Entity Academy Lesson 5 Normal Distribution Notes

.docx

School

Liberty University *

*We aren’t endorsed by this school

Course

BASIC

Subject

Statistics

Date

Jan 9, 2024

Type

docx

Pages

Uploaded by JusticeFogPrairieDog37

The Normal Distribu tion Data distributions come in all shapes and sizes The Normal Distribution is perfectly symmetrical and shaped like a bell. It will always look like a bell and is sometimes called the “Bell curve”. Having been discovered by Gaussian, “Gaussian” as well, so the normal distribution, the bell curve, or the Parameters of Normal Distribution Gaussian distribution all mean the same thing. A very common -1 SD  Mean  +1 SD distribution. Talk about normal distributions in general terms. The Majority shape implies that an infinite number of measurements was taken. of Data The measurements near the center are common, while others aren’t. Though the distribution doesn’t show it, the curved part never to9uches the horizontal axis; it goes left to right forever—otherwise known as a distribution. Descriptive Statistics for the Normal Distribution Mean – the mean of a normally distributed variable is shown graphically as the vertical line in the center of the symmetry or is in the middle of the Bell curve. Standard Deviation – the vertical lines are placed representing “mean +/- x standard deviations”. Much of the data in a normally distributed variable are within one standard deviation of the mean. If you go out to +/- 2 standard deviations from the mean, you have captured most of the data in the distribution. If you go out to +/- 3 standard deviations, you have captured almost all of the data. Median – the median is the “middle value,” or the point at which the half of the area under the curve of the distribution is to the left, and the other ½ is to the right. In a normal distribution, the mean and the media are the same number. Sindy Saintclair Monday, November 28 2021 Lesson 5 – Normal Distribution and the Central Limit Theorem Learning Objectives and Questions Notes and Answers

Range – because the distribution goes on forever to the left and right, there is no min, and there is no max. So, there is no range. Practical Usage of the Normal Distribution Most of the berries measure were between 70 and 130 mg. Interpret that as meaning that the mean of the distribution is 100 mg, and the standard deviation is 10 mg. Can you see how a lot of the berries are between 90 and 110, most are 80 and 120, and virtually all are between 70 and 130? This is the practical interpretation of the mean and standard deviation. Later, you will learn how you can determine the probability of finding a berry in a certain weight range, and how rare it is to see an unusually large or unusually small berry. The Standa rd Normal Distrib ution The mean will always be 0 and the standard deviation will always be 1. Greek symbol stands for sigma. 68% of the values will be within 1 SD of the mean 95% of values will be within 2 SDs of the mean 99.7% of value will be within 3 SDs of the mean

The 69-95-99 Rule Here are a few values for the standard deviation areas on the Standard Normal Distribution: - Area between -3 and -2 = 0.022 - area between -2 and -1 = 0.136 - area between -1 and 0 =0.341 - area between 0 and 1 = 0.341 - area between 1 and 2 = 0.136 - area between 2 and 3 = 0.022 The z- score Example One: Ghana height of a young adult woman, with a mean of 159.0 cm and a standard deviation of about 4.9 cm. Gabianu is a college student originally from Ghana standing at 169.0 cm tall. 10 cm taller than the average woman which is about 4 inches tall—this is considered 2 standard deviations taller (2 x 4.9 = 9.8) than the average woman. So Gabianu is about 2.04 standard deviations taller than the average Ghanian woman. Is she extraordinarily tall or just a bit on the tall side? Rather than depending on subjective declaration of Gabianu’s stature, standardize her height. This is z-score, a great way to measure any individual piece of data relative to the population. To calculate the z- score, you need to know a couple of things:

- population mean - population standard deviation  Though population parameters are technically unknown, treat them as if they are known because many calculations depend on knowing population parameters. Verbiage used to imply that the population parameters are known: - the baseline value of fat content in cheese - the historical mean test score - the agreed upon value for the speed of light - the average lifespan of an incandescent light bulb  mu or µ stands for population mean Z-score is simply the difference  sigma or σ stands for the population standard deviation between x-value and µ  If the value is larger than the mean, the numerator will be positive. If the value is less than the mean, the numerator will be negative, which is fine. A z-score example In the case of Gabianu, the numerator for her z-score is 10 (169.0- 159.0). With sigma in the denominator of the fraction, the difference (in the numerator) is simply getting scaled. Gabianu’s height difference from the mean is 10cm. When you divide that by 4.9 (which is the population SD, or sigma), you are essentially converting the height difference of 10cm and expressing it in terms of the sigma. So, ten divided by 4.9 is about 2.04, which means Gabianu’s height is about 2.04 SDs more than the average height. Gabianu’s friend Rashida from Ghana is 154.8 cm tall. Calculate her z- score: Because Rashida has a z-score of -0.86, she is about one SD shorter than the average Ghanian woman. In short, the z-score is a measure of how many standard deviations your value is away from the population mean.

z- scores in the Standa rd Normal Distrib ution Overlay the Standard Normal Distribution on IQ - the z-score will always be equal to x - unusual events that happen 5% or less - can be used to calculate probabilities, which are equal to the area between the curve and the horizontal axis of the distribution from which the random value is taken. To make the math easy, you can arbitrarily set the value of the area under the entire curve to 1. Berry example – figure out the probability of selecting a single berry at random with the weight between 90 and 110mg. In other words, you would like to figure out the area under the curve between 90 and 110, and compare that to the area under the whole curve. What is the area of the green shaded region, relative to the area under the entire curve (blue and green regions combined)? The z- score for 90 is -1, and the z-score for 110 is 1, because µ=100 and σ=10. The probability of a single berry being between 90 and 110 is the same as the probability of a single z-score being between -1 and 1.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version