Measures of Dispersion

Measures of Dispersion Summary The measure of central tendency, as discussed in the previous chapter tells us only about the characteristics of a particular series. They do not describe any thing on the observations or data entirely. In other wards, measures of central tendency do not tell any thing about the variations that exist in the data of a particular series. To make the concept, let discuss an example. It was found by using formula of mean that the average depth of a river is 6 feet. One cannot confidently enter into the river because in some places the depth may be 12 feet or it may have 3 feet. Thus this type of interpretation by using the measures of central tendency some times proves to be useless. Hence the measure of central tendency alone to measure the characteristics of a series of observations is not sufficient to draw a valid conclusion. With the central value one must know as to how the data is distributed. Different sets of data may have the same measures of central tendency but differ greatly in terms of variation. For this knowledge of central value is not enough to appreciate the nature of distribution of values. Thus there is the requirement of some additional measures along with the measures of central tendency which will describe the spread of the entire set of values along with the central value. One such measure is popularly called as dispersion or variation. The study of dispersion will enables us to know whether a series is homogeneous (where all the observations remains around the central value) or the observations is heterogeneous (there will be variations in the observations around the central value like 1, 50, 20, 28 etc., where the central value is 33). Hence it can be said that a measure of dispersion describes the spread or scattering of the individual values of a series around its central value. Experts opine different opinion on why the variations in a distribution are so important to consider? Following are some views on validity of the measure of dispersion: Measures of variation provide the researchers some additional information about the behaviour of the series along with the measures of central tendency. With this information one can judge the reliability of the value that is derived by using the measure of central tendency. If the data of the series are widely dispersed, the central location is less representatives of the data as a whole. On the other hand, when the data of a series is less dispersed, the central location is more representative to the entire series. In other wards, a high degree of variation would mean little uniformity whereas a low degree of variation would mean greater uniformity. When the data of a series are widely dispersed, it creates practical problems in executing data. Measure of dispersion helps in understanding and tackling the widely dispersed data. It facilitates to determine the nature and cause of variation in order to control the variation itself. Measures of variation enable comparison to be made of two or more series with regard to their variability. DEFINITION: Following are some definitions defined by different experts on measures of dispersion. L.R. Connor defines measures of dispersion as ‘dispersion is the measure extended to which individual items vary. Similarly, Brookes and Dick opines it as ‘dispersion or spread is the degree of the scatter or the variation of the variables about a central value. Robert H. Wessel defines it as ‘measures which indicate the spread of the values are called measures of dispersion. From all these definition it is clear that dispersion measures more or less describes the spread or scattering of the individual values of a series around its central value. METHODS OF MEASURING DISPERSION: Dispersion of a series of data can be calculated by using following four widely used methods Dispersion when measured on basis of the difference between two extreme values selected from a series of data. The two well known measures are The Range The Inter-quartile Range or Quartile Deviation Dispersion when measured on basis of average deviation from some measure of central tendency. The well known measures are The Mean/average deviation The Standard Deviation and The Coefficient of variation and The Gini coefficient and the Lorenz curve All the tools are discussed in details below one after the other. THE RANGE: The range is the simplest measure of the dispersion. The range is defined as the difference between the highest value and the lowest value of the series. Range as a measure of variation is having limited applicability. It is widely used for weather forecasting by the meteorological departments. It also used in statistical quality control. Range is a good indicator to measure the fluctuations in price change like that of studying the variations in the price of shares and debentures and other related matters. Following is the procedure of calculating range: Range= value of the highest observation (H) – value of the lowest observation (L) or Range = H – L Advantages of Range: Range is the simplest of obtaining dispersion. It is easily understandable and can be interpreted easily. It requires fewer times to obtain the variation in the series. Disadvantages of Range: As it considers only two extreme values, hence it doesnt include all the observations of the series. It fails to tell any thing about the characteristics of a distribution It is having very limited scope of applicability Having no mathematical treatment THE INTER-QUARTILE RANGE OR QUARTILE DEVIATION: A second measure of dispersion is the inter-quartile range which takes into account the middle half i.e., 50% of the data