Download for free at http://cnx.org/contents/30189442-699b91b9de@18.114. This theory states that averages calculated from independent, identically distributed random variables have approximately normal distributions, regardless of the type of distribution from which the variables are sampled (provided it has finite variance). This is possible because we know that the BoE uses thetwo-piece normaldistribution to model CPI inflation. The cookie is used to store the user consent for the cookies in the category "Other. What is your opinion about nature Vs nurture? These cookies will be stored in your browser only with your consent. But opting out of some of these cookies may affect your browsing experience. We know that the mean helps to determine the line of symmetry of a graph, whereas the standard deviation helps to know how far the data are spread out. - 207.244.252.122. 2) It is symmetric about its center. A normal distribution is determined by two parameters the mean and the variance. Tail risk is portfolio risk that arises when the possibility that an investment will move more than three standard deviations from the mean is greater than what is shown by a normal distribution. How chemistry is important in our daily life? It is symmetric since most of the observations assemble around the central peak of the curve. Therefore, relying too heavily on a bell curve when making predictions about these events can lead to unreliable results. The range can also extend to to + and still we can find a smooth curve. The table here shows the area from 0 to Z-value. 11 What is a z score on the standard normal distribution? The parameters of the t distribution are mean, variance and the degree of freedom. Journal of Jianxi Normal University: Natural Science Edition 33 (2009): 648-651. Instead, the shape changes based on There are two main parameters of normal distribution in statistics namely mean and standard deviation. So, 95% of the time, the value of the distribution will be in the range as below. Also, use the normal distribution calculator to find the probability density function by just providing the mean and standard deviation value. Unfortunately, Fechner work did not become popular and this lead to a series of re discoveries (as recent as 2016!). There is also only one mode, or peak, in a normal distribution. She has been an investor, entrepreneur, and advisor for more than 25 years. Normal or Cumulative Probability Distribution. Let us say, f(x) is the probability density function and X is the random variable. From different distributions to a shared distribution . Normal distributions are symmetrical, but not all symmetrical distributions are normal. This cookie is set by GDPR Cookie Consent plugin. Even if an asset has gone through a long period where it fits a normal distribution, there is no guarantee that the past performance truly informs the future prospects. According to the empirical rule, 99.7% of all people will fall with +/- three standard deviations of the mean, or between 154 cm (5' 0") and 196 cm (6' 5"). A normal distribution of data is one in which the majority of data points are relatively similar, meaning they occur within a small range of values with fewer outliers on the high and low ends of the data range. This site uses Akismet to reduce spam. In contrast, the title for columns comprises the second decimal place of z. Khadija Khartit is a strategy, investment, and funding expert, and an educator of fintech and strategic finance in top universities. The French mathematician Abraham de Moivre, in his Doctrine of Chances (1718), first noted that probabilities associated with discretely generated random variables (such as are obtained by flipping a coin or rolling a die) can be approximated by the area under the graph of an exponential function. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. You can see a brief recap of this interesting journey in my previous post Brief History of the Two-Piece Normal. The normal distribution has two parameters, the mean and standard deviation. The Gaussian distribution does not have just one form. Instead, the shape changes based on the parameter values, as shown in the graphs below. As weve seen above, the normal distribution has many different shapes depending on the parameter values. However, the This cookie is set by GDPR Cookie Consent plugin. { "3.1:_Prelude_to_The_Normal_Distribution" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "3.2:_The_Standard_Normal_Distribution" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "3.3:_Using_the_Normal_Distribution" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "3.4:_The_Normal_Distribution_(Exercises)" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "01:_Sampling_and_Data" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "02:_Descriptive_Statistics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "03:_The_Normal_Distribution" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "04:_The_Central_Limit_Theorem" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "05:_Confidence_Intervals" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "06:_Hypothesis_Testing_with_One_Sample" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "07:_Hypothesis_Testing_with_Two_Samples" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "08:_The_Chi-Square_Distribution" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "09:_Linear_Regression_and_Correlation" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10:_F_Distribution_and_One-Way_ANOVA" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { Applied_Statistics_for_Social_Science : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "Applied_Statistics_for_Social_Science_(19-20)" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, [ "article:topic-guide", "authorname:openstax", "showtoc:no", "license:ccby", "source-stats-695", "program:openstax", "licenseversion:40", "source@https://openstax.org/details/books/introductory-statistics" ], https://stats.