Mathematics
Measures of Central Tendency
Measures of central tendency are statistical values that represent the center or average of a data set. The three main measures of central tendency are the mean, median, and mode. The mean is the sum of all values divided by the number of values, the median is the middle value when the data is arranged in ascending order, and the mode is the most frequently occurring value.
Written by Perlego with AI-assistance
Related key terms
1 of 5
12 Key excerpts on "Measures of Central Tendency"
eBook - PDFQuantitative 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)
39 3 Measures of Central Tendency 3.1 Introduction One of the most important objectives of statistical analysis is to get one single value that describes the characteristic of the entire mass of unwieldy data. Such a value is called the central value, or an ‘ average ’ . The word average is very commonly used in day-to-day conversation. For example, we often talk of average students in a class, average height or weight, average income and so forth. The objective here is to find one representative value which can be used to locate and summarise the entire set of varying values. This one value can be used to make many deci-sions concerning the entire set. We can define Measures of Central Tendency (or location) to find some central value around which the data tend to cluster. 3.2 Significance of Measures of Central Tendency The significance is to get one single value that describes the characteristics of the entire group. The central value is measured by condensing the mass of data into one single value, enabling us to get a bird ’ s-eye view of the entire data. Thus, one value can represent thou-sands, lakhs and even millions of values. To facilitate comparison, the central value is measured by reducing the mass of data into one single figure, enable comparisons to be made. Comparison can be made either at time points or over a period of time. For example, we can compare the percentage results of the students of different colleges on a certain examination. Measures of Central Tendency, that is considering the mass of data in one single value, enable us to get an idea of the entire data. For example, it is impossible to remember the individual incomes of millions of earning people of any country. But if the average income is obtained, we get one single value that represents the entire population. Measures of Central Tendency also enable us to compare two or more sets of data to facili-tate comparison.
eBook - ePubQuantitative 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)
3Measures of Central Tendency
3.1 Introduction
One of the most important objectives of statistical analysis is to get one single value that describes the characteristic of the entire mass of unwieldy data. Such a value is called the central value, or an ‘average’. The word average is very commonly used in day-to-day conversation. For example, we often talk of average students in a class, average height or weight, average income and so forth.The objective here is to find one representative value which can be used to locate and summarise the entire set of varying values. This one value can be used to make many decisions concerning the entire set. We can define Measures of Central Tendency (or location) to find some central value around which the data tend to cluster.3.2 Significance of Measures of Central Tendency
The significance is to get one single value that describes the characteristics of the entire group. The central value is measured by condensing the mass of data into one single value, enabling us to get a bird’s-eye view of the entire data. Thus, one value can represent thousands, lakhs and even millions of values.To facilitate comparison, the central value is measured by reducing the mass of data into one single figure, enable comparisons to be made. Comparison can be made either at time points or over a period of time. For example, we can compare the percentage results of the students of different colleges on a certain examination.Measures of Central Tendency, that is considering the mass of data in one single value, enable us to get an idea of the entire data. For example, it is impossible to remember the individual incomes of millions of earning people of any country. But if the average income is obtained, we get one single value that represents the entire population.Measures of Central Tendency also enable us to compare two or more sets of data to facilitate comparison. For example, the average sales figures of June may be compared with the sales figures of previous months.
eBook - ePub- Perumal Mariappan(Author)
- 2019(Publication Date)
- Chapman and Hall/CRC(Publisher)
4 Measures of Central Tendency (MCT) 4.1 IntroductionStatistical methods are needed for summarizing and describing the collected numerical data. The main objective of this chapter is to introduce one representative value that can be used to identify and summarize an entire set of data. This representative value is going to be helpful in making decisions based on the data collected. Measures of Central Tendency (MCT) are used to set the central value around which the data are spread over.4.2 MCTThe average of a distribution is its representative size. As most of the items of the series cluster around the average, it is called a ‘measure of central tendency’. The average is computed to reduce the complexity of the data. The entire distribution is reduced to one number, which can be considered typical of an important characteristic of the population, and the same can be used in making comparisons and in examining relations with other distributions.For example, it is not possible to remember the individual’s income in crores of earning people in India. By considering all the data related to income, if the average income is evaluated, we get a single value, which is going to represent the entire population. The commonly used averages are as follows:• Arithmetic mean • Median • Mode • Geometric mean • Harmonic mean 4.2.1 Properties of Best Average It should be • Rigidly defined. • Based on all observations of the series. • Easy to calculate and simple to understand. • Capable of further algebraic treatment. • Free from the extreme values (i.e., it should not be affected by extreme values).The arithmetic mean is ideal in these respects, even though the other averages are also useful in certain specific cases. The median is quite useful for studying data not capable of direct quantitative measurement like skin colour, etc. The measure mode is good when the extreme values are not well defined.
