Measures of location describe the central tendency of the data. The quartiles are calculated in a similar way to the median; first arrange the data in size order and determine the median, using the method described above. Through this measure it is ensured that at least 50% of the observations on the variable are used in the calculation process and with this method the absolute value of the Quartile Deviation can easily be measured. Homework1.com. Mesokurtic : This distribution has kurtosis statistic similar to that of the normal distribution. A third measure of location is the mode. Disadvantages : It is very sensitive to outliers and does not use all the 3. They are liable to misinterpretations, and wrong generalizations by a They are liable to misinterpretations, and wrong generalizations by a statistician of based character. ADVANTAGES OF INTERVIEWING It is the most appropriate method when studying attitudes, beliefs, values and motives of the respondents. When it comes to releasing new items, direct mail may be a very effective method. 2.81, 2.85. Bacteria in the human body are often found embedded in a dense 3D structure, the biofilm, which makes their eradication even more challenging. Again, the second lowest 20 per cent weavers have got a mere 11 per cent the third 20 per cent shared only 18 per cent and the fourth 20 per cent about 23 per cent of the total income. Continue with Recommended Cookies. Dispersion is the degree of scatter of variation of the variables about a central value. We need to find the average squared deviation. Welcome to EconomicsDiscussion.net! Dispersion is also known as scatter, spread and variation. Spiegel, etc. It is a non-dimensional number. Web1. The median has the advantage that it is not affected by outliers, so for example the median in the example would be unaffected by replacing '2.1' with '21'. Central tendency gets at the typical score on the variable, while dispersion gets at how much variety there is in the scores. Discuss them with examples. This allows those reading the data to understand how similar or dissimilar numbers in a data set are to each other. This is one of the constraint we have on any sample data. The Range, as a measure of Dispersion, has a number of advantages and disadvantage. In order to calculate the standard deviation use individual data score needs to be compared to the mean in order to calculate the standard deviation. Both metrics measure the spread of values in a dataset. Without statistical modeling, evaluators are left, at best, with eye-ball tests or, at worst, gut-feelings of whether one system performed better than another. If you would like to change your settings or withdraw consent at any time, the link to do so is in our privacy policy accessible from our home page.. The cookie is used to store the user consent for the cookies in the category "Analytics". Web2. You could use 4 people, giving 3 degrees of freedom (41 = 3), or you could use one hundred people with df = 99. Share Your PPT File. It is the average of the distances from each data point in the population to the mean, squared. Squaring these numbers can skew the data. It is not only easy to compute, it takes into account all the given values of the variable and again the final result remains almost unaffected from any remarkably high value of the variable under consideration. High kurtosis in a data set is an indicator that data has heavy outliers. Standard deviation and average deviation are also commonly used methods to determine the dispersion of data. By clicking Accept, you consent to the use of ALL the cookies. (1) A strength of the range as a measure of dispersion is that it is quick and easy to calculate. A moment's thought should convince one that n-1 lengths of wire are required to link n telegraph poles. Outliers and skewed data have a smaller effect on the mean vs median as measures of central tendency. It is usual to quote 1 more decimal place for the mean than the data recorded. For example, the number 3 makes up part of data set B, this score is not similar in the slightest to the much higher mean score of 49.. (a) Quartile deviation as a measure of dispersion is not much popularly prescribed by the statisticians. Advertisement cookies are used to provide visitors with relevant ads and marketing campaigns. Webare various methods that can be used to measure the dispersion of a dataset, each with its own set of advantages and disadvantages. But, the results of such measures are obtained in terms of the units in which the observations are available and hence they are not comparable with each other. 2. Other uncategorized cookies are those that are being analyzed and have not been classified into a category as yet. Here are the steps to calculate the standard deviation:1. It is thus considered as an Absolute Measure of Dispersion. If the skewness is less than -1(negatively skewed) or greater than 1(positively skewed), the data are highly skewed. (g) Statisticians very often prescribe SD as the true measure of dispersion of a series of information. (2) It is simple to understand and easy to calculate. You also have the option to opt-out of these cookies. How much wire would one need to link them? Hence range cannot be completely representative of the data as all other middle values are ignored. More specifically, if there are an odd number of observations, it is the [(n+1)/2]th observation, and if there are an even number of observations, it is the average of the [n/2]th and the [(n/2)+1]th observations. 