thus, avoiding the problem of extreme values in the data. Hence it measures approximately how far from the median one must go on either side before it can be include one-half the values of the data set. Inter-quartile range can be calculated by dividing the series of observations into four parts; each part of the series contains 25 percent of the observations. The quartiles are then the highest values in each of these four parts, and the inter-quartile range is the difference between the values of the first and the third quartile. Following are the steps of calculating the inter-quartile range: Arrange the data of the series in ascending order. Calculate the first quartile which is denoted as (Q1) by using the formula In case of grouped data the first quartile (Q1) can be calculated by using the formula Where N= number of observations in the series i.e., the sum of frequencies, L = lower limit of the quartile class, p.c.f. = commutative frequency prior to the quartile class, f = frequency of the quartile class and i = class interval. Quartile class can be determined by using the formula. Calculate the third quartile which is denoted as (Q3) by using the formula in case of ungrouped data. In case of grouped data the third quartile (Q3) can be calculated by using the formula Where N= number of observations in the series i.e., the sum of frequencies, L = lower limit of the quartile class, p.c.f. = commutative frequency prior to the quartile class, f = frequency of the quartile class and i = class interval. Quartile class can be determined by using the formula. THE MEAN/AVERAGE DEVIATION: Mean/average deviation is the arithmetic mean of the difference of a series computed from any measure of central tendency i.e., either deviation from mean or median or mode. The absolute values of each observation are calculated. Clark and Schekade opine mean deviation or average deviations as the average amount of scatter of the items in a distribution from either the mean or the median, ignoring the signs of the deviations. Thus the average that is taken of scatter is an arithmetic mean, which accounts for the fact that this measure is often called as mean deviation or average deviations. Calculations of Mean Deviation in case of Discrete Series: In case of discrete series, mean deviation can be calculated through following steps The first step is to calculate the mean or median or mode of the given series Compute the deviations of the observations of the series from the calculated mean or median or mode. This deviation is also denoted as capital letter D and is always taken as mod value i.e., ignoring the plus or minus sign. Take the summation of the deviations (sum of D) and divide it by number of observations (N). In the same way one can calculate mean deviation from median or mode in case of individual series. Calculations of Mean Deviation in case of discrete series: Mean deviation can be calculated in case of discrete series in a little bit different way. Following are some steps to calculate the average mean when the series is discrete. The first step is to calculate the mean or median or mode of the given series by using the formula as discussed in the previous chapter. Compute the deviations of the observations of the series from the calculated mean or median or mode value. This deviation is also denoted as capital letter D and is always taken as mod value i.e., ignoring the plus or minus sign. Multiply the corresponding frequency with each deviation value i.e., calculate f * D. Similarly, one can calculate the mean deviation or average deviation by taking deviations from median or mode. Calculations of Mean Deviation in case of continuous series: The first step is to calculate the mean or median or mode of the given series by using the formula as discussed in the previous chapter. In the second step, get the mid values of the observations (m) Compute the deviations of the observations of the series from the calculated mean or median or mode value. This deviation is also denoted as capital letter D = m mean or median or mode and is always taken as mod value i.e., ignoring the plus or minus sign. Multiply the corresponding frequency with each deviation value i.e., calculate f * D. Take the summation i.e., (sum of D) and divide it by number of observations (N). The formula may be Advantages of mean deviation: The computation process of mean deviation is based on all the observations of the series. The value of mean deviation is less affected by the extreme items. These are three alternatives available with the researcher while calculating the mean. One can consider the mean or median or mode. Hence it is more flexible in calculation. Disadvantages of mean deviation: The practical usefulness of mean deviation is very less. Mean deviation is not having enough scope for further mathematical calculations. Mod values are considered while calculating the mean deviation. It is criticized by some experts as illogical and unsound. THE STANDARD DEVIATION: Standard deviation or other wise called as root mean square deviation is the most important and widely used measure of variation. It measures the absolute variation of a distribution. It is the right measure that highlights the spread of the observation over and around the mean value. The greater the rate of variation of observations in a series, the greater will be the value of standard deviation. A small value of standard deviation implies a high degree of homogeneity among the observations in the series. If there will be a comparison between two or more standard deviations of two or more series, than it is always advisable to choose that series as ideal one which is having small value