eBook - PDF- Sally Caldwell(Author)
- 2012(Publication Date)
- Cengage Learning EMEA(Publisher)
On the other hand, don’t suspend your concentra-tion on what you’re reading. There’s likely to be some new material to digest. Accordingly, let me urge you to take whatever time is necessary to develop a thorough understanding of the various concepts. In many ways, they represent essential building blocks in the field of statistics. Measures of Central Tendency To a statistician, the mean (or more correctly, the arithmetic mean) is only one of several Measures of Central Tendency. The purpose behind any measure of central tendency is to get an idea about the center, or typicality, of a distribu-tion. As it turns out, though, the idea of the center of a distribution and what that really reflects depends on several factors. That’s why statisticians have several Measures of Central Tendency. The Mean The one measure of central tendency that you’re probably most familiar with is the one I mentioned earlier—namely, the mean. The mean is calculated Copyright 2012 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. Measures of Central Tendency 23 by adding all the scores in a distribution and dividing the sum by the number of scores. If you’ve ever calculated your test average in a class (based on a number of test scores over the semester), you’ve calculated the mean. I doubt there is anything new to you about this, so let’s move along without a lot of commentary. Now let’s have a look at the symbols that make up the formula for the mean. Remember: All that’s involved is summing all the scores (or values) and then dividing the total by the number of scores (or values).
- Bruce M. King, Patrick J. Rosopa, Edward W. Minium(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
4 Central Tendency When you have finished studying this chapter, you should be able to: • Understand what is meant by “measure of central tendency”; • Calculate the mode, median, and mean for ungrouped and grouped distributions; • Appreciate the unique properties of the mode, median, and the mean; • Describe where each measure of central tendency falls in symmetrical and skewed distribu- tions; and • Understand how each of the Measures of Central Tendency is affected by adding a constant to each score, subtracting a constant from each score, or multiplying or dividing each score by a constant. Where, on the scale of possible scores, is a point that best represents the overall level (high, medium, low) of the set of scores? Measures of this type are called measure of central tendency. measure of central ten- dency a single summary figure that describes the central location of an entire distribution of observations Their purpose is to provide a single summary figure that best describes the central location of an entire distribution of observations. Two distributions that differ only in central tendency are shown in Figure 4.1. A measure of central tendency is helpful in comparing the performance of a group with that of a standard reference group. It also helps to simplify the comparison of two or more groups tested under different conditions. For example, we may be interested in whether video games can improve mental or physical functioning in the elderly. In one study, psychologists administered an IQ test to 11 volunteers in their 60s and 70s, then gave them 30 minutes of play on a video game twice a week for 2 months, and then tested them again. The average IQ was 101.8 before the 2-month intervention and 108.3 afterward (Drew & Waters, 1985). We would need to know more than this to conclude that video games improve mental functioning in the elderly, but as a first step, a comparison of the group averages suggests that it might.
- Harold O. Kiess, Bonnie A. Green(Authors)
- 2019(Publication Date)
- Cambridge University Press(Publisher)
C H A P T E R 4 Looking at Data: Measures of Central Tendency Sample Mean Calculating the Sample Mean Characteristics of the Sample Mean Population Mean Median Calculating the Median with an Odd Number of Scores Calculating the Median with an Even Number of Scores Characteristics of the Median Mode Journal Presentation of Measures of Central Tendency Comparing Measures of Central Tendency Descriptive Uses of Measures of Central Tendency Inferential Uses of Measures of Central Tendency Summary Key Terms and Symbols Review Questions CHAPTER 4 Looking at Data: Measures of Central Tendency 71 C hances are that you took an exam recently either in this class or another one. When grades were returned, you probably wanted to know how the class as a whole did on the exam. Most likely, the professor indicated what the typical grade was and how much the grades varied from one another. In this instance, the professor pro- vided you with some descriptive statistics to help you better understand your grade in the context of the other grades on the exam. In the last chapter, we introduced how to organize data by using frequency distributions and graphs. Although frequency distributions and graphs present a great deal of information about the shape of a distribution of scores and the range of scores, a scientist often needs to describe a distribution of scores with only one number. These plain but mighty numbers are called descriptive statistics for they describe the raw data with a single number. There are two types of descriptive statistics that are commonly used. One type, called Measures of Central Tendency, indicates the typical score obtained, and the other type, called measures of variability or measures of disper- sion, indicates the amount of variation or dispersion around the typical score.
