Determination of Probability Using Normal Distribution

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The graph of the normal distribution depends on two factors - the mean and the standard deviation. The mean of the distribution determines the location of the center of the graph, and the standard deviation determines the height and width of the graph. All normal distributions look like a symmetric, bell-shaped curve, as shown below.


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When the standard deviation is small, the curve is tall and narrow; and when the standard deviation is big, the curve is short and wide see above. The normal distribution is a continuous probability distribution. This has several implications for probability. Additionally, every normal curve regardless of its mean or standard deviation conforms to the following "rule". Collectively, these points are known as the empirical rule or the Clearly, given a normal distribution, most outcomes will be within 3 standard deviations of the mean.

To find the probability associated with a normal random variable, use a graphing calculator, an online normal distribution calculator, or a normal distribution table. In the examples below, we illustrate the use of Stat Trek's Normal Distribution Calculator , a free tool available on this site.

The Role of Probability

In the next lesson, we demonstrate the use of normal distribution tables. The z -table shows a z-score of approximately 1. Thus, we can write the following:. The 90 th percentile is This means that 90 percent of the test scores fall at or below To get this answer on the calculator, follow this next step. Find the 70 th percentile—that is, find the score k such that 70 percent of scores are below k and 30 percent of the scores are above k.

An R Introduction to Statistics

Draw a new graph and label it appropriately. The 70 th percentile is This means that 70 percent of the test scores fall at or below The golf scores for a school team were normally distributed with a mean of 68 and a standard deviation of three. A personal computer is used for office work at home, research, communication, personal finances, education, entertainment, social networking, and a myriad of other things. Suppose that the average number of hours a household personal computer is used for entertainment is two hours per day.

Normal Distribution Calculator with step by step explanation

Assume the times for entertainment are normally distributed and the standard deviation for the times is half an hour. Find the probability that a household personal computer is used for entertainment between 1. Find P 1. First, calculate the z -scores for each x -value.

Normal Distribution: Calculating Probabilities/Areas (z-table)

Now, use the Z -table to locate the area under the normal curve to the left of each of these z -scores. The area to the left of the z -score of 1. The area between these scores will be the difference in the two areas, or 0. The probability that a household personal computer is used between 1. Find the maximum number of hours per day that the bottom quartile of households uses a personal computer for entertainment. The maximum number of hours per day that the bottom quartile of households uses a personal computer for entertainment is 1.

Find the probability that a golfer scored between 66 and The Z -table shows the area to the left of a z -score with an absolute value of 1 to be 0. It shows the area to the left of a z -score of 2 to be 0. The difference in the two areas is 0.

How to Find Statistical Probabilities in a Normal Distribution

This is slightly different than the area given by the calculator, due to rounding. Find the 80 th percentile of this distribution, and interpret it in a complete sentence. Use the information in Example 6. Using this information, answer the following questions—round answers to one decimal place:. Two thousand students took an exam.

A citrus farmer who grows mandarin oranges finds that the diameters of mandarin oranges harvested on his farm follow a normal distribution with a mean diameter of 5. Find the probability that a randomly selected mandarin orange from this farm has a diameter larger than 6.

Sketch the graph. So, the middle 20 percent of mandarin oranges have diameters between 5.

Probability

This process is called the probability density function. Academics, financial analysts and fund managers alike may determine a particular stock's probability distribution to evaluate the possible expected returns that the stock may yield in the future. The stock's history of returns, which can be measured from any time interval, will likely be composed of only a fraction of the stock's returns, which will subject the analysis to sampling error.

By increasing the sample size, this error can be dramatically reduced. There are many different classifications of probability distributions. Some of them include the normal distribution, chi square distribution, binomial distribution , and Poisson distribution. The different probability distributions serve different purposes and represent different data generation processes.

Normal distribution and it’s characteristics

The binomial distribution, for example, evaluates the probability of an event occurring several times over a given number of trials and given the event's probability in each trial. Another typical example would be to use a fair coin and figuring the probability of that coin coming up heads in 10 straight flips. A binomial distribution is discrete , as opposed to continuous, since only 1 or 0 is a valid response. The most commonly used distribution is the normal distribution, which is used frequently in finance, investing, science, and engineering.

The normal distribution is fully characterized by its mean and standard deviation, meaning the distribution is not skewed and does exhibit kurtosis. This makes the distribution symmetric and it is depicted as a bell-shaped curve when plotted. A normal distribution is defined by a mean average of zero and a standard deviation of 1. Unlike the binomial distribution, the normal distribution is continuous, meaning that all possible values are represented as opposed to just 0 and 1 with nothing in between.


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Stock returns are often assumed to be normally distributed but in reality, they exhibit kurtosis with large negative and positive returns seeming to occur more than would be predicted by a normal distribution. In fact, because stock prices are bounded by zero but offer a potential unlimited upside, the distribution of stock returns has been described as log-normal. This shows up on a plot of stock returns with the tails of the distribution having greater thickness.