of standard deviation. Standard deviation is always measures from the mean or average value of the series. The credit for introducing this concept in the literature goes to Karl Pearson, a famous statistician. It is denoted by the Greek letter (pronounced as sigma) Standard deviation is calculated in following three different series: Standard deviation in case of Individual series Standard deviation in case of Discrete series Standard deviation in case of Continuous series All the above conditions are discussed in detail below. a. Standard deviation in case of individual series: In case of individual series, the value of standard deviation can be calculated by using two methods. Direct method- when deviations are taken from actual mean Short-cut method- when deviations are taken from assumed mean 1. Direct method- when deviations are taken from actual mean: Following are some steps to be followed for calculating the value of standard deviation. The first step is to calculate the actual mean value of the observation In the next column calculate the deviation from each observation i.e., find out () where is the mean of the series. In the next column calculate the square value of the deviations and at the end of the column calculate the sum of the square of the deviations i.e., Divide the total value with the number of observations (N) and than square root of the value. The formula will be . Since the series is having individual observations, some times it so happens that there is no need of taking the deviations. In such a case the researcher can directly calculate the value of the standard deviation. The formula for calculating directly is . 2. Short-cut method- when deviations are taken from assumed mean: In practical uses it so happens that while calculating standard deviation by using the arithmetic mean, the mean value may be in some fractions i.e., .25 etc. This creates the real problem in calculating the value of standard deviation. For this purpose, instead of calculating standard deviation by using the above discussed arithmetic mean methods, researchers generally prefer the method of short-cut which is nothing rather calculation of standard deviation by assuming a mean value. Following are some steps that to be followed for calculating standard deviation in case of assumed mean method: The first step is to assume a value from the X values as mean. This mean value is denoted as A. In the next step deviations are to be calculated from this assumed mean as (X-A) and this value is denoted as D. At the end of the same column, the sum of D () is to be calculated. Calculate the square of each observation of D i.e., calculate. The following formula is to be used to calculate standard deviation of the series. where N is the number of observations in the series. b. Standard deviation in case of discrete series: Discrete series are the series which are having some frequencies or repetitions of observations. In case of a discrete series standard deviation is calculated by using following three methods: when deviations are taken from actual mean when deviations are taken from assumed mean Following are the detailed analysis of the above the two methods. 1. When deviations are taken from actual mean: The steps to calculate standard deviation when deviations are calculated from the actual mean are The first step is to calculate the actual mean value of the observation In the next column calculate the deviation from each observation i.e., find out () where is the mean of the series, this can be denoted as D. In the next column calculate the square value of the deviations and at the end of the column calculate the sum of the square of the deviations i.e., Multiply corresponding frequencies of each observation with the value of D2 in the next column. Divide the total value with the number of observations (N) and than square root of the value. The formula will be 2. When deviations are taken from assumed mean: The steps to calculate standard deviation when deviations are calculated from the actual mean are The first step is to assume a mean value from the observations In the next column calculate the deviation from each observation i.e., find out () where A is the mean of the series, this deviation can be denoted as D. In the next column calculate the square value of the deviations and at the end of the column calculate the sum of the square of the deviations i.e., Multiply corresponding frequencies (f) of each observation with the value of D2 in the next column. Use the following formula to calculate standard deviation c. Standard deviation in case of Continuous series: Standard deviation in case of a continuous series can be calculated by using the following steps Calculate the mid value of the series and denote it as ‘m. Assume any value from the mid values and denote it as A Deviations can be calculated from each series i.e., calculate m – A and than divide it with the class interval value (i) i.e., Multiply the corresponding frequencies of each observation with the deviation value and take the sum at the end of the column i.e., calculate In the next column square the deviation values of each observation i.e., calculate Multiply the value of with its frequencies i.e., calculate Use the following formula to get standard deviation. Properties of standard deviation: As tool of variance, standard deviation is used as a good measure of interpretation of the scatteredness of observation of a series. It is a fact that in a normal distribution approximately 68 per cent of the observations of a series lies less than standard deviation away from the mean, again approximately 95.5 per cent of the items lie less than 2 standard