Mathematics
Median
The median is a measure of central tendency in a set of numbers. It is the middle value when the numbers are arranged in ascending or descending order. If there is an even number of values, the median is the average of the two middle numbers.
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11 Key excerpts on "Median"
- David Howell(Author)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
2 The definition of the Median is another one of those things over which statisticians love to argue. The defini-tion given here, in which the Median is defined as a point on a distribution of numbers, is the one most critics prefer. It is also in line with the statement that the Median is the 50th percentile. On the other hand, there are many who are perfectly happy to say that the Median is either the middle number in an ordered series (if N is odd) or the average of the two middle numbers (if N is even). Reading these arguments is a bit like going to a faculty meeting when there is nothing terribly important on the agenda. The less important the issue, the more there is to say about it. 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. 66 Chapter 4 Measures of Central Tendency Thus, for five numbers the Median location 5 (5 1 1)/2 5 3, which simply means that the Median is the third number in an ordered series. For 12 numbers the Median location 5 (12 1 1)/2 5 6.5; the Median falls between, and is the average of, the sixth and seventh numbers. For the data on reaction times in Table 3.1, the Median location 5 (600 1 1)/2 5 300.5. When the data are arranged in order, the 300.5th score is in the interval of 1.50–1.59, and so we take the midpoint of that interval (1.55) as our Median. For the data on intrusive thoughts in the breast cancer patients presented in Figure 3.13, there are 85 scores, and the Median location is (85 1 1)/2 5 43. We can tell from the stem-and-leaf display that the 43rd score is 15, which is the Median.
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)
A second measure of central tendency is the Median. Median is that value which divides the distribution into two equal parts. Fifty percent of the observations in the distribution are above the value of Median, and the other 50% of the observations are below this value. The Median is the value of the middle observation when the series is arranged in order of size or magnitude. If the number of observations is odd, then the Median is equal to one of the original observations. If the number of observations is even, then the Median is the arithmetic mean of the two middle observations; for example if the incomes of seven persons are 1100, 1200, 1350, 1500, 1550, 1600 and 1800, then the Median income would be Rs. 1500. Suppose one more person joins and his income is Rs. 1850; then the Median income of eight persons would be(Since the number of observations is even, the Median is the arithmetic mean of the fourth and fifth persons.)= 15251500 + 15502As it is distinct from the arithmetic mean, which is calculated from the value of every item in the series, it is called a positional average. The term position refers to the place of value in the series. The Median is just the 50th percentile value below which 50% of the values in a sample fall. The object of Median is therefore not merely to fix a value that shall be representative of a set, but also to establish a dividing line separating the higher from the lower values.- It is especially useful in the case of open-ended classes since only the position and not the values of items must be known.
- It is not influenced by the magnitude of extreme deviations from it. For example, the Median of 10, 20, 30, 40 and 50 would be 30, whereas the mean is 50. Hence, very often when extreme values are present in a set of observations, the Median is a more satisfactory measure of the central tendency than the mean.
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)
Fifty percent of the observations in the distribution are above the value of Median, and the other 50% of the observations are below this value. The Median is the value of the middle observation when the series is arranged in order of size or magnitude. If the number of observations is odd, then the Median is equal to one of the original observations. If the number of observations is even, then the Median is the arithmetic mean of the two middle observations; for example if the incomes of seven persons are 1100, 1200, 1350, 1500, 1550, 1600 and 1800, then the Median income would be Rs. 1500. Suppose one more person joins and his income is Rs. 1850; then the Median income of eight persons would be 1500 1550 2 + = 1525 TABLE 3.10 Calculation of Weighted Arithmetic Mean Grade (X) Weight (W) WX Final examination 80 50 4000 First assignment 95 25 2375 Second assignment 85 25 2125 ∑ W = 100 ∑ WX = 8500 51 Measures of Central Tendency (Since the number of observations is even, the Median is the arithmetic mean of the fourth and fifth persons.) As it is distinct from the arithmetic mean, which is calculated from the value of every item in the series, it is called a positional average. The term position refers to the place of value in the series. The Median is just the 50th percentile value below which 50% of the values in a sample fall. The object of Median is therefore not merely to fix a value that shall be representative of a set, but also to establish a dividing line separating the higher from the lower values. 1. It is especially useful in the case of open-ended classes since only the position and not the values of items must be known. 2. It is not influenced by the magnitude of extreme deviations from it. For example, the Median of 10, 20, 30, 40 and 50 would be 30, whereas the mean is 50. Hence, very often when extreme values are present in a set of observations, the Median is a more satisfactory measure of the central tendency than the mean.
