UNIT 4: Learning Journal

UNIT 4 LEARNING JOURNAL The Gaussian distribution, sometimes referred to as the normal distribution, is a symmetric probability distribution with respect to the mean. Since many statistical techniques presume that the data have a normal distribution, it is one of the most popular distributions in statistics. The bell-shaped curve of the distribution shows that the majority of the data points are grouped around the mean, with fewer points appearing as the values deviate from the mean. A particular instance of the normal distribution in which the standard deviation is one and the mean is zero is known as the standard normal distribution. Importance in Statistics In statistics, the normal distribution is important for a number of reasons. It is a vital model for portraying data from the actual world because, first of all, it is a natural occurrence that occurs in many disciplines, including biology, economics, and engineering. The Central Limit Theorem states that regardless of the variables' initial distribution, the average or sum of many independent random variables, each with a different distribution, will typically follow a normal distribution. This makes the normal distribution a useful tool for statistical analysis since it can be used to describe a wide range of naturally occurring phenomena. Second, statistical analysis is made easier by the well-known characteristics of the normal distribution. In a normal distribution, for instance, 99.7% of the data fall within three standard deviations, 95% fall within two, and roughly 68% fall within one standard deviation of the mean. When evaluating probabilities and drawing conclusions about the population from sample data, these characteristics are useful. Lastly, a lot of statistical procedures, such as regression analysis, confidence intervals, and hypothesis testing, are based on the normal distribution. Sound statistical analysis requires a grasp of the data's features because many statistical techniques rely on the assumption that the data follows a normal distribution. Example of a Normal Distribution Scenario The height of adult females in a population is a typical illustration of a scenario that may be represented using a normal distribution. Assume that, with a mean height of 64 inches and a standard deviation of 3 inches, the heights of adult women in a given nation exhibit a normal distribution. The likelihood that a woman is a given height or falls within a given range of heights can be determined using the normal distribution. For instance, we can figure out the likelihood that a lady chosen at random from this group will be between 61 and 67 inches tall. We can standardize these values and determine the associated probabilities using the Z-score calculation because the distribution is normal. 1.​ Z-score Calculation: The Z-score formula is given by: where: ●​ X is the value (height), ●​ μ is the mean (64 inches), ●​ σ is the standard deviation (3 inches). For 61 inches: Z = (61 − 64) / 3 ​= −3​/ 3 = −1 For 67 inches: Z = (67 − 64) / 3 ​= 3 / 3 ​= 1 Finding Probabilities: Using a standard normal distribution table (Z-table), we can find the probabilities corresponding to these Z-scores. For a Z-score of -1, the probability is approximately 0.1587, and for a Z-score of 1, the probability is approximately 0.8413. To find the probability of a woman being between 61 inches and 67 inches tall, we subtract the probability at Z = -1 from the probability at Z = 1: P (61 ≤ X ≤ 67) = P (Z = 1) − P (Z = −1) = 0.8413 − 0.1587 = 0.6826 Therefore, the probability that a randomly selected woman from this population is between 61 inches and 67 inches tall is approximately 68.26%. Graphical Representation The following graph represents the normal distribution for the heights of adult women in this example: /\ / / \ \ ----|----|----|----|----|----|----|----| 58 61 64 67 Mean = 64 inches 70 73 76 This graph illustrates the bell-shaped curve of the normal distribution, where the majority of women have heights near the mean (64 inches), and fewer women have heights further away from the mean (either shorter or taller). Conclusion Because it makes calculating probabilities easier and represents a wide range of naturally occurring occurrences, the normal distribution is a fundamental component of statistical analysis. Regression analysis, confidence intervals, and hypothesis testing all make heavy use of its characteristics. The normal distribution offers a straightforward and effective method for drawing statistical conclusions in practical contexts, such as determining the likelihood that a woman falls within a particular height range. Anyone interested in data analysis and decision-making has to understand the fundamentals of the normal distribution. REFERENCES: ●​ Field, A. (2013). Discovering statistics using IBM SPSS statistics (4th ed.). Sage Publications. ●​ Montgomery, D. C., & Runger, G. C. (2014). Applied statistics and probability for engineers (6th ed.). Wiley.