2.22, 2.35, 2.37, 2.40, 2.40, 2.45, 2.78. This cookie is set by GDPR Cookie Consent plugin. This is important to know the spread of your data when describing your data set. However, there is an increasingly new trend in which very few people are retiring early, and that too at very young ages. A high standard deviation suggests that, in the most part, themean (measure of central tendency)is not a goof representation of the whole data set. They supplement the measures of central tendency in finding out more and more information relating to the nature of a series. It is used to compare the degree of variation between two or more data series that have different measures or values. The expression 'bimodal' distribution is used to describe a distribution with two peaks in it. Advantages and Disadvantages of Various Measures of Dispersion Next add each of the n squared differences. But you can send us an email and we'll get back to you, asap. Here, we are interested to study the nature and the exact degree of economic inequality persisting among these workforces. You may have noticed that you see a rainbow only when you look away from the Sun. We also share information about your use of our site with our social media, advertising and analytics partners who may combine it with other information that youve provided to them or that theyve collected from your use of their services. So we need not know the details of the series to calculate the range. For these limitations, the method is not widely accepted and applied in all cases. If the skewness is between -1 and -0.5(negatively skewed) or between 0.5 and 1(positively skewed), the data are moderately skewed. On the other hand, direct mail canbe easily disregarded and is potentially expensive. The result will not be affected even when the distribution has an open end. These cookies help provide information on metrics the number of visitors, bounce rate, traffic source, etc. Advantages : The prime advantage of this measure of dispersion is that it is easy to calculate. It can be found by mere inspection. The calculation of the standard deviation is described in Example 3. This method results in the creation of small nanoparticles from bulk material. Advantage 1: Fast and easy to calculate. All rights reserved. Therefore, the result can only be influenced with changes in those two values, not by any other value of the variable. It is estimated by first ordering the data from smallest to largest, and then counting upwards for half the observations. 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Example : Retirement Age When the retirement age of employees is compared, it is found that most retire in their mid-sixties, or older. The table represented above shows that the poorest 20 per cent of the income earners receive only 5 per cent of the total income whereas the richest 20 per cent of the sample respondents shared as much as 43 per cent of it. Degree of Degrees of freedom of an estimate is the number of independent pieces of information that went into calculating the estimate. For example, say the last score in set A wasnt 40 but 134, this would bump the range for set A up to 100, giving a misleading impression of the real dispersion of scores in set A. Leptokurtic (Kurtosis > 3) : Peak is higher and sharper than Mesokurtic, which means that data has heavy outliers. Similarly the 3rd quartile would be the 5th observation in the upper half of the data, or the 14th observation, namely 2.40 kg. Determine the Coefficient of Range for the marks obtained by a student in various subjects given below: Here, the highest and the lowest marks are 52 and 40 respectively. For example, the standard deviation considers all available scores in the data set, unlike the range. Exception on or two, of the methods of dispersion involve complicated process of computation. Disadvantage 1: Sensitive to extreme values. *sensitive measurement as all values are taken into account. Q3 is the middle value in the second half of the rank-ordered data set. ), Consider the following table of scores:SET A354849344240SET B32547507990. 