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.
- 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.
- 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- Frederick Gravetter, Larry Wallnau(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
■ ■ The Median, the Mean, and the Middle Earlier, we defined the mean as the “balance point” for a distribution because the distances above the mean must have the same total as the distances below the mean. One conse-quence of this definition is that the mean is always located inside the group of scores, somewhere between the smallest score and the largest score. You should notice, however, that the concept of a balance point focuses on distances rather than scores. In particular, it is possible to have a distribution in which the vast majority of the scores are located on one side of the mean. Figure 3.6 shows a distribution of N = 6 scores in which 5 out of 6 scores have values less than the mean. In this figure, the total of the distances above the mean is 8 points and the total of the distances below the mean is 8 points. Thus, the mean is located in the middle of the distribution if you use the concept of distance to define the “middle.” However, you should realize that the mean is not necessarily located at the exact center of the group of scores. The Median, on the other hand, defines the middle of the distribution in terms of scores. In particular, the Median is located so that half of the scores are on one side and half are on the other side. For the distribution in Figure 3.6, for example, the Median is located at X = 2.5, with exactly 3 scores above this value and exactly 3 scores below. Thus, it is pos-sible to claim that the Median is located in the middle of the distribution, provided that the term “middle” is defined by the number of scores. In summary, the mean and the Median are both methods for defining and measuring central tendency. 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.
eBook - PDFHistory by Numbers
An Introduction to Quantitative Approaches
- Pat Hudson, Mina Ishizu(Authors)
- 2016(Publication Date)
- Bloomsbury Academic(Publisher)
There is a fairly even spread of observations above and below the mean, tapering off symmetrically. For further discussion of the properties of bell-shaped distributions, see p. 112. The Median The Median is the observation that lies at the centre or middle of a distribution when all of the observations or values are ranked in size order. It can be used with ordinal or interval data. When there is an even number of cases the Median is the mean of the two middle ranking values. The advantages of using the Median are: (a) it is immune from the influence of extreme values (b) it has some measures of dispersion associated with it (see below pp. 101–104). The Median is a better way of calculating the average level of tax paid from Table 3.17, than was the calculation of the arithmetic mean, above, because it is immune from the influence of the two largest and untypical tax payers (outliers) (Sir Wats Horton, Figure 4.2 Height distribution of US passport applicants, 1830–1857. Source: John Komlos, ‘On the nature of the Malthusian threat in the eighteenth century’, Economic History Review , 52, 4 (1999), p. 736. In this figure the histogram is compared for analytical purposes with the shape of the normal distribution (see pp. 112–113). Summarizing Data: Averages and Distributions 91 Gentleman, who held 24 separate parcels of land and George Stansfield, merchant, who owned 15). The Median tax payment is £1 0s. 6d. compared with the mean of £4 6s. 1d. It is possible to estimate the Median of a grouped frequency distribution which is useful if the original figures are not available. This is done by assuming that the values of items in the class containing the Median are distributed evenly, that is that the Median falls in the middle of that class. The probability of this occurring increases with the size of the dataset so this may be a decisive consideration in adopting this calculation.