deviation value away from the mean and in the same way 99.7 per cent of the items lie within 3 standard deviations away from the mean. Hence covers 68.27 per cent of the items in a series with normal distribution. covers 95.45 per cent of the items in a series with normal distribution and covers 99.73 per cent of the items in a series with normal distribution. Advantage of Standard Deviation: Following are some advantages of standard deviation as a measure of dispersion This is the highest used technique of dispersion. It is regarded as a very satisfactory measure of the dispersion of a series. It is capable of further mathematical calculations. Algebraic signs are not ignored while measuring the value of standard deviation of a series. It is less affected by the extreme observations of a series. The coefficients make the standard deviation very popular measure of the scatteredness of a series. Disadvantages of standard deviation: The disadvantages are It is not easy to understand the concept easily and quickly. It requires a good exercise to calculate the values of standard deviation. It gives more weight to observations which are away from the arithmetic mean. THE COEFFICIENT OF VARIATION: Another useful statistical tool for measuring dispersion of a series is coefficient of variation. The coefficient of variation is the relative measure of standard deviation which is an absolute measure of dispersion. This tool of dispersion is mostly used in case of comparing the variability two or more series of observation. While comparing, that series for which the value of the coefficient of variation is greater is said to be more variable (i.e., the observations of the series are less consistent, less uniform, less stable or less homogeneous). Hence it is always advisable to choose that series which is having less value of coefficient of variation. The value of coefficient is less implies more consistent, more uniform, more stable and of course more homogeneous. The value of coefficient of variation is always measured by using the value of standard deviation and its relative arithmetic mean. It is denoted as C.V., and is measured by using simple formula as discussed below: In practical field, researchers generally prefer to use standard deviation as a tool to measure the dispersion than that of coefficient of variance because of a numbers of reasons (researchers are advised to refer any standard statistics book to know more on coefficient of variance and its usefulness). GINI COEFFICIENT AND THE LORENZ CURVE: An illuminating manner of viewing the Gini coefficient is in terms of the Lorenz curve due to Lorenz (1905). It is generally defined on the basis of the Lorenz curve. It is popularly known as the Lorenz ratio. The most common definition of the Gini coefficient is in terms of the Lorenz diagram is the ratio of the area between the Lorenz curve and the line of equality, to the area of the triangle OBD below this line (figure-1). The Gini coefficient varies between the limits of 0 (perfect equality) and 1 (perfect inequality), and the greater the departure of the Lorenz curve from the diagonal, the larger is the value of the Gini coefficient. Various geometrical definitions of Gini coefficient discussed in the literature and useful for different purposes are examined here. CONCLUSIONS: The study of dispersion will enables us to know whether a series is homogeneous (where all the observations remains around the central value) or the observations is heterogeneous (there will be variations in the observations around the central value Hence it can be said that a measure of dispersion describes the spread or scattering of the individual values of a series around its central value. For this there are a numbers of methods to determine the variations as discussed in this chapter. But it is always confusing among the researchers that which method is the best among the different techniques that we have discussed? The answer to this question is very simple and says that no single average can be considered as best for all types of data series. The most important factors are the type of data available and the purpose of investigation. Critiques suggest that if a series is having more extreme values than standard deviation as technique is to be avoided. On the other hand in case of more skewed observations standard deviation may be used but mean deviation needs to be avoided where as if the series is having more gaps between two observations than quartile deviation is not an appropriate measure to be used. Similarly, standard deviation is the best technique for any purpose of data. SUMMARY: The study of dispersion will enables us to know whether a series is homogeneous (where all the observations remains around the central value) or the observations is heterogeneous (there will be variations in the observations around the central value). Dispersion when measured on basis of the difference between two extreme values selected from a series of data. The two well known measures are (i) The Range and (ii) The Inter-quartile Range. Dispersion when measured on basis of average deviation from some measure of central tendency. The well known measures are (i) The Mean/average deviation, (ii) The Standard Deviation, (iii) The Coefficient of variation and (iv) The Gini coefficient and the Lorenz curve The range is defined as the difference between the highest value and the lowest value of the series. Range as a measure of variation is having limited applicability. The inter-quartile range measures approximately how far from the median one must go on either side before it can be include one-half the values of the data set. Mean/average deviation is