests, the scores can be represented by a1 , a2 , …, an . The mean of these n values of a1 , a2 , …, an is found by adding these values and then dividing this sum by n
- Nigel Walford(Author)
- 2011(Publication Date)
- Wiley(Publisher)
4 Statistical Measures (or quantities)4.1 Descriptive statisticsLearning outcomes This chapter will enable readers to:Chapter 4 concentrates on the basic descriptive measures or quantities that can be derived to compare and contrast sets of numbers in their role as measuring quantifiable differences between attributes and variables. Attention is directed towards summary measures referring to numbers, such as the median, mean, range, and standard deviation, and to the location of geographical phenomena, such as mean centre and standard distance. This chapter establishes the characteristics of the basic measures and quantities that may be subject to statistical testing.- explain the purpose of spatial and nonspatial descriptive statistics for comparing and contrasting numerical data;
- identify the appropriate summary measures to use with variables recorded on different measurement scales;
- start to plan how to use descriptive statistics in an independent research investigation in Geography, Earth Science and related disciplines.
Descriptive statistics are one of two main groups of statistical techniques, the other is known as inferential statistics or hypothesis testing. Descriptive statistics are about obtaining a numerical measure (or quantity), or a frequency count to describe the characteristics of attributes and variables of observations in a dataset. We will look at frequency distributions, the principles of hypothesis testing and inferential statistics in Chapter 5, but for the present the latter can be defined as a group of analytical techniques that are used to find out if the descriptive statistics for the variables and attributes in a dataset can be considered as important or significant. Descriptive statistics are useful in their own right and in most geographical investigations they provide an initial entry point for exploring the complex mass of numbers and text that makes up a dataset. Some research questions can satisfactorily be answered using descriptive statistics on their own. However, in many cases this proves insufficient because of an unfortunate little problem known as sampling error,
- David Anderson, Dennis Sweeney, Thomas Williams, Jeffrey Camm(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
Robert Dant/Alamy Stock Photo *The authors are indebted to John A. McCarthy, President of Small Fry Design, for providing this Statistics in Practice. Copyright 2020 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. 3.1 Measures of Location 105 In Chapter 2 we discussed tabular and graphical presentations used to summarize data. In this chapter, we present several numerical measures that provide additional alternatives for summarizing data. We start by developing numerical summary measures for data sets consisting of a single variable. When a data set contains more than one variable, the same numerical measures can be computed separately for each variable. However, in the two-variable case, we will also develop measures of the relationship between the variables. Numerical measures of location, dispersion, shape, and association are introduced. If the measures are computed for data from a sample, they are called sample statistics . If the measures are computed for data from a population, they are called population parameters . In statistical inference, a sample statistic is referred to as the point estimator of the corres-ponding population parameter. 3.1 Measures of Location Mean Perhaps the most important measure of location is the mean , or average value, for a variable. The mean provides a measure of central location for the data. If the data are for a sample, the mean is denoted by x ; if the data are for a population, the mean is denoted by the Greek lower case letter mu, or m .
No longer available |Learn more- 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. 2.1 Measures of Center 55 Introduction Graphs can help you describe the basic shape of a data distribution; “a picture is worth a thousand words.” There can be problems, however, with graphs. Suppose you need to display your data to a group of people and the bulb on the data projector blows out! Or you might need to describe your data during a conference call—no way to display the graphs! You need to find another way to convey a mental picture of the data to your audience. A second problem is that graphs are somewhat imprecise for use in statistical infer-ence. For example, suppose you want to use a sample histogram to make inferences about a population histogram. How can you measure the similarities and differences between the two histograms in some concrete way? If they were identical, you could say “They are the same!” But, if they are different, it is difficult to describe the “degree of difference.” One way to solve these problems is to use numerical measures , which can be calculated for either a sample or a population of measurements. You can use the data to calculate a set of numbers that will give you a good mental picture of the frequency distribution. These measures are called parameters when associated with the population, and they are called statistics when calculated from sample measurements. DEFINITION Numerical measures associated with a population of measurements are called parameters ; those calculated from sample measurements are called statistics . Figure 2.1 Center of the birth weight data Relative Frequency Birth Weights Center 5.6 6.1 6.6 7.1 7.6 8.1 8.6 9.1 9.6 8/30 7/30 6/30 5/30 4/30 3/30 2/30 1/30 0 2.1 Measures of Center In Chapter 1, we used dotplots, stem and leaf plots, and histograms to describe a set of measurements taken on a quantitative variable x .