No longer available |Learn moreData Analytics
Systems Engineering - Cybersecurity - Project Management
- Christopher Greco(Author)
- 2021(Publication Date)
- Mercury Learning and Information(Publisher)
3 STATISTICS REVIEW – MEASURES OF THE CENTRAL TENDENCYIt is interesting when someone draws a circle. The circle takes a shape, but if the person wants a perfect circle, they should use a protractor and start with a center point to ensure that the area of the circle is the same in all directions. This is the same as the concept of the “central tendency in statistics” (or central tendency). The idea of finding the center of the data, much like having the center of the circle, is the perfect beginning to the analysis of that data. These measures are not only vital to determining the center of the data, but essential for the fundamentals of data science. The first measure of the central tendency that most people have some familiarity with using is the mean, or average. Before delving into this concept, remember back in high school when the teacher was handing out the test results? She may have described the average of the test. This is not only a measure of center, but also a measure of how well the student did in relation to other students. An average or mean is the focal point of many analytic beginnings and should always be considered at the beginning of the analysis because it helps describe the data. Some analysts gloss over the mean as just a forced calculation to get to the more detailed analysis, but in actuality, the mean is probably the best way to determine some basics about the data. For example, in many convenience stores, there is a height chart so that the police can make a quick description of an individual should that individual rob the store. Think of the mean as this chart, a place to quickly observe and describe data now and in the future.3.1 MeanMost people use the mean and do not even realize they are doing so. They prefer using the mean because it is relatively straightforward and gives a quick result. However, and this is important, the mean is impacted by high and low numbers. These very high and low numbers in a dataset are called outliers and are described in a handbook by a federal agency (www.itl.nist.gov/div898/handbook/prc/section1/prc16.htm
eBook - PDF- Dawn A. Willoughby(Author)
- 2016(Publication Date)
- Wiley(Publisher)
C H A P T E R • 5 Measures of Central Tendency OBJECTIVES This chapter explains how to: • calculate the mode, mean and median for - raw data sets - frequency distributions - grouped frequency distributions • calculate a combined mean and a weighted mean • understand when it is appropriate to use each measure of central tendency KEY TERMS arithmetic mean average bimodal combined mean mean measure of central tendency median mode multimodal resistant weighted mean Introduction In Chapter 3, we explored a variety of techniques that allowed us to represent data in a pictorial form. These graphical methods are important for showing structure and present- ing data clearly, but it can sometimes be helpful to use a numerical value to summarise a data set. A measure of central tendency describes a set of data by identifying a ‘typical value’ that is representative of the set. Such a value could be typical because it occurs most frequently in the data set (mode), because it is the average for the data set (mean), or because it is the middle point when the data set is ordered (median). Each measure requires a different type of calculation and has advantages and disadvantages associated with its use. The Mode The mode of a set of data is the value which occurs most frequently. If no data value appears more frequently than any other values in the set, then there will be no mode. A data set with two modes is known as bimodal, while a data set with more than two modes is described as multimodal. When finding the mode of a set of raw data, it can be useful to put the data in order so that values which are the same will appear together in the list. In a frequency distribution, we look for the data value which has the highest associated frequency. For grouped frequency distributions, we aim to identify the modal class rather than a specific value for the mode because the individual data values are not available in this type of distribution.
- David Howell(Author)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
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. 4.7 Summary In this chapter we considered several measures used to describe the center of a distribu-tion. The most frequently used measure is the mean, often represented by the symbol X , which is simply what most of us learned in elementary school as the “average.” You add up all of the scores and then divide by the number of scores, which is usually denoted as N . We went a bit further with the mean and discussed the trimmed mean, which is simply the mean of the data you have left when you drop some percentage of values at each end of the distribution. Trimming 10% at each end often proves useful. A third very useful measure of central tendency is the median, which is the middle value when you ar-range your data in ascending or descending order. (If there are an even number of values, the median is the average of the two middle values.) Finally, we have the mode, which is that value (or set of values) that occurs most frequently in the distribution of outcomes. The mean is the most commonly used measure of central tendency, but the median is very useful when you want to minimize the effect of extreme scores. When speaking about the salaries of sports figures, for example, the median salary is a more meaningful measure of how much players make, because, unlike the mean, it is not influenced by the huge salaries that a few players receive.
