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## The Standard Normal Distribution & Applications

This guide is designed to introduce students to the fundamentals of statistics with special emphasis on the major topics covered in their STA 2023 class including methods for analyzing sets of data, probability, probability distributions and more.

## The Standard Normal Distribution

### Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution . It is the distribution that occurs when a normal random variable has a mean of zero and a standard deviation of one.

The normal random variable of a standard normal distribution is called a standard score or a  score. Every normal random variable  can be transformed into a  score via the following equation:

= ( - μ) / σ

where  is a normal random variable, μ is the mean, and σ is the standard deviation.

## Normal Distribution

Normal Distribution What is the Normal Distribution?

The normal distribution refers to a family of continuous probability distributions described by the normal equation.

## The Normal Equation

The normal distribution is defined by the following equation:

Normal equation. The value of the random variable  is:

Y = { 1/[ σ * sqrt(2π) ] } * e-(x - μ)2/2σ2

where  is a normal random variable, μ is the mean, σ is the standard deviation, π is approximately 3.14159, and  is approximately 2.71828.

The random variable  in the normal equation is called the normal random variable. The normal equation is the probability density function for the normal distribution.

## The Normal Curve

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. When the standard deviation is large, the curve is short and wide; when the standard deviation is small, the curve is tall and narrow. All normal distributions look like a symmetric, bell-shaped curve, as shown below.

The curve on the left is shorter and wider than the curve on the right, because the curve on the left has a bigger standard deviation.

## Probability and the Normal Curve

The normal distribution is a continuous probability distribution. This has several implications for probability.

The total area under the normal curve is equal to 1.

The probability that a normal random variable  equals any particular value is 0.

The probability that  is greater than  equals the area under the normal curve bounded by and plus infinity (as indicated by the non-shaded area in the figure below).

The probability that  is less than  equals the area under the normal curve bounded by  and minus infinity (as indicated by the shaded area in the figure below).

Additionally, every normal curve (regardless of its mean or standard deviation) conforms to the following "rule".

About 68% of the area under the curve falls within 1 standard deviation of the mean.

About 95% of the area under the curve falls within 2 standard deviations of the mean.

About 99.7% of the area under the curve falls within 3 standard deviations of the mean.

Collectively, these points are known as the empirical rule or the 68-95-99.7 rule. 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. In the next lesson, we demonstrate the use of normal distribution tables.

## Normal Distribution Calculator

The normal calculator solves common statistical problems, based on the normal distribution. The calculator computes cumulative probabilities, based on three simple inputs. Simple instructions guide you to an accurate solution, quickly and easily. If anything is unclear, frequently-asked questions and sample problems provide straightforward explanations. The calculator is free. It can be found under the Stat Tables tab, which appears in the header of every Stat Trek web page.

Normal Calculator

Example 1

An average light bulb manufactured by the Acme Corporation lasts 300 days with a standard deviation of 50 days. Assuming that bulb life is normally distributed, what is the probability that an Acme light bulb will last at most 365 days?

Given a mean score of 300 days and a standard deviation of 50 days, we want to find the cumulative probability that bulb life is less than or equal to 365 days. Thus, we know the following:

The value of the normal random variable is 365 days.

The mean is equal to 300 days.

The standard deviation is equal to 50 days.

We enter these values into the Normal Distribution Calculator and compute the cumulative probability. The answer is: P( X < 365) = 0.90. Hence, there is a 90% chance that a light bulb will burn out within 365 days.

Example 2

Suppose scores on an IQ test are normally distributed. If the test has a mean of 100 and a standard deviation of 10, what is the probability that a person who takes the test will score between 90 and 110?

Here, we want to know the probability that the test score falls between 90 and 110. The "trick" to solving this problem is to realize the following:

P( 90 <  < 110 ) = P( X < 110 ) - P( X < 90 )

स्रोत : guides.fscj.edu

## The Standard Normal Distribution

In the standard normal distribution, the mean is 0 and the standard deviation is 1. A normal distribution can be standardized using z-scores.

## The Standard Normal Distribution | Examples, Explanations, Uses

Published on November 5, 2020 by Pritha Bhandari. Revised on January 16, 2023.