- David Anderson, Dennis Sweeney, Thomas Williams, Jeffrey Camm(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
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. 3.1 Measures of Location 109 In Chapter 2 we discussed tabular and graphical presentations used to summarize data. In this chapter, we present several numerical measures that provide additional alternatives for summarizing data. We start by developing numerical summary measures for data sets consisting of a single variable. When a data set contains more than one variable, the same numerical measures can be computed separately for each variable. However, in the two-variable case, we will also develop measures of the relationship between the variables. Numerical measures of location, dispersion, shape, and association are introduced. If the measures are computed for data from a sample, they are called sample statistics . If the measures are computed for data from a population, they are called population parameters . In statistical inference, a sample statistic is referred to as the point estimator of the corresponding population parameter. In the chapter appendixes we show how statistical software can be used to compute the numerical measures described in the chapter. 3.1 Measures of Location Mean Perhaps the most important measure of location is the mean , or average value, for a variable. The mean provides a measure of central location for the data. If the data are for a sample, the mean is denoted by x ; if the data are for a population, the mean is denoted by the Greek letter m . In statistical formulas, it is customary to denote the value of variable x for the first observation by x 1 , the value of variable x for the second observation by x 2 , and so on.
- Harold O. Kiess, Bonnie A. Green(Authors)
- 2019(Publication Date)
- Cambridge University Press(Publisher)
CHAPTER 5 Looking at Data: Measures of Variability 99 Shape of the Frequency Distribution The shape of the frequency distribution determines the choice of the descriptive statistics only when the dependent variable is measured with an interval or ratio scale of measure- ment. If the distribution of scores is approximately symmetrical, then the sample mean and standard deviation are the preferred statistics. But if the distribution of scores is consider- ably skewed, then the median and range may be the descriptive statistics of choice. Further Data Analysis The choice of measures of central tendency and variability is also determined by whether the descriptive statistics are to be used for statistical inference. For inferential purposes, the mean, estimated population standard deviation, and estimated population variance are used because they are related to the known characteristics of the normal distribution. These relationships are discussed in Chapter 6. Hence, behavioral scientists may use the sample mean and standard deviation to summarize their data when considerations of level of measurement and shape of the frequency distribution might dictate against it. The issue of making inferences from the data sometimes overrides these other considerations. For this reason, you will find that the mean and standard deviation are the most frequently used descriptive statistics in behavioral research. 1. For each of the following distributions of scores, indicate the measure of central ten- dency and variability that would provide the most appropriate description of the scores. a. The scores represent interval measurement, and the distribution is heavily nega- tively skewed. b. The scores represent ordinal measurement, and the distribution is approximately symmetric. c. The scores represent ratio measurement, and the distribution is approximately symmetric. d. The scores represent ordinal measurement, and the distribution is heavily positively skewed.