eBook - PDF- Frederick Gravetter, Larry Wallnau(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
3 b. 3.5 c. 4 d. 5 1. A, 2. A, 3. A LEARNING CHECK Number of Absences f 5 1 4 2 3 7 2 5 1 3 0 2 X f 5 1 4 3 3 4 2 2 1 1 ANSWER S SECTION 3.4 | The Mode 85 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. 3.5 Selecting a Measure of Central Tendency 8. Explain when each of the three Measures of Central Tendency—mean, median, and mode—should be used, identify the advantages and disadvantages of each, and describe how each is presented in a report of research results. You usually can compute two or even three Measures of Central Tendency for the same set of data. Although the three measures often produce similar results, there are situations in which they are predictably different (see Section 3.6). Deciding which measure of central tendency is best to use depends on several factors. Before we discuss these factors, how-ever, note that whenever the scores are numerical values (interval or ratio scale) the mean is usually the preferred measure of central tendency. Because the mean uses every score in the distribution, it typically produces a good representative value. Remember that the goal of central tendency is to find the single value that best represents the entire distribu-tion. Besides being a good representative, the mean has the added advantage of being closely related to variance and standard deviation, the most common measures of vari-ability (Chapter 4). This relationship makes the mean a valuable measure for purposes of inferential statistics.
- Alan R. Jones(Author)
- 2018(Publication Date)
- Routledge(Publisher)
Measures of Central Tendency: Means, Modes, Medians 2 As estimators, we are faced with the dilemma of what value we should choose as being the most representative of the task, activity or, if you prefer, ‘ thing’ for which we have been asked to provide an estimate. Whether we like to admit it or not, in making such a decision, we are delving into the world of statistics. 2.1 ‘S’ is for shivers, statistics and spin The mere mention of the ‘S’ word sends shivers down some people’s backs, and yet everyone uses them to some degree. The reputation of statistics is often that they can be used to bamboozle, confound or confuse, and are the tools of confidence tricksters and political spin doctors and the like.That is not a new perception, and it has prevailed since Victorian days at least as the views of Benjamin Disraeli demonstrate. For the estimator, and other number jug- glers, they provide a means of making an informed decision based on the evidence available. They either summarise and describe an aspect or characteristic of the evidence, or they provide a way of weighting how likely some future event or thing will happen based on the evidence of the past. However, on the assumption that many of us may still be in the ‘shiver’ camp, we’ll start our journey with what are probably A word (or two) from the wise? ‘There are three kinds of lies: lies, damned lies, and statistics’. Benjamin Disraeli (1804–1881) UK Prime Minister 16 | Measures of Central Tendency ( in the statistical sense) the most used statistics of all; collectively these are referred to as the Measures of Central Tendency ( see, even the terminology infers something dark and mysterious, almost as if the data could exercise free will and telekinetic properties!) 2.1.1 Cutting through the mumbo-jumbo: What is or are statistics? Statistics can be defined as either or both: 1.
No longer available |Learn moreUnderstandable Statistics
Concepts and Methods, Enhanced
- Charles Henry Brase, Corrinne Pellillo Brase(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
Copyright 201 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. 132 Chapter 3 AVERAGES AND VARIATION CHAPTER REVIEW SUMMARY IMPORTANT WORDS & SYMBOLS Section 3.1 Average 90 Mode 90 Median 91 Mean 93 Summation symbol, S 93 Sample mean, x 93 Population mean, m 93 Resistant measure 94 Trimmed mean 94 Weighted average 96 Harmonic mean 101 Geometric mean 102 Section 3.2 Range 102 Variance 103 Standard deviation 103 Sum of squares, © 1 x 2 x 2 2 103 Sample variance, s 2 103 Sample standard deviation, s 104 Population variance, s 2 107 Population standard deviation, s 107 Population size, N 107 Coefficient of variation, CV 109 Chebyshev’s theorem 110 Outlier 112 Mean of grouped data 117 Standard deviation of grouped data 117 Moving average 118 Section 3.3 Percentile 122 Quartiles 123 Interquartile range, IQR 123 Five-number summary 125 Box-and-whisker plot 125 Whisker 126 To characterize numerical data, we use both measures of center and of spread. • Commonly used measures of center are the arithmetic mean, the median, and the mode. The weighted average and trimmed mean are also used as appropriate. • Commonly used measures of spread are the variance, the standard deviation, and the range. The variance and standard deviation are measures of spread about the mean. • Chebyshev’s theorem enables us to estimate the data spread about the mean. • The coefficient of variation lets us compare the relative spreads of different data sets. • Other measures of data spread include per-centiles, which indicate the percentage of data falling at or below the specified percentile value.
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.
