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Advanced Algebra II

Advanced Algebra II. Normal Distribution. In probability theory , the normal (or Gaussian ) distribution is a continuous probability distribution that has a bell-shaped probability density function , known as the Gaussian function or informally the bell curve : [ nb 1]

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Advanced Algebra II

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  1. Advanced Algebra II Normal Distribution

  2. In probability theory, the normal (or Gaussian) distribution is a continuous probability distribution that has a bell-shaped probability density function, known as the Gaussian function or informally the bell curve:[nb 1] Note that a normally-distributed variable has a symmetric distribution about its mean.

  3. Dark blue is less than one standard deviation from the mean. For the normal distribution, this accounts for about 68% of the set, while two standard deviations from the mean (medium and dark blue) account for about 95%, and three standard deviations (light, medium, and dark blue) account for about 99.7%.

  4. Figure 1. A simple bimodal distribution, in this case a mixture of two normal distributions with the same variance but different means. The figure shows the probability density function (p.d.f.), which is an average of the bell-shaped p.d.f.s of the two normal distributions

  5. The bimodal distribution of sizes of weaver ant workers shown in Figure 2 arises due to existence of two distinct classes of workers, namely major workers and minor workers.

  6. Bimodal distributions are a commonly used example of how summary statistics such as the mean, median, and standard deviation can be deceptive when used on an arbitrary distribution. For example, in the distribution in Figure 1, the mean and median would be about zero, even though zero is not a typical value. The standard deviation is also larger than deviation of each normal distribution.

  7. μ is the mean of the population; σ is the standard deviation of the population.

  8. In statistics, a standard score indicates by how many standard deviations an observation or datum is above or below the mean. It is a dimensionless quantity derived by subtracting the population mean from an individual raw score and then dividing the difference by the populationstandard deviation. Standard scores are also called z-values, z-scores, normal scores, and standardized variables; the use of "Z" is because the normal distribution is also known as the "Z distribution".

  9. Standard Normal Distributions and Z Scores A normal distribution that is standardized (so that it has a mean of 0 and a SD of 1) is called the standard normal distribution, or the normal distribution of z-scores. If we know the mean m ("mu"), and standard deviation s ("sigma") of a set of scores which are normally distributed, we can standardize each "raw" score, x, by converting it into a z score by using the following formula on each individual score:

  10. If a Z-Score…. ü      Has a value of 0, it is equal to the group mean. ü      Is positive, it is above the group mean. ü      Is negative, it is below the group mean. ü      Is equal to +1, it is 1 Standard Deviation above the mean. ü      Is equal to +2, it is 2 Standard Deviations above the mean. ü      Is equal to -1, it is 1 Standard Deviation below the mean. ü      Is equal to -2, it is 2 Standard Deviations below the mean.

  11. Q1: Suppose that SAT scores among U.S. college students are normally distributed with a mean of 500 and a standard deviation of 100. What is the probability that a randomly selected individual from this population has an SAT score at or below 600?

  12. Using the formula to calculate the z value, we find z = (x - m)/s = (600 - 500)/100 = +1.00. Recall that the z-score is the number of standard deviations that the score of interest differs from the mean. A score of 600 is one standard deviation above the mean, so it has a z value of 1.00. Using either a z table or the p-z converter, we find that the probability that a randomly selected z score in a normal distribution will exceed z = 1.00 is .159, the right-tailed p value, or about 16%. The probability that z will exceed 1.00 for a randomly selected score is equivalent to the probability that a randomly selected individual from this population will have an SAT score over 600, about 16%. The probability that a randomly selected individual from this population will have an SAT score below 600 is 100% - 16% = 84%

  13. http://www.youtube.com/watch?annotation_id=annotation_202153&feature=iv&src_vid=1xhCL5m4nI0&v=fXOS4Q3nJQYhttp://www.youtube.com/watch?annotation_id=annotation_202153&feature=iv&src_vid=1xhCL5m4nI0&v=fXOS4Q3nJQY

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