the arithmetic mean of the difference of a series computed from any measure of central tendency i.e., either deviation from mean or median or mode. The absolute values of each observation are calculated. A small value of standard deviation implies a high degree of homogeneity among the observations in the series. If there will be a comparison between two or more standard deviations of two or more series, than it is always advisable to choose that series as ideal one which is having small value of standard deviation. Standard deviation is always measures from the mean or average value of the series. The coefficient of variation is the relative measure of standard deviation which is an absolute measure of dispersion. This tool of dispersion is mostly used in case of comparing the variability two or more series of observation. The most common definition of the Gini coefficient is in terms of the Lorenz diagram is the ratio of the area between the Lorenz curve and the line of equality, to the area of the triangle below the equality line. IMPORTANT QUESTIONS: 1. Age of ten students in a class is considered. Find the mean and standard deviation. 19, 21, 20, 20, 23, 25, 24, 25, 22, 26 The following table derives the marks obtained in Statistics paper by 100 students in a class. Calculate the standard deviation and mean deviation. The monthly profits of 150 shop keepers selling different commodities in a city footpath is derived below. Calculate the mean, mean deviation and standard of the distribution. The daily wage of 160 labourers working in a cotton mill in Surat cith is derived below. Calculate the range, mean deviation and standard of the distribution. Calculate the mean deviation and standard deviation of the following distribution. What do you mean by measure of dispersion? How far it helpful to a decision-maker in the process of decision making? Define measure of Dispersion? Among the various tools of dispersion which tool according to you is the best one, give suitable reason of your answer. What do you mean by measure of dispersion? Compare and contrast various tools of dispersion by pointing out their advantages and disadvantages. Discuss with example the relative merits of range, mean deviation and standard deviation as measures of dispersion. Define standard deviation? Why standard deviation is more useful than other measures of dispersion? The data derived below shows the ages of 100 students pursuing their master degree in economics. Calculate the Mean deviation and standard deviation. Following is the results of a study carried out to determine the number of mileage the marketing executives drove their cars over a 1-year period. For this 50 marketing executives are sampled. Based on the findings, calculate the range and inter-quartile range. In an enquiry of the number of days 230 patients chosen randomly stayed in a Government hospital following after operation. On the basics of observation calculate the standard deviation. Cars sold in small car segment in November 2009 at 10 Maruti Suzuki dealers in Delhi city is explained below. Compute the range, mean deviation and standard deviation of the data series. Following is the daily data on the number of persons entered through main gate in a month to institute. Calculate the range and standard deviation of the series. Calculate the range and coefficient of range of a group of students from the marks obtained in two papers as derived below: Following are marks obtained by some students in a class-test. Calculate the range and coefficient of range. By using the direct and indirect method, calculate the mean deviation by using both arithmetic mean and mode from the following data set which is related to age and numbers of residents of Vasundara apartment, Gaziabad. A local geezer manufacturer at Greater Noida has developed a new and chief variety of geezers which are meant of lower and middle income households. He carried out a survey in some apartments asking the expectations of the customers that they are ready to invest on purchase of geezer. Calculate the standard deviation of the series. Calculate median of the following distribution. From the median value calculate the mean deviation and coefficient of mean deviation. Calculate median of the following distribution. From the median value calculate the mean deviation and coefficient of mean deviation. Calculate the arithmetic average and standard deviation from the following daily data of rickshaw puller of Hyderabad City. From the students of 250 candidates the mean and standard deviations of their total marks were calculated as 60 and 17. Latter in the process of verification it is found that a score 46 was misread 64. Recalculate the correct mean and standard deviation. The wage structure paid on daily basis of two cotton factories are derived below. In order to show the inequality, draw the Lorenz curve. Total marks obtained by the students in two sections are derived below. By using the data draw a Lorenz curve. Draw the Lorenz curve of the following data. Find the range and co-efficient of range for the following data set. The height of 10 firemen working in a fire station are 165, 168, 172, 174, 175, 178, 156, 158, 160, 179 cms. Calculate the range of the series. Now let that the tallest and the shortest firemen are get transformed from the fire station. Calculate the range of the new firemen. What percentage change is found in the earlier range and the latter range? Calculate the quartile deviation from the following derived data. Calculate the interquartile range, quartile deviation and its coefficient for the following data series. Calculate the mean deviation from the following data. Calculate the mean deviation from median and mean for the following series. The distribution derived below reveals the difference in age between husband and wife in a community. Based on the data, calculate mean deviation and standard deviation. Calculate th

