<meta http-equiv="refresh" content="1; url=/nojavascript/">

# Standard Deviation of a Normal Distribution

%
Progress
Practice Standard Deviation of a Normal Distribution
Progress
%
Types of Distributions

distribution is a description of the possible values of a random variable and the possible occurrences of these values.

A probability distribution is a graph or table that specifies the probablity associated with each possible value that the random variable can assume.

• Mean Value or Expected Value

The mean value, or expected value , of a discrete random variable $X$ is given by the following equation:

$\mu=E(x)=\sum_{} xp(x)$

• Variance

The variance of a discrete random variable is given by the following formula:

$\sigma^2 = \sum_{} (x-\mu)^2 P(x)$

• Standard Deviation

The square root of the variance, or, in other words, the square root of $\sigma^2$ , is the standard deviation of a discrete random variable:

$\sigma=\sqrt{\sigma^2}$

A sampling distribution is the probability distribution of a given statistic based on a random sample.

Normal Distribution

A normal distribution is called a bell curve because its shape is comparable to a bell. It has this shape because the majority of the data is concentrated at the middle and slowly decreases symmetrically on either side.  A normal distribution can be described by the mean and standard deviation of the data.

Here is an example of a normal distribution:

[Figure1]

The line in the middle represents the mean, and the tick lines along the bottom represent the standard deviation.

Binomial Distribution

Characteristics of a Binomail Probability Distribution

• The experiment consists of a fixed number of independent trials
• Each trial results in one of two outcomes: success or failure
• The probability of success p is the same for each trial
• The binomial random variable $X$ is defined as the number of successes in n trials

The binomial distribution is found by calculating the binomial probabilities for k = 0, 1, 2,..., n.  It can be represented in a histogram with k along the x-axis and P(k) along the y-axis.

When n * p > 10 and n(1-p) > 10, the following statements are all true:

• the normal distribution will provide a good approximation of the binomial distribution
• Mean for the binomial distribution: μ = np
• Standard deviation for the binomial distribution: $\sigma=$ √(np(1-p))

The binompdf function on the TI-83/84 calculator can be used to solve problems involving the probability of a precise number of successes out of a certain number of trials.  The syntax for the binompdf function is binompdf $(n, p, a)$ , where $n$ is the number of trials, $p$ is the probability of success for any particular trial, and $a$ is the number of successes.

The binomcdf function on the TI-83/84 calculator can be used to solve problems involving the probability of less than or equal to a number of successes out of a certain number of trials.  The syntax for the binomcdf function is binomcdf $(n, p, a)$.

Multinomial Distribution

The multinomial distribution gives the probability of any particular combination of numbers of successes for the various categories.

This probability is given by:

$P & = \frac{n!}{n_1!n_2!n_3!\ldots n_k!} \times \left (p_1{^{n_1}} \times p_2{^{n_2}} \times p_3{^{n_3}} \ldots p_k{^{n_k}} \right )$

where:

$n$ is the number of trials.

$p$ is the probability for each possible outcome.

$k$ is the number of possible outcomes.

Geometric Distribution

Characteristics of a Geometric Probability Distribution

• The experiment consists of a sequence of independent trials
• Each trial results in one of two outcomes: success or failure
• The probability of success p is the same for each trial
• The geometric random variable $X$ is defined as the number of trials until the first success is observed

The geometric distribution is found by calculating the geometric probabilities for k = 0, 1, 2, ..., ∞ .  It can be represented in a histogram with k along the x-axis and P(k) along the y-axis.  Note: As k increases, P(k) approaches 0, so as this happens you can stop your calculations of P(k) for the purpose of creating the distribution.

Mean for the binomial distribution: μ = 1/p

Standard deviation for the binomial distribution: $\sigma=$ √((1-p)/p2)

Poisson Distribution

In a binomial distribution, if the number of trials, $n$ , gets larger and larger as the probability of success, $p$ , gets smaller and smaller, we obtain a Poisson distribution.

Characteristics of a Poisson distribution:

• The experiment consists of counting the number of events that will occur during a specific interval of time or in a specific distance, area, or volume.
• The probability that an event occurs in a given time, distance, area, or volume is the same.
• Each event is independent of all other events.

Poisson Random Variable:

The probability distributionmean, and variance of a Poisson random variable are given as follows:

$p(x) &= \frac{\lambda^x e^{-\lambda}}{x!} \quad x=0, 1, 2, 3, \ldots\\\mu &= \lambda\\\sigma^2 &= \lambda$

where:

$\lambda=$ the mean number of events in the time, distance, volume or area

$e=$ the base of the natural logarithm

Student's T Distribution

When you use $s$ to estimate $\sigma$ , you must use $t$ instead of $z$ to complete the significance test for a mean.

In calculating the $t-$ test statistic, we use the formula:

$t=\frac{\bar{x}-\mu_0}{\frac{s}{\sqrt{n}}}$

where:

$t$ is the test statistic and has $n-1$ degrees of freedom.

$\bar{x}$ is the sample mean

$\mu_0$ is the population mean under the null hypothesis.

$s$ is the sample standard deviation

$n$ is the sample size

$\frac{s}{\sqrt{n}}$ is the estimated standard error

df = n - 1