eBook - PDF- Dawn A. Willoughby(Author)
- 2016(Publication Date)
- Wiley(Publisher)
4. Apply the formula to the Median class. The Median is suitable for discrete or continuous quantitative data; it is not suitable for a qualitative data set which does not contain numerical values. It can be described as a resistant measure because it is not particularly influenced by extreme data values that are unusual in numerical size compared to the rest of the data set. EXAMPLES: Median Median – RAW DATA The table below shows the average price in pence for a litre of unleaded petrol in the UK in 2010. 111.8 112.1 116.1 120.5 121.5 118.1 117.5 116.1 115.2 111.7 119.1 122.1 132 A N E S S E N T I A L G U I D E T O B U S I N E S S S T A T I S T I C S Putting the data in numerical order, we have: 111.7 111.8 112.1 115.2 116.1 116.1 117.5 118.1 119.1 120.5 121.5 122.1 There is an even number of values in the data set, 12, so the Median is calculated as the average of the two values closest to the middle. Using 1 2 12 and 1 2 12 1 identifies the 6th and 7th data values; the 6th value is 116.1 and the 7th value is 117.5, so the Median is calculated as 116:1 117:5 2 116:8 Median – FREQUENCY DISTRIBUTION The quality control department in a factory records the number of defective items produced by a particular machine each day for two months. The data collected are shown below: Number of defective items x Frequency f Cumulative frequency 0 5 5 1 16 21 2 10 31 3 10 41 4 12 53 5 4 57 6 4 61 P f 61 There is an odd number of values in the data set, 61, so the Median is the middle data value. Using 1 2 61 1 identifies the 31st data value as the middle one. The cumulative frequency column shows that the first five data values are 0, the 6th to 21st data values are 1, and the 22nd to 31st data values are 2. The 31st value, and hence the Median, is 2 for this distribution. Median – GROUPED FREQUENCY DISTRIBUTION The frequency table shows the time, in minutes, that a sample of users spent browsing a website during a single visit.
- Alan R. Jones(Author)
- 2018(Publication Date)
- Routledge(Publisher)
The problem with the term ‘bias’ is that it has a very negative connotation in general use, implying an emotional response that leans towards or away from a particular sce- nario, circumstance, group etc, based on some prior experiential or irrational condition- ing. This sort of emotional bias may be easy to identify but is often difficult to change. In its statistical sense, bias can be the opposite; it can sometimes be difficult to spot, but we can try to compensate for it in the choices we make, because there is no emotion attached (. . . well estimator frustration aside!) 2.2 Measures of Central Tendency We can describe the middle ground of a set of data in a variety of ways: • Midpoint: The value that is halfway between the largest and smallest values • Mean: The value which represents a ‘balance’ of all the other values taking account of their individual values and how often they occur 18 | Measures of Central Tendency • Mode: The value that occurs the most frequently • Median: The value for which there are an equal number of other values greater than and less than it The one that is the least useful to us as esti- mators is the midpoint, as it actually tells us nothing of real value about the majority of the data as a ‘ central value’; it could be second smallest or largest value! From now on, we will ignore it. . . . Central Tendency: Like the data had a choice . . . Some definitions of Central Tendency refer to data points clustering around some ‘middle’ value almost like they had a choice where they positioned themselves. This unfortunate phrasing is not helped by use of the word ‘tendency’, which suggests an element of choice, or an inclination towards a particular value or behavioural characteristic. Furthermore, we may also see expressions, or even use them ourselves ( yes, I plead ‘guilty as charged’) such as ‘ the data is well-behaved’ .
No longer available |Learn moreUnderstandable Statistics
Concepts and Methods, Enhanced
- Charles Henry Brase, Corrinne Pellillo Brase(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
Percentiles and Box-and-Whisker Plots FOCUS POINTS • Interpret the meaning of percentile scores. • Compute the Median, quartiles, and five-number summary from raw data. • make a box-and-whisker plot. Interpret the results. • Describe how a box-and-whisker plot indicates spread of data about the Median. We’ve seen measures of central tendency and spread for a set of data. The arithme-tic mean x and the standard deviation s will be very useful in later work. However, SECTION 3.3 30 (a) n 1 < 34; n 2 < 69; n 3 < 47. (b) m < 88.43. a 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. 122 Chapter 3 AVERAGES AND VARIATION because they each utilize every data value, they can be heavily influenced by one or two extreme data values. In cases where our data distributions are heavily skewed or even bimodal, we often get a better summary of the distribution by utilizing relative position of data rather than exact values. Recall that the Median is an average computed by using relative position of the data. If we are told that 81 is the Median score on a biology test, we know that after the data have been ordered, 50% of the data fall at or below the Median value of 81. The Median is an example of a percentile; in fact, it is the 50th percentile. The general definition of the P th percentile follows. For whole numbers P (where 1 # P # 99 ), the P th percentile of a distribution is a value such that P % of the data fall at or below it and 1 100 2 P 2 % of the data fall at or above it. In Figure 3-3, we see the 60th percentile marked on a histogram.
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