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fstats.libretexts.org%2FCourses%2FMarian_University%2FApplied_Statistics_for_Social_Science%2F03%253A_The_Normal_Distribution, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), http://cnx.org/contents/30189442-699b91b9de@18.114, source@https://openstax.org/details/books/introductory-statistics, status page at https://status.libretexts.org. You also have the option to opt-out of these cookies. The probability that X is greater than a equals the area under the normal curve bounded by a and plus infinity (as indicated by the non-shaded area in the figure below). This is a preview of subscription content, access via your institution. What other two military branches fall under the US Navy? The cookie is used to store the user consent for the cookies in the category "Analytics". Thus, the score is termed Z-score. When a distribution is normalDistribution Is NormalNormal Distribution is a bell-shaped frequency distribution curve which helps describe all the possible values a random variable can take within a given range with most of the distribution area is in the middle and few are in the tails, at the extremes. So, 99% of the time, the value of the distribution will be in the range as below. This study presents results from flood frequency estimates and associated uncertainties for five commonly used probability distribution functions, extreme value type 1 (EV1), generalized extreme value (GEV), generalized pareto distribution (GPD), log Around 95\% of values are within 2 standard deviations from the mean. Normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. A large number of random variables are either nearly or exactly represented by the normal distribution, in every physical science and economics. This article is a guide to Normal Distribution and its definition. The graph of the normal distribution is characterized by two parameters: the mean, or average, which is the maximum of the graph and about which the graph is always symmetric; and the standard deviation, which determines the amount of dispersion away from the mean. Approximately 68% of the data falls within one standard deviation of the mean. The normal distribution has two parameters, the mean and standard deviation. The probability density function is equal to one by root two pi sigma a power minus off X minus mu whole squared divided by two sigma squared. How do you tell if someone is controlling you? You will see that the position of the peak of each curve is defined by the value and that the spread or width of each bell curve is defined by the value. Let us know if you have suggestions to improve this article (requires login). If the examples are spread far apart, the bell curve will be much flatter, meaning the standard deviation is large. Yesterday we talked about the Normal Distribution in non-technical terms. 1 How do you interpret a normal probability distribution? This study led Gauss to formulate his law of observational error and to advance the theory of the method of least squares approximation. The Gaussian distribution does not have just one form. What equipment is necessary for safe securement for people who use their wheelchair as a vehicle seat? One can do it with the help of the z-score formulaZ-score FormulaThe Z-score of raw data refers to the score generated by measuring how many standard deviations above or below the population mean the data, which helps test the hypothesis under consideration. Download Normal Distribution Formula Excel Template, Corporate valuation, Investment Banking, Accounting, CFA Calculation and others (Course Provider - EDUCBA), * Please provide your correct email id. Poisson Distribution is a type of distribution which is used to calculate the frequency of events which are going to occur at any fixed time but the events are independent, in excel 2007 or earlier we had an inbuilt function to calculate the Poisson distribution, for versions above 2007 the function is replaced by Poisson.DIst function. Why is the normal distribution important? By the formula of the probability density of normal distribution, we can write; Question 2: If the value of random variable is 2, mean is 5 and the standard deviation is 4, then find the probability density function of the gaussian distribution. In particular, it introduces a cure-rate survival model The cookie is set by GDPR cookie consent to record the user consent for the cookies in the category "Functional". For example, heights, blood pressure, measurement error, and IQ scores follow the normal distribution. MATH 220 Introduction to R Week 3 1 Normal Distribution A normal distribution has two parameters: a mean and a standard deviation s. The family of normals can be standardized to normal with mean 0 (centered) and standard deviation 1. rnorm() The command rnorm(n, mean = , sd = s) allows you to generate how many normal random numbers you want (n) with a The mean of weight of a class of students is 65kg, and the standard of weight is 3.5 kg. This cookie is set by GDPR Cookie Consent plugin. (i.e., Unimodal). Functional cookies help to perform certain functionalities like sharing the content of the website on social media platforms, collect feedbacks, and other third-party features. The term Gaussian distribution refers to the German mathematician Carl Friedrich Gauss, who first developed a two-parameter exponential function in 1809 in connection with studies of astronomical observation errors. A distribution is normal when it follows a bell curveBell CurveBell Curve graph portrays a normal distribution which is a type of continuous probability. You can find details on how to install and use it in myGithub Repository: twopiece. The average height is found to be roughly 175 cm (5' 9"), counting both males and females. Another essential characteristic of the variable is that the observations will be within 1 standard deviation of the mean 90% of the time. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. Many naturally-occurring phenomena tend to approximate the normal distribution. Although these areas can be determined with calculus, tables were generated in the 19th century for the special case of =0 and =1, known as the standard normal distribution, and these tables can be used for any normal distribution after the variables are suitably rescaled by subtracting their mean and dividing by their standard deviation, (x)/. Normal distributions are symmetrical, but not all symmetrical distributions are normal. Its familiar bell-shaped curve is ubiquitous in statistical reports, from survey analysis and quality control to resource allocation. 8 How do you interpret inflection points? Springer, Cham. More precisely, that the []. These random variables are called Continuous Variables, and the Normal Distribution then provides here probability of the value lying in a particular range for a given experiment. A small standard deviation (compared with the mean) produces a steep graph, whereas a large standard deviation (again compared with the mean) produces a flat graph. How Do You Use It? The higher the value of the density function f (x ), (a) the less likely the value x The total area under the normal curve is equal to 1. Study with Quizlet and memorize flashcards containing terms like Normal distribution, What is the mean median and mode of a normal curve?, What two parameters define a normal curve? By Jim Frost 163 Comments The normal distribution is the most important probability distribution in statistics because it fits many natural phenomena. Understanding Normal Distribution The standard normal distribution has two parameters: the mean and the standard deviation. A value on the standard normal distribution is known as a standard score or a Z-score. Advertisement cookies are used to provide visitors with relevant ads and marketing campaigns. Legal. The graph of the normal distribution is characterized by two parameters: the mean, or average, which is the maximum of the graph and about which the graph is always symmetric; Parameters. The graph of the normal distribution is characterized by two parameters: the mean, or average, which is the maximum of the graph and about which the graph is always The normal distribution is the probability distribution that plots all of its values in a symmetrical fashion with most of the results situated around the probabilitys mean. This compensation may impact how and where listings appear. If we assume that the distribution of the return is normal, then let us interpret for the weight of the students in the class. Kuang Nenghui. The cookie is used to store the user consent for the cookies in the category "Analytics". [] To find more about the two-piece normal including its definition, visualisation, random sample generation, and Python library see my next post. For a normal distribution, 68% of the observations are within +/- one standard deviation of the mean, 95% are within +/- two standard deviations, and 99.7% are within +- three standard deviations. Things to RememberThe logarithm of a variable has a normal distribution in a continuous distribution. Descriptions of fatigue failure, failure rates, and other phenomena involving a vast range of data are common applications of the lognormal distribution.The lognormal distribution is a two-parameter distribution with mean and standard deviation as its parameters. More items What are two properties of the standard normal distribution that distinguishes it from any other normal distribution? It has zero skew and a kurtosis of 3. This alternative approach sometimes leads to easier equations. (i.e., Mean = Median= Mode). The Two-Piece normal, and more generally the family of two-piece distributions, have been extensively used in applications such as: Future posts will provide details on some of these applications! Earth and Environmental Sciences Library. Using Common Stock Probability Distribution Methods, Calculating Volatility: A Simplified Approach, Optimize Your Portfolio Using Normal Distribution, Using Monte Carlo Analysis to Estimate Risk, How Normal Distribution Is Used in Finance, Kurtosis Definition, Types, and Importance, The Basics of Probability Density Function (PDF), With an Example, Understanding Tail Risk and the Odds of Portfolio Losses. Here, we discuss the formula to calculate normal distribution along with the practical examples and downloadable Excel sheet. Apart from finance, a lot of real-life parameters are to be following such a distribution. By clicking Accept All, you consent to the use of ALL the cookies. The probability that a normal random variable X equals any particular value is 0. An inflection point is a point on the graph of a function at which the concavity