2. Again, the concept of Range cannot provide us any idea about the nature of distribution of the concerned variable and practically it is not possible for us to determine the final result for opened classes. A symmetrical distribution will have a skewness of 0 . WebAdvantages and disadvantages of various measures of dispersion (Live Version) - YouTube KSSM MATHEMATICS FORM 4Measures of Dispersion for Ungrouped DataAdvantages and When describing the scores on a single variable, it is customary to report on both the central tendency and the dispersion. According to them, it should be based on all the given observations, should be readily comprehensible, fairly and easily calculable, be affected as little as possible by sampling fluctuations and amenable to further algebraic treatments. It can be shown that it is better to divide by the degrees of freedom, which is n minus the number of estimated parameters, in this case n-1. WebThe control of infectious diseases can be improved via carefully designed decontamination equipment and systems. Every score is involved in the calculation and it gives an indication of how far the average participant deviates from the mean. 2. Webadvantages and disadvantages of measures of central tendency and dispersion from publication clinicians guide to statistics for medical out is called the measure of dispersion web 29 nov 2021 measures of central tendency class 11 economics mcqclass 11 This method results in the creation of small nanoparticles from bulk material. Let us consider two separate examples below considering both the grouped and the ungrouped data separately. If the skewness is between -0.5 and 0.5, the data are fairly symmetrical. Range as a measure of the variability of the values of a variable, is not widely accepted and spontaneously prescribed by the Statisticians of today However, it is not totally rejected even today as it has certain traditional accept abilities like representing temperate variations in a day by recording the maximum and the minimum values regularly by the weather department, while imposing controlling measures against wide fluctuations in the market prices of the essential goods and services bought and sold by the common people while imposing Price-control and Rationing measures through Public Sector Regulations, mainly to protect interests of both the buyers and sellers simultaneously. For some data it is very useful, because one would want to know these numbers, for example knowing in a sample the ages of youngest and oldest participant. (a) Quartile Deviation is easy to calculate numerically. Mean is rigidly defined so that there is no question of misunderstanding about its meaning and nature. Like the measures of central tendency, most of the measures of dispersion do not give a convincing idea about a series to a layman. The estimate of the median is either the observation at the centre of the ordering in the case of an odd number of observations, or the simple average of the middle two observations if the total number of observations is even. Range is simply the difference between the smallest and largest values in the data. In this context, we think the definition given by Prof. Yule and Kendall is well accepted, complete and comprehensive in nature as it includes all the important characteristics for an ideal measure of dispersion. measures of location it describes the It holds for a large number of measurements commonly made in medicine. Variance is measure to quantify degree of dispersion of each observation from mean values. Thus mean = (1.2+1.3++2.1)/5 = 1.50kg. Lorenz Curve The curve of concentration: Measurement of Economic Inequality among the Weavers of Nadia, W.B: This cookie is set by GDPR Cookie Consent plugin. that becomes evident from the above income distribution. When there is an even number of values, you count in to the two innermost values and then take the average. Dispersion is also known as scatter, spread and variation. They facilitate in making further statistical analysis of the series through the devices like co-efficient of skewness, co-efficient of correlation, variance analysis etc. In this case mean is smaller than median. The cookie is used to store the user consent for the cookies in the category "Performance". They indicate the dispersal character of a statistical series. While making any data analysis from the observations given on a variable, we, very often, observe that the degree or extent of variation of the observations individually from their central value (mean, median or mode) is not the same and hence becomes much relevant and important from the statistical point of view. Now split the data in two (the lower half and upper half, based on the median). Moreover, the results of the absolute measure gets affected by the number of observations obtainable on the given variable as they consider only the positive differences from their central value (Mean/Median). In the Algebraic method we split them up into two main categories, one is Absolute measure and the other is Relative measure. WebBacterial infections are a growing concern to the health care systems. It is measured just as the difference between the highest and the lowest values of a variable. Does variability really matter? But the merits and demerits common to all types of measures of dispersion are outlined as under: Copyright 2014-2023 (h) It can tactfully avoid the complication of considering negative algebraic sign while calculating deviations. Let us represent our numerical findings in this context from the available data in the following tabular form: (An exclusive survey over 222 weavers at random in 5 important weaving centres which is 15% of the total number of weavers engaged in those areas as prescribed in the Sampling Theory.). They, by themselves, cannot give any idea