eBook - PDFEssential Maths
for Business and Management
- Clare Morris(Author)
- 2007(Publication Date)
- Bloomsbury Academic(Publisher)
167 Statistics: summarising data CHAPTER 9 Intended learning outcomes By the end of your work on this chapter you should: ɀ understand why it is desirable to calculate summary measures for a set of data, and what features of the data these measures might represent ɀ be able to define the mean, median and mode of a set of data, and to calculate these by hand in simple cases ɀ be able to define the standard deviation, quartiles and mode of a set of data, and to calculate these by hand in simple cases ɀ be able to interpret the summary measures in practical terms, and to make an informed choice of the best measure to use in a given situation ɀ know what is meant by skewness, and be able to describe in qualitative terms the skewness or symmetry of a distribution. Prerequisites In order to obtain full benefit from your work on this chapter, you should be confident in your understanding of the material covered in Chapter 8. 168 Essential Maths The tables and charts which we constructed in Chapter 8 were very helpful in enabling data to be presented in a simple and meaningful way. However, for many purposes, further simplification is required. For example, if we want to compare two or more sets of numbers – perhaps comparing the ‘number of items purchased’ data from the XYZ Supermarket survey of Chapter 8 with similar data from another supermarket in a differ-ent location – doing so by looking at a chart or a table is not easy. We need to be able to reduce the large number of individual measurements to just two or three numbers which will sum up the key features of the data. But just what are those key features of a dataset? We can get some ideas by looking at the histograms which we constructed in Chapter 8 – for example, the one shown in Figure 8.7 for the data giving the length of customers’ visits to the supermarket.
eBook - PDF- Paul J. Mitchell(Author)
- 2022(Publication Date)
- Wiley-Blackwell(Publisher)
5 Descriptive statistics; measures to describe and summarise data sets I argued in Chapter 1 that statistical analysis may be divided into Descriptive and Inferential Statistics. So, let ’ s take a set of values and examine how such data may be described by Descriptive Statistics . The following discussion refers to height data obtained from a group of young adults to demonstrate how Descriptive Statistics may be used to describe and summarise experimental data. Table 5.1 below lists the various heights of 40 female undergraduate students obtained in 2016 and 2017; I ’ ve only included female stu-dents since, as we shall see later, sex is important even in statistics! There are some descriptive values which we can obtain easily from such a set of data; • Minimum : the lowest value in the set. Here the minimum value is 1.482 m. • Maximum : the highest value in the set. Here the maximum value is 1.794 m. • Range : the range is the difference between the minimum and maximum values. Here the range is 1.794 -1.482 = 0.312 m. However, these values provide little useful information about the data set, especially if we wish to compare this set to the heights of another set of students. For example, Table 5.2 lists the corresponding heights of the male students taken from the same student cohort. The minimum, maximum, and range vales of these data are, 1.670 m, 1.920 m, and 0.250 m, respectively. Comparison of these two sets of data suggests that the shortest and tallest of the male stu-dents are both taller than their equivalent counterparts in the female cohort, while the range of the male heights is slightly less than that of the female heights. Does this infer that the males are generally taller than the females? If so, then how can we be sure? One descriptive statistic we could easily obtain is where the cen-tre of the frequency distribution lies (this is known as a measure of central tendency of the data sets).
eBook - PDF- Sally L.D. Katary, Katary(Authors)
- 2018(Publication Date)
- Routledge(Publisher)
If a distribution contains extreme values, the mean will be pulled toward these values and so lose its typicality as a measure of central tendency. The greatest advantage of the mean, however, is that it can be used as a basis for a number of valuable computations such as variance and standard deviation which will be examined later in this chapter. The median is another average which is useful in describing and understanding distributions of quantitative data. It is the value that divides a set of observations ordered with respect to the 31 Chapter 2 - Statistical Measures and Tests magnitude of values so that the number of items above it is equal to the number of items below it. The median is therefore called the position average. The great utility of the median in contrast to the mean lies in the fact that it is the most representative average of any distribution. This is because the aggregate distance between the median and each of the values in the distribution is less than from any other point. The mode of a distribution is that value obtained most frequently in a series of observations, i.e., the value at which the concentration of items is most dense. The mode is also called the probability average because it is the most probable value in any given distribution. While the mode provides the most probable value in the distribution, it does not reveal the degree of probability of that value. A second limitation lies in the fact that the mode is not easily manipulated (i.e., does not lend itself to algebraic manipulation) and cannot be used as a basis for further computation as the mean can be. Since it is influenced by the interval width employed, the mode may also be said to lack stability. The usefulness of the mode, however, in many ways out weighs its disadvantages provided that it is used in conjunction with other measures of central tendency. The modal value is appropriate where it is important to know the predominate class, size, or quantity.
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