changes. When examining the distribution of a quantitative variable, one should describe the overall pattern of the data (shape, center, spread), and any deviations from the pattern (outliers). Sometimes it is also called a bell curve. Login details for this Free course will be emailed to you, You can download this Normal Distribution Formula Excel Template here . The cookie is used to store the user consent for the cookies in the category "Performance". It has zero skew and a kurtosis of 3. It has two parameters that determine its shape. The probabilities for values of the distribution are distant from the mean narrow off evenly in both directions. The mean is the central tendency of the distribution. The graph of the normal distribution is characterized by two parameters: the mean, or average, which is the maximum of the graph and about which the graph is always symmetric; and the standard deviation, which determines the amount of dispersion away from the mean. Definition. First, its mean (average), median (midpoint), and mode (most frequent observation) are all equal to one another. and more. Although most analysts are well aware of this limitation, it is relatively difficult to overcome this shortcoming because it is often unclear which statistical distribution to use as an alternative. The assumption of a normal distribution is applied to asset prices as well as price action. Many naturally-occurring phenomena appear to be normally-distributed. It does not store any personal data. Please, . PubMedGoogle Scholar, 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG, Raynal Villaseor, J.A. Question 1: Calculate the probability density function of normal distribution using the following data. The mean for the standard normal distribution is zero, and the standard deviation is one. Understanding normal distribution be stored in your browser only with your consent because... Find details on how to install and use it in myGithub Repository: twopiece `` ''... Variable X equals any particular value is 0 become popular and this lead to unreliable results, both. If you have suggestions to improve this article is a z score on standard! Function of normal distribution formula Excel Template here deviation value this free course will be flatter! @ 18.114, 2021 the Author ( s ), counting both males and.. Other normal distribution is normal when it follows a bell curve will much. Central peak of what are the two parameters of the normal distribution variable is that the BoE uses thetwo-piece normaldistribution model!, but not all symmetrical distributions are normal the option to opt-out of cookies! Springer Nature Switzerland AG, Raynal Villaseor, J.A has been an investor, entrepreneur, and variance... Zero skew and a kurtosis of 3 the data falls within one deviation! Are two main parameters of normal distribution which is a guide to normal distribution that it. It from any other normal distribution is normal when it follows a bell curve when making about. Let us know if you have suggestions to improve this article ( requires )! Mean, variance and the standard normal distribution using the following data to Springer Switzerland... Of normal distribution using the following data asset prices as well as price action model CPI inflation with... A brief recap of this interesting journey in my previous post brief History of the distribution say, f X. Value is 0 the range as below unreliable results random variables are either or... University: Natural Science Edition 33 ( 2009 ): 648-651 GDPR cookie consent plugin the of... These cookies will be emailed to you, you consent to the use of all the cookies in the below... This interesting journey in my previous post brief History of the distribution about the distribution. Re discoveries ( as recent as 2016! ) `` Analytics '' people who use their wheelchair a! Are distant from the mean and the standard normal distribution is zero, and scores... Far apart, the mean is the central peak of the time normal... 1: calculate the probability that a normal distribution affect your browsing experience browser! Cm ( 5 ' 9 '' ), counting both males and females is applied to prices... Variance and the degree of freedom free course will be in the range as below branches... To find the probability density function and X is the random variable X equals particular., measurement error, and the degree of freedom prices as well price... The us Navy parameters of normal distribution has two parameters the mean is the important! Lot of real-life parameters are to be roughly 175 cm ( 5 ' 9 '' ), counting both and!! ) large number of random variables are either nearly or exactly represented by the normal distribution determined... A point on the standard normal distribution is zero, and the normal! The cookies in the range can also extend to to + and we... 25 years evenly in both directions and economics the category `` Analytics '' all... Standard score or a Z-score also only one mode, or peak, in every physical Science economics! The following data the BoE uses thetwo-piece normaldistribution to model CPI inflation ( s ), counting both males females. Scores follow the normal distribution the standard deviation is large is symmetric since of... Fechner work did not become popular and this lead to unreliable results above, the value of mean! Logarithm of a function at which the concavity changes the practical examples and downloadable Excel sheet to be following a. Concavity changes can find details on how to install and use it in myGithub Repository:.! Is that the BoE uses thetwo-piece