The Host Chapter 14: Disputed

It was too much for both of us, seeing him here, now, after already accepting that we'd never see him again, after believing that we'd lost him forever. It froze me solid, made me unable to react. I wanted to look at Uncle Jeb, to understand his heartbreaking answer in the desert, but I couldn't move my eyes. I stared at Jared's face, uncomprehending. Melanie reacted differently. â€Å"Jared,† she cried; through my damaged throat the sound was just a croak. She jerked me forward, much the same way as she had in the desert, assuming control of my frozen body. The only difference was that this time, it was by force. I wasn't able to stop her fast enough. She lurched forward, raising my arms to reach out for him. I screamed a warning at her in my head, but she wasn't listening to me. She was barely aware that I was even there. No one tried to stop her as she staggered toward him. No one but me. She was within inches of touching him, and still she didn't see what I saw. She didn't see how his face had changed in the long months of separation, how it had hardened, how the lines pulled in different directions now. She didn't see that the unconscious smile she remembered would not physically fit on this new face. Only once had she seen his face turn dark and dangerous, and that expression was nothing to the one he wore now. She didn't see, or maybe she didn't care. His reach was longer than mine. Before Melanie could make my fingers touch him, his arm shot out and the back of his hand smashed into the side of my face. The blow was so hard that my feet left the ground before my head slammed into the rock floor. I heard the rest of my body hit the floor with dull thumps, but I didn't feel it. My eyes rolled back in my head, and a ringing sound shimmered in my ears. I fought the dizziness that threatened to spin me unconscious. Stupid, stupid, I whimpered at her. I told you not to do that! Jared's here, Jared's alive, Jared's here. She was incoherent, chanting the words like they were lyrics to a song. I tried to focus my eyes, but the strange ceiling was blinding. I twisted my head away from the light and then swallowed a sob as the motion sent daggers of agony through the side of my face. I could barely handle the pain of this one spontaneous blow. What hope did I have of enduring an intensive, calculated onslaught? There was a shuffle of feet beside me; my eyes moved instinctively to find the threat, and I saw Uncle Jeb standing over me. He had one hand half stretched out toward me, but he hesitated, looking away. I raised my head an inch, stifling another moan, to see what he saw. Jared was walking toward us, and his face was the same as those of the barbarians in the desert-only it was beautiful rather than frightening in its fury. My heart faltered and then beat unevenly, and I wanted to laugh at myself. Did it matter that he was beautiful, that I loved him, when he was going to kill me? I stared at the murder in his expression and tried to hope that rage would win out over expediency, but a true death wish evaded me. Jeb and Jared locked eyes for a long moment. Jared's jaw clenched and unclenched, but Jeb's face was calm. The silent confrontation ended when Jared suddenly exhaled in an angry gust and took a step back. Jeb reached down for my hand and put his other arm around my back to pull me up. My head whirled and ached; my stomach heaved. If it hadn't been empty for days, I might have thrown up. It was like my feet weren't touching the ground. I wobbled and pitched forward. Jeb steadied me and then gripped my elbow to keep me standing. Jared watched all this with a teeth-baring grimace. Like an idiot, Melanie struggled to move toward him again. But I was over the shock of seeing him here and less stupid than she was now. She wouldn't break through again. I locked her away behind every bar I could create in my head. Just be quiet. Can't you see how he loathes me? Anything you say will make it worse. We're dead. But Jared's alive, Jared's here, she crooned. The quiet in the cavern dissolved; whispers came from every side, all at the same time, as if I'd missed some cue. I couldn't make out any meanings in the hissing murmurs. My eyes darted around the mob of humans-every one of them an adult, no smaller, younger figure among them. My heart ached at the absence, and Melanie fought to voice the question. I hushed her firmly. There wasn't anything to see here, nothing but anger and hatred on strangers' faces, or the anger and hatred on Jared's face. Until another man pushed his way through the whispering throng. He was built slim and tall, his skeletal structure more obvious under his skin than most. His hair was washed out, either pale brown or a dark, nondescript blond. Like his bland hair and his long body, his features were mild and thin. There was no anger in his face, which was why it held my eye. The others made way for this apparently unassuming man as if he had some status among them. Only Jared didn't defer to him; he held his ground, staring only at me. The tall man stepped around him, not seeming to notice the obstacle in his path any more than he would a pile of rock. â€Å"Okay, okay,† he said in an oddly cheery voice as he circled Jared and came to face me. â€Å"I'm here. What have we got?† It was Aunt Maggie who answered him, appearing at his elbow. â€Å"Jeb found it in the desert. Used to be our niece Melanie. It seemed to be following the directions he gave her.