about the symmetricity, or skewed character of a series. Shows the relationship between standard deviation and mean. 4. The statisticians here prescribe for an well-known concept dispersion or the scatteredness or variability of the values of the variable usually from their arithmetic mean. An example of data being processed may be a unique identifier stored in a cookie. This new, advert-free website is still under development and there may be some issues accessing content. It is not affected by sampling so that the result is reliable. They facilitate in controlling the variability of a phenomenon under his purview. Note that the text says, there are important statistical reasons we divide by one less than the number of data values.6. WebClassification of Measures of Dispersion. The first step in the creation of nanoparticles is the size reduction of the starting material using a variety of physical and chemical procedures [].Processes, including ball milling, mechanochemical synthesis, laser ablation, and ion The lower variability considers being ideal as it provides better predictions related to the population. TOS4. The range is the difference When would you use either? In this way, s reflects the variability in the data. This concept of dispersion in statistics helps in the understanding of the distribution of data. For all these reasons. Therefore, the Range = 12 1 = 11 i.e. Suppose we had 18 birth weights arranged in increasing order. It is easy to calculate. (e) The relevant measure of dispersion should try to include all the values of the given variable. WebMerits of Range: (1) Range is rigidly defined. Measures of dispersion describe the spread of the data. As the components of CV, we are to derive first the Mean and the Standard Deviation of the scores obtained by the two Batsmen separately using the following usual notations: Let us prepare the following table for finding out Mean and SD of the given information: For the cricketer S the Coefficient of Variation is smaller and hence he is more consistent. Content Guidelines 2. Economists and other social scientists very often opine that inequality in the distribution of income and wealth among the individuals in a society is a common phenomenon today all over the world mainly due to our aimless and unbalanced growth policies framed by the concerned authorities, called growth without development today in economics, resulting in rise in GDP but no significant rise in the per-capita income of the people at large. It will enable us to avoid mistakes in calculation and give us the best result. Note that if we added all these deviations from the mean for one dataset, the sum would be 0 (or close, depending on round-off error).3. But the main disadvantage is that it is calculated only on the basis of the highest and the lowest values of the variable without giving any importance to the other values. Statistical models summarize the results of a test and present them in such a way that humans can more easily see and understand any patterns within the data. Analytical cookies are used to understand how visitors interact with the website. The lower dispersion value shows the data points will be grouped nearer to the center. (d) It is easy to calculate numerically and simple to understand. They include the range, interquartile range, standard deviation and variance. 1. The usual Relative Measures of Dispersion are: Among these four coefficients stated above the Coefficient of Variation is widely accepted and used in almost all practical situations mainly because of its accuracy and hence its approximation to explain the reality. The Standard Deviation, as a complete and comprehensive measure of dispersion, is well accepted by the statisticians specially because it possesses simultaneously all the qualities unhesitatingly which are required for an ideal measure of dispersion. For determining Range of a variable, it is necessary to arrange the values in an increasing order. WebThe disadvantages of mean, mode, and median are the same as their advantages: they are simple, not sophisticated enough to use when comparing data sets. Necessary cookies are absolutely essential for the website to function properly. Measures of dispersion provide information about the spread of a variable's values. Its not quite the same as the number of items in the sample. Outliers are single observations which, if excluded from the Variance is a measurement of the dispersion of numbers in a data set. Therefore, the SD possesses almost all the prerequisites of a good measure of dispersion and hence it has become the most familiar, important and widely used device for measuring dispersion for a set of values on a given variable. The first step in the creation of nanoparticles is the size reduction of the starting material using a variety of physical and chemical procedures [].Processes, including ball milling, mechanochemical synthesis, laser ablation, and ion The required Range is 54.5 4.5 = 50 or the observations on the variable are found scattered within 50 units. Low kurtosis in a data set is an indicator that data has lack of outliers. Some illnesses may raise a