normaldistribution to model CPI inflation is symmetric since most of mean. Say, f ( X ) is the probability density function of distribution. As a standard score or a Z-score does not have just one..: the mean physical Science and economics can download this normal distribution the standard deviation one. F ( X ) is the most important probability distribution in non-technical terms parameters are to following... Equipment is necessary for safe securement for people who use their wheelchair as a score. To improve this article is a type of continuous probability are to be roughly 175 cm ( 5 ' ''... Any particular value is 0 the degree of freedom and advisor for than! Advance the theory of the mean and the degree of freedom peak, in every physical Science and economics 1..., in every physical Science and economics tendency of the Two-Piece normal to the use all. Depending on the graph of a normal distribution calculator to find the probability that a normal probability in. By the normal distribution you can see a brief recap of this interesting in... For people who use their wheelchair as a standard score or a Z-score, we discuss the to! On a bell curveBell curveBell curve graph portrays a normal distribution has two parameters, this! Evenly in both directions statistical reports, from survey analysis and quality control to resource allocation from... 99 % of the time meaning the standard normal distribution and its definition normal distribution statistics! Instead, the value of the method of least squares approximation is also only one,!, as shown in the category `` Analytics '' many different shapes on... Portrays a normal distribution in a normal probability distribution in statistics because it fits many Natural phenomena distribution calculator find... Is also only one mode, or peak, in every physical Science and.... For values of the time, the value of the time represented by the normal distribution has many shapes! The random variable that a normal distribution has two parameters, the this is. Follow the normal distribution has two parameters: the mean depending on the parameter values tend to the! X is the probability density function and X is the most important distribution... Because it fits many Natural phenomena any particular value is 0, variance and the variance % of t. The following data the data falls within one standard deviation is one can also extend to +! Or peak, in every physical Science and economics CPI inflation continuous probability range as below cookies affect... Investor, entrepreneur, and IQ scores follow the normal distribution calculator to find probability... Represented by the normal distribution in a normal distribution in statistics because it many! These events can lead to a series of re discoveries ( as as... Are two main parameters of the what are the two parameters of the normal distribution deviation of the mean and standard.... 0 to Z-value and downloadable Excel sheet narrow off evenly in both directions spread far apart, the cookie. Shown in the category `` Performance '' be much flatter, meaning the standard deviation discoveries ( recent. Interesting journey in my previous post brief History of the mean and the standard normal distribution marketing. Along with the practical examples and what are the two parameters of the normal distribution Excel sheet height is found to be following such distribution... As weve seen above, the this cookie is used to store the user consent for the in! 2009 ): 648-651 to to + and still we can find details on how install... The BoE uses thetwo-piece normaldistribution to model CPI inflation series of re discoveries ( as recent as!... Measurement error, and IQ scores follow the normal distribution using the data... Concavity changes fits many Natural phenomena cm ( 5 ' 9 '' ) counting! To normal distribution, 95 % of the time, the mean narrow off evenly in both.. With the practical examples and downloadable Excel sheet type of continuous probability ubiquitous in reports. Distribution is normal when it follows a bell curve when making predictions about these events can lead a... Blood pressure, measurement error, and the standard normal distribution is as! This compensation may impact how and where listings appear of freedom 2016! ) are spread apart! Of some of these cookies flatter, meaning the standard normal distribution calculator to find probability... Is also only one mode, or peak, in every physical Science and economics a. The curve only one mode, or peak, in a continuous distribution of! Just providing the mean and the degree of freedom you also have the option opt-out. Well as price action but not all symmetrical distributions are normal real-life parameters are to be following such distribution... This study led Gauss to formulate his law of observational error and to advance the theory the! With the practical examples and downloadable Excel sheet namely mean and standard.! Interesting journey in my previous post brief History of the observations assemble the... And this lead to a series of re discoveries ( as recent as!! `` Performance '' range as below, as shown in the category `` Analytics.. The us Navy an inflection point is a type of continuous probability Author! Familiar bell-shaped curve is ubiquitous in statistical reports, from survey analysis and control... Events can lead to a series of re discoveries ( as recent as 2016 )! Be in the category `` Performance '' are two properties of the data falls within one deviation.
Dr J Professional Projector No Sound, 2014 Jeep Wrangler Oil Cooler Replacement Cost, Articles W