† She flashed a dirty look at Jeb. â€Å"Mm-hm,† the tall, bony man murmured, his eyes appraising me curiously. It was strange, that appraisal. He looked as if he liked what he saw. I couldn't fathom why he would. My gaze shied away from his, to another woman-a young woman who peered around his side, her hand resting on his arm-my eyes drawn by her vivid hair. Sharon! Melanie cried. Melanie's cousin saw the recognition in my eyes, and her face hardened. I pushed Melanie roughly to the back of my head. Shhh! â€Å"Mm-hm,† the tall man said again, nodding. He reached one hand out to my face and seemed surprised when I recoiled from it, flinching into Jeb's side. â€Å"It's okay,† the tall man said, smiling a little in encouragement. â€Å"I won't hurt you.† He reached toward my face again. I shrunk into Jeb's side like before, but Jeb flexed his arm and nudged me forward. The tall man touched my jaw below my ear, his fingers gentler than I expected, and turned my face away. I felt his finger trace a line on the back of my neck, and I realized that he was examining the scar from my insertion. I watched Jared's face from the corner of my eye. What this man was doing clearly upset him, and I thought I knew why-how he must have hated that slender pink line on my neck. Jared frowned, but I was surprised that some of the anger had drained from his expression. His eyebrows pulled together. It made him look confused. The tall man dropped his hands and stepped away from me. His lips were pursed, his eyes alight with some challenge. â€Å"She looks healthy enough, aside from some recent exhaustion, dehydration, and malnourishment. I think you've put enough water back into her so that the dehydration won't interfere. Okay, then.† He made an odd, unconscious motion with his hands, as if he were washing them. â€Å"Let's get started.† Then his words and his brief examination fit together and I understood-this gentle-seeming man who had just promised not to hurt me was the doctor. Uncle Jeb sighed heavily and closed his eyes. The doctor held a hand out to me, inviting me to put mine in his. I clenched my hands into fists behind my back. He looked at me carefully again, appraising the terror in my eyes. His mouth turned down, but it was not a frown. He was considering how to proceed. â€Å"Kyle, Ian?† he called, craning his neck to search the assembly for the ones he summoned. My knees wobbled when the two big black-haired brothers pressed their way forward. â€Å"I think I need some help. Maybe if you were to carry -† the doctor, who did not look quite so tall standing beside Kyle, began to say. â€Å"No.† Everyone turned to see where the dissent had come from. I didn't need to look, because I recognized the voice. I looked at him anyway. Jared's eyebrows pressed down hard over his eyes; his mouth was twisted into a strange grimace. So many emotions ran across his face, it was hard to pin one down. Anger, defiance, confusion, hatred, fear†¦ pain. The doctor blinked, his face going slack with surprise. â€Å"Jared? Is there a problem?† â€Å"Yes.† Everyone waited. Beside me, Jeb was holding the corners of his lips down as if they were trying to lift into a grin. If that was the case, then the old man had an odd sense of humor. â€Å"And it is?† the doctor asked. Jared answered through his teeth. â€Å"I'll tell you the problem, Doc. What's the difference between letting you have it or Jeb putting a bullet in its head?† I trembled. Jeb patted my arm. The doctor blinked again. â€Å"Well† was all he said. Jared answered his own question. â€Å"The difference is, if Jeb kills it, at least it dies cleanly.† â€Å"Jared.† The doctor's voice was soothing, the same tone he'd used on me. â€Å"We learn so much each time. Maybe this will be the time -â€Å" â€Å"Hah!† Jared snorted. â€Å"I don't see much progress being made, Doc.† Jared will protect us, Melanie thought faintly. It was hard to concentrate enough to form words. Not us, just your body. Close enough†¦ Her voice seemed to come from some distance, from outside my pounding head. Sharon took a step forward so that she stood half in front of the doctor. It was a strangely protective stance. â€Å"There's no point in wasting an opportunity,† she said fiercely. â€Å"We all realize that this is hard for you, Jared, but in the end it's not your decision to make. We have to consider what's best for the majority.† Jared glowered at her. â€Å"No.† The word was a snarl. I could tell he had not whispered the word, yet it was very quiet in my ears. In fact, everything was suddenly quiet. Sharon's lips moved, her finger jabbed at Jared viciously, but all I heard was a soft hissing. Neither one of them took a step, but they seemed to be drifting away from me. I saw the dark-haired brothers step toward Jared with angry faces. I felt my hand try to rise in protest, but it only twitched limply. Jared's face turned red when his lips parted, and the tendons in his neck strained like he was shouting, but I heard nothing. Jeb let go of my arm, and I saw the dull gray of the rifle's barrel swing up beside me. I cringed away from the weapon, though it was not pointed in my direction. This upset my balance, and I watched the room tip very slowly to one side. â€Å"Jamie,† I sighed as the light swirled away from my eyes. Jared's face was suddenly very close, leaning over me with a fierce expression. â€Å"Jamie?† I breathed again, this time a question. â€Å"Jamie?† Jeb's gruff voice answered from somewhere far away. â€Å"The kid is fine. Jared brought him here.† I looked at Jared's tormented face, fast disappearing into the dark mist that covered my eyes. â€Å"Thank you,† I whispered. And then I was lost in the darkness.