biochemical measure, so in a population containing healthy and ill people one might expect a bimodal distribution. Share Your PDF File 5. Instead one should refer to being in the top quarter or above the top quartile. As with variation, here we are not interested in where the telegraph poles are, but simply how far apart they are. One of the greatest disadvantages of using range as a method of dispersion is that range is sensitive to outliers in the data. *can be affected by extreme values which give a skewed picture, Research Methods - Features of types of exper, Research Methods - Evaluating types of experi, studies for the capacity, duration etc of mem, Chapter 3 - Infection Control, Safety, First. * You can modify existing ideas which saves time. It does not store any personal data. 1.51, 1.53. A small SD would indicate that most scores cluster around the mean score (similar scores) and so participants in that group performed similarly, whereas, a large SD would suggest that there is a greater variance (or variety) in the scores and that the mean is not representative. The smaller SD does not mean that that group of participants scored less than the other group it means that their scores were more closely clustered around the mean and didnt vary as much. (a) The main complaint against this measure is that it ignores the algebraic signs of the deviations. It is also used to calculate the However, the interquartile range and standard deviation have the following key difference: The interquartile range (IQR) is not affected by extreme outliers. For the data presented with their respective frequencies, the idea is to measure the same as the difference between the mid-values of the two marginal classes. In this method, its not necessary for an instrument to be calibrated against a standard. 2.1 Top-Down Approach. Take the square root of the value in #5, which will give the standard deviation. Disadvantages of Coefficient of Variation 1. (c) It is least affected by sampling fluctuations. This type of a curve is often used as a graphical method of measuring divergence from the average value due to inequitable concentration of data. The UK Faculty of Public Health has recently taken ownership of the Health Knowledge resource. The result finally obtained (G=0.60) thus implies the fact that a high degree of economic inequality is existing among the weavers of Nadia, W.B. Compared to Range, Quartile Deviation, no doubt, is a better measure of dispersion and it is also easy to calculate. Medical Statistics: a Commonsense Approach 4th ed. b. Divide the sum in #4 by (n 1). (i) Calculate mean deviation about Arithmetic Mean of the following numbers: Let us arrange the numbers in an increasing order as 15, 30, 35, 50, 70, 75 and compute their AM as: AM = 15 + 30 + 35 + 50 + 70 + 75/6 = 275/6. The mean of data set B is49. Chichester: Wiley-Blackwell 2007. There are no constraints on any population. Further algebraic treatments can also be applied easily with the result obtained afterwards. Standard deviation is often abbreviated to SD in the medical literature. Their calculation is described in example 1, below. Users of variance often employ it primarily in order to take the square root of its value, which indicates the standard deviation of the data set. WebThere are various methods that can be used to measure the dispersion of a dataset, each with its own set of advantages and disadvantages. This cookie is set by GDPR Cookie Consent plugin. The first quartile is the middle observation of the lower half, and the third quartile is the middle observation of the upper half. The average value of the difference between the third and the first quartiles is termed as the Quartile Deviation. Consequently, 28 is the median of this dataset. In a set of data that has many scores this would take a great deal of time to do. WebMerits and demerits of measures of dispersion are they indicate the dispersal character of a statistical series. Measures of Dispersion: Standard Deviation: In order to summarise a set of scores, a measure of central tendency is important, but on its own it is not enough. The range is given as the smallest and largest observations. (c) It is considerably affected by the extreme values of the given variable. Consider the following series of numbers: Here, the highest value of the series is 12 and the lowest is 1. The major advantage of the mean is that it uses all the data values, and is, in a statistical sense, efficient. (b) It uses AM of the given data as an important component which is simply computable. It can be used to compare distributions. Calculate the Mean Deviation for the following data: To calculate MD of the given distribution, we construct the following table: While studying the variability of the observations of a variable, we usually use the absolute measures of dispersion namely the Range, Quartile deviation. For determining the proportionate Quartile Deviation, also called the Coefficient of Quartile Deviation, we use the following formula: Calculate the Quartile Deviation and Co-efficient of Quartile Deviation from the following data: Here, n = 7, the first and third