The standard normal distribution, also called the z-distribution, is a special normal distribution where the mean is 0 and the standard deviation is 1.

Any normal distribution can be standardized by converting its values into z scores. Z scores tell you how many standard deviations from the mean each value lies. Converting a normal distribution into a z-distribution allows you to calculate the probability of certain values occurring and to compare different data sets.

## Standard normal distribution calculator

You can calculate the standard normal distribution with our calculator below.

## Normal distribution vs the standard normal distribution

All normal distributions, like the standard normal distribution, are unimodal and symmetrically distributed with a bell-shaped curve. However, a normal distribution can take on any value as its mean and standard deviation. In the standard normal distribution, the mean and standard deviation are always fixed.

Every normal distribution is a version of the standard normal distribution that’s been stretched or squeezed and moved horizontally right or left.

The mean determines where the curve is centered. Increasing the mean moves the curve right, while decreasing it moves the curve left.

The standard deviation stretches or squeezes the curve. A small standard deviation results in a narrow curve, while a large standard deviation leads to a wide curve. A (M = 0, SD = 1)

Standard normal distribution

B (M = 0, SD = 0.5)

Squeezed, because SD < 1

C (M = 0, SD = 2)

Stretched, because SD > 1

D (M = 1, SD = 1)

Shifted right, because M > 0

E (M = –1, SD = 1)

Shifted left, because M < 0

## Standardizing a normal distribution

When you standardize a normal distribution, the mean becomes 0 and the standard deviation becomes 1. This allows you to easily calculate the probability of certain values occurring in your distribution, or to compare data sets with different means and standard deviations.

While data points are referred to as x in a normal distribution, they are called z or z scores in the z distribution. A z score is a standard score that tells you how many standard deviations away from the mean an individual value (x) lies:

A positive z score means that your x value is greater than the mean.

A negative z score means that your x value is less than the mean.

A z score of zero means that your x value is equal to the mean.

Converting a normal distribution into the standard normal distribution allows you to:

Compare scores on different distributions with different means and standard deviations.

Normalize scores for statistical decision-making (e.g., grading on a curve).

Find the probability of observations in a distribution falling above or below a given value.

Find the probability that a sample mean significantly differs from a known population mean.

### How to calculate a z score

To standardize a value from a normal distribution, convert the individual value into a z-score:

Subtract the mean from your individual value.

Divide the difference by the standard deviation.

x = individual value

μ = mean

σ = standard deviation

Example: Finding a z score

You collect SAT scores from students in a new test preparation course. The data follows a normal distribution with a mean score (M) of 1150 and a standard deviation (SD) of 150. You want to find the probability that SAT scores in your sample exceed 1380.

To standardize your data, you first find the z score for 1380. The z score tells you how many standard deviations away 1380 is from the mean.

Step 1: Subtract the mean from the x value.

x = 1380 M  = 1150

x – M = 1380 − 1150 = 230

Step 2: Divide the difference by the standard deviation.

SD = 150

z = 230 ÷ 150 = 1.53

The z score for a value of 1380 is 1.53. That means 1380 is 1.53 standard deviations from the mean of your distribution.

Next, we can find the probability of this score using a z table.

## Use the standard normal distribution to find probability

The standard normal distribution is a probability distribution, so the area under the curve between two points tells you the probability of variables taking on a range of values. The total area under the curve is 1 or 100%.

Every z score has an associated p value that tells you the probability of all values below or above that z score occuring. This is the area under the curve left or right of that z score. स्रोत : www.scribbr.com

## Standard Normal Distribution

Standard normal distribution: a normal distribution with mean of zero and standard deviation of one. How to use standard normal table. Problems with solutions.

## Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution. It is the distribution that occurs when a normal random variable has a mean of zero and a standard deviation of one.

## Standard Score (aka, z-score)

The normal random variable of a standard normal distribution is called a standard score or a z-score. Every normal random variable can be transformed into a score via the following equation:

= ( - μ) / σ

where is a normal random variable, μ is the mean of , and σ is the standard deviation of .