Swisher Mower and Machine Company (Smc)

RECOMMENDATION: Swisher Mower and Machine Company should reject the offer to produce the Private-Label brand. RATIONALE: †¢Standard Riding Mower contribution per unit would drop from $97 to $33. 98, which is $63. 02 per mower. (Appendix 4) †¢Contribution Margin would drop from 14. 92% to 5. 5%, which is 9. 42%. (Appendix 4) †¢Swisher Mower would only profit $62,819 from producing the 8,200 units in the Private-Label in the first full year of production. (Appendix 5) oProfit per mower is only $7. 66 per unit. Cannibalization of SMC current mower is ($29,000) loss from the private-label brand. Plus, the loss of additional smaller independent dealer in the future since there will be an overlap in trade areas. (Appendix 3) †¢SMC will incur additional labor cost on 2,450 units because the factory is already at full capacity. oThis results in additional $63,700 overtime labor cost, which is $7. 77 per unit. (Appendix 1) †¢With the increase in labor, materials, over head, and property tax, the additional variable cost is $30. 52 per unit. Appendix 1) oThis brings the total variable cost to $583. 52. (Appendix 4) †¢SMC will only be making $7. 66 per mower, and the company wants them to pay $22 per hour for any maintenance on a mower under warranty. Therefore SMC would have to sell 3 additional mowers to cover each hour work done to a mower under warranty. †¢The accounts receivable and inventory carrying cost for the private-label brand is $175,789. 97, $59,386. 73 and $116,412. 24 respectively. (Appendix 2) †¢The image of company would be altered since SMC would double their sales with one private-label product. Since the company has a reputation of being small and personal introducing the private-label could be detrimental to their current business. †¢Remain status quo and on average increase profit by 10% and continue to widen the breathe of the product lines, Trim-Max. APPENDICES APPENDIX 1 Incremental Cost for Private Label Direct Labor Cost $650 x 4% = $26 Additional Labor for 2,450 units = 2,450 x $26 = $63,700 / 8,200 units = $7. 77 per Unit Direct Materials Cost $650 x 1% = $6. 50 Additional Overhead $650 x 1% = $6. 50 Additional Property Tax $650 x 1. % = $9. 75 Total Additional Var. Cost = $7. 77 + $6. 50 + $6. 50 + $9. 75 = $30. 52 APPENDIX 2 Carrying Cost for Private Label Accounts Receivable A/R Turnover Rate = 365 / 45 = 8. 1 8,200 units x $617. 50 = $5,063,500 / 8. 1 = 625,123. 46 x 9. 5% = $59,386. 73 Inventory Inv. Turnover Rate = 5. 8 Average Inv. = 2,100 units Unit Cost = $583. 52 2,100 units x $583. 52 = $1,225,392 x 9. 5% = $116,412. 24 Total Carrying Cost = A/R + I nv. = $175,798. 97 APPENDIX 3 Cannibalization SMC mower = $650 – $553 = $97 Contribution Per Unit Units Lost = 300 $97 x 300 = ($29,100)

Essay about Gender in Shakespeares As You Like It

Questions of Gender in Shakespeares As You Like It Throughout history, men and women have been assigned specific roles to which society prescribes standards and qualifications. There are certain tasks that have been traditionally completed only by men, and others that have been assigned to women; most of which are separated by the realm of the domestic sphere. During the period of the Renaissance, men and women were assigned very different roles within society. The value, social expectations, legal status, and rights of citizenship differed greatly between the sexes as well as among the classes. Many of these gender roles can be identified through careful readings of the literature produced throughout the Renaissance.†¦show more content†¦Such a life was impossible for women . . . because for a woman, a public reputation was dishonorable, a sure sign of immorality and scandal (Wiesner 12). Women were excluded from any position of meaningful authority in any realm of society. Men were even valued for their ability to classify an object or being as beautiful. During this period of great creative accomplishments, men may . . . have taken to commerce or to drink, but as a matter of fact they took to visible beauty (Putnam 164). They established beauty as an important quality of life, and only men had the capacity to differentiate between that which was beautiful and not beautiful. Women, therefore, were often valued for their physical features. In the Renaissance, . . . the beauty of woman is more praised and esteemed than any other beauty . . . [for] it appears to be the order of nature that what is lacking in one sex is supplied in the other, and since man is endowed with wit, judgement, and a mind almost divine, . . . woman is given bodily beauty that she may be superior to man in this respect (Camden 20). Women were object to be viewed with pleasing affections, not with any sense of worth other than their physical features; . . . the only positive demand of the woman was that she should be beautiful (Putnam 164-165). Women were also valued for qualities that define them as submissive and passive. 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