normal distribution

Examples of normal distribution in the following topics:

• The Standard Normal Distribution

  • The standard normal distribution is a normal distribution of standardized values called z-scores.
  • For example, if the mean of a normal distribution is 5 and the standard deviation is 2, the value 11 is 3 standard deviations above (or to the right of) the mean.
  • The mean for the standard normal distribution is 0 and the standard deviation is 1.
  • The transformation z = (x − µ)/σ produces the distribution Z ∼ N ( 0,1 ) .
  • The value x comes from a normal distribution with mean µ and standard deviation σ.

• Introduction to Normal Distributions

  • The normal distribution is the most important and most widely used distribution in statistics.
  • As you will see in the section on the history of the normal distribution, although Gauss played an important role in its history, de Moivre first discovered the normal distribution.
  • Strictly speaking, it is not correct to talk about "the normal distribution" since there are many normal distributions.
  • Figure 1 shows three normal distributions.
  • The mean, median, and mode of a normal distribution are equal.

• Areas Under Normal Distributions

  • State the proportion of a normal distribution within 1 standard deviation of the mean
  • Areas under portions of a normal distribution can be computed by using calculus.
  • Figure 1 shows a normal distribution with a mean of 50 and a standard deviation of 10.
  • Figure 2 shows a normal distribution with a mean of 100 and a standard deviation of 20.
  • The normal distributions shown in Figures 1 and 2 are specific examples of the general rule that 68% of the area of any normal distribution is within one standard deviation of the mean.

• Introduction

  • The normal, a continuous distribution, is the most important of all the distributions.
  • Some of your instructors may use the normal distribution to help determine your grade.
  • Most IQ scores are normally distributed.
  • Often real estate prices fit a normal distribution.
  • In this chapter, you will study the normal distribution, the standard normal, and applications associated with them.

• The Normal Distribution

  • Normal distributions are a family of distributions all having the same general shape.
  • The normal distribution is a continuous probability distribution, defined by the formula:
  • If $\mu = 0$ and $\sigma = 1$, the distribution is called the standard normal distribution or the unit normal distribution, and a random variable with that distribution is a standard normal deviate.
  • The simplest case of normal distribution, known as the Standard Normal Distribution, has expected value zero and variance one.
  • Many sampling distributions based on a large $N$ can be approximated by the normal distribution even though the population distribution itself is not normal.

• Student Learning Outcomes

• History of the Normal Distribution

  • The importance of the normal curve stems primarily from the fact that the distributions of many natural phenomena are at least approximately normally distributed.
  • Laplace showed that even if a distribution is not normally distributed, the means of repeated samples from the distribution would be very nearly normally distributed, and that the larger the sample size, the closer the distribution of means would be to a normal distribution.
  • Because the distribution of means is very close to normal, these tests work well even if the original distribution is only roughly normal.
  • The normal approximation to the binomial distribution for 12 coin flips.
  • The smooth curve is the normal distribution.

• Introduction to normal distribution

  • Among all the distributions we see in practice, one is overwhelmingly the most common.
  • Indeed it is so common, that people often know it as the normal curve or normal distribution, shown in Figure 3.1.
  • Variables such as SAT scores and heights of US adult males closely follow the normal distribution.
  • Many variables are nearly normal, but none are exactly normal.
  • Thus the normal distribution, while not perfect for any single problem, is very useful for a variety of problems.

• Probability Histograms and the Normal Curve

  • Many different types of distributions can be approximated by the normal curve.
  • When constructing probability histograms, one often notices that the distribution may closely align with the normal distribution.
  • The occurrence of the normal distribution in practical problems can be loosely classified into three categories: exactly normal distributions, approximately normal distributions, and distributions modeled as normal.
  • This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.
  • This is a sample of size 50 from a normal distribution, plotted as a normal probability plot.

• The Normal Approximation to the Binomial Distribution

  • The process of using the normal curve to estimate the shape of the binomial distribution is known as normal approximation.
  • The importance of the normal curve stems primarily from the fact that the distribution of many natural phenomena are at least approximately normally distributed.
  • Laplace showed that even if a distribution is not normally distributed, the means of repeated samples from the distribution would be very nearly normal, and that the the larger the sample size, the closer the distribution would be to a normal distribution.
  • Because the distribution of means is very close to normal, these tests work well even if the distribution itself is only roughly normal.
  • The smooth curve is the normal distribution.