## Standard Normal Distribution Table

A standard normal distribution table shows a cumulative probability associated with a particular z-score. Table rows show the whole number and tenths place of the z-score. Table columns show the hundredths place. The cumulative probability (often from minus infinity to the z-score) appears in the cell of the table.

For example, a section of the standard normal table is reproduced below. To find the cumulative probability of a z-score equal to -1.31, cross-reference the row of the table containing -1.3 with the column containing 0.01. The table shows that the probability that a standard normal random variable will be less than -1.31 is 0.0951; that is, P(Z < -1.31) = 0.0951.

z 0.00 0.01 0.02 0.03 0.04 0.05

-3.0 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011

... ... ... ... ... ... ...

-1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735

-1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885

-1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056

... ... ... ... ... ... ...

3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989

Of course, you may not be interested in the probability that a standard normal random variable falls between minus infinity and a given value. You may want to know the probability that it lies between a given value and plus infinity. Or you may want to know the probability that a standard normal random variable lies between two given values. These probabilities are easy to compute from a normal distribution table. Here's how.

Find P(Z > a). The probability that a standard normal random variable (z) is greater than a given value (a) is easy to find. The table shows the P(Z < a). The P(Z > a) = 1 - P(Z < a).

Suppose, for example, that we want to know the probability that a z-score will be greater than 3.00. From the table (see above), we find that P(Z < 3.00) = 0.9987. Therefore, P(Z > 3.00) = 1 - P(Z < 3.00) = 1 - 0.9987 = 0.0013.

Find P(a < Z < b). The probability that a standard normal random variables lies between two values is also easy to find. The P(a < Z < b) = P(Z < b) - P(Z < a).

For example, suppose we want to know the probability that a z-score will be greater than -1.40 and less than -1.20. From the table (see above), we find that P(Z < -1.20) = 0.1151; and P(Z < -1.40) = 0.0808. Therefore, P(-1.40 < Z < -1.20) = P(Z < -1.20) - P(Z < -1.40) = 0.1151 - 0.0808 = 0.0343.

In school or on the Advanced Placement Statistics Exam, you may be called upon to use or interpret standard normal distribution tables. Standard normal tables are commonly found in appendices of most statistics texts.

## The Normal Distribution as a Model for Real-World Events

Often, phenomena in the real world follow a normal (or near-normal) distribution. This allows researchers to use the normal distribution as a model for assessing probabilities associated with real-world phenomena. Typically, the analysis involves two steps.

Transform raw data. Usually, the raw data are not in the form of z-scores. They need to be transformed into z-scores, using the transformation equation presented earlier: = ( - μ) / σ.

Find probability. Once the data have been transformed into z-scores, you can use standard normal distribution tables, online calculators (e.g., Stat Trek's free normal distribution calculator), or handheld graphing calculators to find probabilities associated with the z-scores.

The problem in the next section demonstrates the use of the normal distribution as a tool to model real-world events.

Problem 1

Molly earned a score of 940 on a national achievement test. The mean test score was 850 with a standard deviation of 100. What proportion of students had a higher score than Molly? (Assume that test scores are normally distributed.)

(A) 0.10 (B) 0.18 (C) 0.50 (D) 0.82 (E) 0.90

Solution

The correct answer is B. As part of the solution to this problem, we assume that test scores are normally distributed. In this way, we use the normal distribution to model the distribution of test scores in the real world. Given an assumption of normality, the solution involves three steps.

First, we transform Molly's test score into a z-score, using the z-score transformation equation.

= ( - μ) / σ = (940 - 850) / 100 = 0.90

Then, using an online calculator (e.g., Stat Trek's free normal distribution calculator), a handheld graphing calculator, or the standard normal distribution table, we find the cumulative probability associated with the z-score. In this case, we find P(Z < 0.90) = 0.8159.

Therefore, the P(Z > 0.90) = 1 - P(Z < 0.90) = 1 - 0.8159 = 0.1841.

Thus, we estimate that 18.41 percent of the students tested had a higher score than Molly.

स्रोत : stattrek.com

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Mohammed 10 day ago

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