Mathematics
Mean Median and Mode
Mean, median, and mode are measures of central tendency used in statistics. The mean is the average of a set of numbers, calculated by adding them together and dividing by the total count. The median is the middle value when the numbers are arranged in ascending order. The mode is the number that appears most frequently in the set.
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10 Key excerpts on "Mean Median and Mode"
- Bruce M. King, Patrick J. Rosopa, Edward W. Minium(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
The median shows the point along the scale of possible scores that divides the lower half of scores from the upper half. It is another name for P 50 . The median is responsive to the number of scores above or below it but not to their magnitude and thus is less affected by extreme scores than the mean. The mean is the sum of all the scores divided by the total num- ber of scores. It is the balance point of the distribution (i.e., the point around which the sum of the negative deviations equals that of the positive deviations). The mean is responsive to the exact position (magnitude) of each score in the distribution and is there- fore more sensitive to extreme scores than are the median and the mode. The three measures of central tendency are affected differently by the symmetry of a distribution. For a bell-shaped, normal dis- tribution, mean, median, and mode have the same value, but for a lopsided (skewed) distribution, their values will differ. Among the three measures, the mean varies the least from sample to sam- ple, which is why it is the measure of choice in most behavioral science applications. Nevertheless, there are some situations (e.g., open-ended distributions and those that are strongly skewed or have a few very deviant scores) in which the median may have significant advantages. Finally, we note that changing each score by adding or sub- tracting a constant amount increases or decreases each of the three measures by that same amount. Similarly, if we multiply or divide each score by a constant, it has the effect of multiplying or dividing these measures of central tendency by the same amount. Mathematical Note Note 4.1 The Mean as a Balance Point (Ref.: Section 4.5) The algebraic way of saying that the mean is the balance point of the distribution is Σ(X − X) = 0.
- Brase/Brase, Charles Henry Brase, Corrinne Pellillo Brase(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
For skewed-right distributions, the mode is the smallest value, the median is the next largest, and the mean is the largest. Figure 3-1 shows the general relationships among the mean, median, and mode for different types of distributions. CRITICAL THINKING continued The ideas at the right can be used to review levels of measurement and link some of those concepts to the material in this section. Distribution shapes and averages (a) Mound-shaped symmetrical Mean Median Mode (b) Skewed left Mode Mean Median (c) Skewed right Mode Mean Median FIGURE 3-1 Distribution Types and Averages © Cengage Learning ® © Cengage Learning ® © Cengage Learning ® 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. Section 3.1 Measures of Central Tendency: Mode, Median, and Mean 95 W EIGHTED AVERAGE Suppose your midterm test score is 83 and your final exam score is 95. Using weights of 40% for the midterm and 60% for the final exam, compute the weighted average of your scores. If the minimum average for an A is 90, will you earn an A? SOLUTION: By the formula, we multiply each score by its weight and add the results together. Then we divide by the sum of all the weights. Converting the percentages to decimal notation, we get Weighted average 5 83 1 0.40 2 1 95 1 0.60 2 0.40 1 0.60 5 33.2 1 57 1 5 90.2 Your average is high enough to earn an A. The TI-84Plus/TI-83Plus/TI-n spire calculators directly support weighted averages. Both Excel 2010 and Minitab can be programmed to provide the averages.
eBook - PDF- Sally Caldwell(Author)
- 2012(Publication Date)
- Cengage Learning EMEA(Publisher)
28 CHAPTER 2 Describing Data and Distributions of the median. Also, the formula rests on the assumption that the scores in the distribution are in ascending or descending order. LEARNING CHECK Question: What is the median, and how is it determined? Answer: The median is a measure of central tendency; it is the score that cuts a distribution in half. The formula locates the position of the median in a distribution, provided the scores in distribution have been arranged in ascending or descending order. The Mode In addition to the mean and the median, there’s another measure of central tendency to consider—namely, the mode. The mode is generally thought of as the score, value, or response that appears most frequently in a distribu-tion. For example, a distribution containing the values 2, 3, 6, 1, 3, and 7 would produce a mode of 3. The value of 3 appears more frequently than any other value. Table 2-5 Locating the Median Scores/Values Scores/Values 1 2 4 8 12 Median = 4 1 2 4 8 120 Median = 4 Scores/Values Scores/Values 3 5 7 9 Median = 6 10 15 25 80 Median = 20 Scores/Values Scores/Values 10 10 14 23 23 80 100 Median = 23 17 27 34 34 34 59 62 Median = 34 © Cengage Learning 2013 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 29 A distribution containing the values 2, 3, 6, 1, 3, 7, and 7 would be referred to as a bimodal distribution because it has two modes—3 and 7. Both values (3 and 7) appear an equal number of times, and both appear more frequently than any of the other values.