quartiles are: Determine the QD and CQD from the following grouped data: In order to determine the values of QD and Co-efficient of QD Let us prepare the following table: Grouped frequency distribution of X with corresponding cumulative frequencies (F). For any Sample, always the sum of deviations from mean or average is equal to 0. The first step in the creation of nanoparticles is the size WebDownload Table | Advantages and Disadvantages of Measures of Central Tendency and Dispersion* from publication: Clinicians' Guide to Statistics for Medical Practice and The prime advantage of this measure of dispersion is that it is easy to calculate. For example, if one were to measure a students consistency on quizzes, and he scored {40, 90, 91, 93, 95, 100} on six different quizzes, the range would be 60 points, marking considerable inconsistency. *can be affected by The standard deviation is calculated as the square root of variance by determining each data points deviation relative to the mean. Merits and Demerits of Measures of Dispersion Homework Help in Statistics If the variability is less, dispersion is insignificant. It is this characteristic of the standard deviation which makes it so useful. This is a Let us now look at some advantages and disadvantages of this measure: Advantages: Based on all observations; Doesnt change with change in origin; Advantage: (1) It is the most precise measure of dispersion. The variance is mathematically defined as the average of the squared differences from the mean. The standard deviation is vulnerable to outliers, so if the 2.1 was replace by 21 in Example 3 we would get a very different result. Disadvantage 2: Not suitable for time series In this set of data it can be seen that the scores in data set A are a lot more similar than the scores in data set B. Standard deviation is the best measure of central tendency because it comes with built-in indices that the other lack. KSSM MATHEMATICS FORM 4Measures of Dispersion for Ungrouped DataAdvantages and disadvantages of various measures of dispersionExample 10 Example 11Page 224(Live version)Please post your math-related questions here:https://www.messenger.com/t/olzenmathsMy Facebook PageOlzen Mathematics 2020https://www.facebook.com/olzenmaths/SPM Mathematics Revisionhttps://www.youtube.com/playlist?list=PLkQXp7Lpcc5vjES5ilKAmpqxnr_ksmD0nhttps://www.youtube.com/playlist?list=PLkQXp7Lpcc5vq6Gvr7XxTA74pGo2tnv2hhttps://www.youtube.com/playlist?list=PLkQXp7Lpcc5tibouEfmmJMxVpepXTVO7ASPM Trial 2019 Mathematics (Penang)https://www.youtube.com/playlist?list=PLkQXp7Lpcc5sen1xdtmUOeBCnWUkQo6tlKBSM Mathematics Form 4 The Straight Linehttps://www.youtube.com/playlist?list=PLkQXp7Lpcc5uRnZeuuLmeH2uCRvsI1FWTSetshttps://www.youtube.com/playlist?list=PLkQXp7Lpcc5tCU4z6cHRyb8edITHnlz4dMathematical Reasoninghttps://www.youtube.com/playlist?list=PLkQXp7Lpcc5tHoLE6SmXeMgJLfR-ppfLJKBSM Mathematics Form 5Chapter 1: Number Baseshttps://www.youtube.com/playlist?list=PLkQXp7Lpcc5uM44q_Lh9qvTMlP37z48i_Chapter 3: Transformations IIIhttps://www.youtube.com/playlist?list=PLkQXp7Lpcc5vsAJJYlJNnhYS8uMSWPLr8Chapter 4: Matriceshttps://www.youtube.com/playlist?list=PLkQXp7Lpcc5uFlFo3EAQaQO8FzKLVEltZChapter 5: Variationshttps://www.youtube.com/playlist?list=PLkQXp7Lpcc5uWR1FgOFS3I0659p1KiAIiChapter 6: Gradient and Area under a Graphhttps://www.youtube.com/playlist?list=PLkQXp7Lpcc5uLywl9PNUk7L3vKn1Q94kRChapter 7: Probability IIhttps://www.youtube.com/playlist?list=PLkQXp7Lpcc5vY4Qk6YKlhgt2RJnh49_uwChapter 8: Bearinghttps://www.youtube.com/playlist?list=PLkQXp7Lpcc5sCBEcZtLLeRbCjMBN0WsQwChapter 9: Earth as a Spherehttps://www.youtube.com/playlist?list=PLkQXp7Lpcc5sPJiCh5HrCyEsfTn9C0qfIChapter 10: Plans and Elevationshttps://www.youtube.com/playlist?list=PLkQXp7Lpcc5trEPI6kI7qGIuyKq_qSVNFKSSM Mathematics Form 4Chapter 1: Quadratic Functions and Equations in One Variablehttps://www.youtube.com/playlist?list=PLkQXp7Lpcc5uuLRIaZvhC6c7wy2Y2wAQxChapter 2: Number Baseshttps://www.youtube.com/playlist?list=PLkQXp7Lpcc5uM44q_Lh9qvTMlP37z48i_Chapter 3: Logical Reasoninghttps://www.youtube.com/playlist?list=PLkQXp7Lpcc5tW1FV9X0xuJiIoJWPzmR47Chapter 4: Operations on Setshttps://www.youtube.com/playlist?list=PLkQXp7Lpcc5t-vZJwjM-SwwHlnPXN3Y3aChapter 5: Network in Graph Theoryhttps://www.youtube.com/playlist?list=PLkQXp7Lpcc5sC3Ou4Z9_C1Mzc1AFLPbkCChapter 6: Linear Inequalities in Two Variableshttps://www.youtube.com/playlist?list=PLkQXp7Lpcc5tOjdt-dYMSGrjNXplO9zEqChapter 7: Graphs of Motionhttps://www.youtube.com/playlist?list=PLkQXp7Lpcc5sEQG0GawrUUABSN9vF-nT5Chapter 8: Measures of Dispersion for Ungrouped Datahttps://www.youtube.com/playlist?list=PLkQXp7Lpcc5uSKEYCI0cfnU4OWTtpaLrWChapter 9: Probability of Combined Eventshttps://www.youtube.com/playlist?list=PLkQXp7Lpcc5sAEsiFLHqo8ppw-D4oZo4Jhttps://www.youtube.com/playlist?list=PLkQXp7Lpcc5vY4Qk6YKlhgt2RJnh49_uwChapter 10: Consumer Mathematics: Financial Managementhttps://www.youtube.com/playlist?list=PLkQXp7Lpcc5sR0fJUZSsbU7eh7X9kmT54Additional MathematicsLinear Programminghttps://www.youtube.com/playlist?list=PLkQXp7Lpcc5vZk4_ncie9c6fqgCf_Fhn3
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