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)
2. It is not affected by extreme values. It can also be used even when the classes are open-ended. 3. It is usually an actual value, as it occurs most frequently in the series. 4. Its value can be determined graphically. Demerits of mode 1. At times, a data set contains no value that occurs more than once. Further, all val-ues in a data set might occur an equal number of times; that is all items have the same frequency. 2. It cannot always be determined, as some data sets contain two, three or many modes, making it difficult to interpret them. 3. It is not based on all observations. 4. It is not capable of further algebraic treatment. 3.8 Relationship among Mean, Median and Mode A distribution in which mean, median and mode coincide is known as a symmetrical (bell-shaped) distribution. If a distribution is skewed (i.e. not symmetrical), then mean, median and mode are not equal. In a moderately skewed distribution, a very interesting relation-ship exists among mean, median and mode. In such type of distributions, it can be proved that the distance between the mean and median is approximately one-third of the distance between the mean and mode. This is shown below for two types of such distributions. In the case of a symmetrical distribution, the mean, median and mode coincide. However, according to Karl Pearson, if the distribution is moderately asymmetrical, the mean, median and mode are related in the following manner: Mean – Median = (Mean – Mode)/3 Thus, Mode = 3 Median – 2 Mean Similarly, we can express the approximate relationship for median in terms of mean and mode. Also, this can be expressed for mean in terms of median and mode. Thus, if 62 Quantitative Techniques in Business, Management and Finance we know any of the two values of the averages, the third average value can be determined from this approximate relationship.
- 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)
Although they both define the middle of the distribution, they use differ-ent definitions of the term “middle.” 1 2 1 2 Frequency 3 3 4 m 5 4 5 6 7 8 9 10 11 12 13 X F I G U R E 3.6 A population of N 5 6 scores with a mean of μ 5 4. Notice that the mean does not neces-sarily divide the scores into two equal groups. In this example, 5 out of the 6 scores have values less than the mean. 1. What is the median for the following set of scores: Scores: 3, 4, 6, 8, 9, 10, 11 a. 7 b. 7.5 c. 8 d. 8.5 LEARNING CHECK 82 CHAPTER 3 | Central Tendency 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. 3.4 The Mode 7. Define and identify the mode(s) for a distribution, including the major and minor modes for a binomial distribution. The final measure of central tendency that we will consider is called the mode . In its com-mon usage, the word mode means “the customary fashion” or “a popular style.” The statis-tical definition is similar in that the mode is the most common observation among a group of scores. In a frequency distribution, the mode is the score or category that has the greatest frequency. As with the median, there are no symbols or special notation used to identify the mode or to differentiate between a sample mode and a population mode. In addition, the defini-tion of the mode is the same for a population and for a sample distribution. The mode is a useful measure of central tendency because it can be used to deter-mine the typical or most frequent value for any scale of measurement, including a nominal scale (see Chapter 1).
- Harold O. Kiess, Bonnie A. Green(Authors)
- 2019(Publication Date)
- Cambridge University Press(Publisher)
For example, the median is often reported for the typical income for a population because incomes are heavily positively skewed. There are only a few people who have incomes in the many millions of dollars (i.e., outliers). If the numerical values of these incomes were included in the measure of typical income, it would be too high to represent an average person. The median, however, is not affected by these few large incomes and, thus, pro- vides a measure of central tendency representative of a typical income. As another example, the typical sale price for houses in a community is usually reported as a median, for the sale of one very expensive house in a community of more modest houses might consider- ably skew the distribution of sale prices. The median is also often used as a measure of central tendency when not all entries in a distribution have a numerical value. For example, for teams in the National Football League, it typically takes a median of 8 years after being purchased by a new owner to win a championship (Healy, 2008). This median, however, is based only on the 11 teams that have won a Super Bowl since being purchased by a new owner. The other 21 teams in the league have never won the Super Bowl, so there is no numerical value to attach to the number of years before winning the Super Bowl for these teams. Thus the median is used as a measure of central tendency for this distribution. Mode The mode is the most frequently occurring score in a distribution of scores. The mode is found by inspecting a simple frequency distribution and finding which score occurs most often. For example, in Table 3.2 (page 38), the most frequently occurring Implicit View of Intelligence score is 31, with five occurrences. Accordingly, the mode for this distribution is 31. Because there is only one mode for this set of scores, this distribution is said to be unimodal.
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.
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)
However, harmonic mean cannot be computed when one or more observations have a zero value or when there are either positive or negative observations. In dealing with business problems, harmonic mean is rarely used.TABLE 3.23 Calculations for Computing Harmonic Mean3.12 Summary
Central tendency is a statistical measure that takes one number as the representative of a group. It is an economical estimate of the general characteristics of the group. There are three main measures: mean, median and mode. Mean is the average of all the observations. The median is the middle of a distribution: half the observations are above the median and half are below the median. The median is less sensitive to extreme scores than the mean. This makes it a better measure than the mean for a data set with extreme values. The mode is the most frequently occurring observation in a distribution.Measures of central tendency give one of the very important characteristics of data. Any one of the various measures of central tendency may be chosen as the most representative or typical measure. The arithmetic mean is widely used and understood as a measure of central tendency. The concepts of weighted arithmetic mean, geometric mean and harmonic mean are useful for specific types of applications. The median is generally a more representative measure for open-ended distributions and highly skewed distributions. The most demanded or customary value is needed.REVIEW QUESTIONS
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- Distinguish between arithmetic mean and median?
- Define measure of central te ndency.
- What are the applications of central tendency?
- State the importance of median.
- What are the various measures of central tendency studied in this unit? Explain the difference between them.
- Discuss the mathematical properties of arithmetic mean and median.
- Review the advantages and disadvantages of each measure of central tendency.
- Explain how you will decide which average to use in a particular problem.
- What are quantiles? Explain and illustrate the concepts of quartiles, deciles and percentiles.
SELF-PRACTICE PROBLEMS
- Median = 139.69, mean = 139.51. Calculate the mode.
- Find the mean of the following:
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.
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