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Distributions Menu

The Distributions menu selection is used to calculate critical values and probabilities for various distributions. The most common distributions are the z (normal) distribution, t distribution, F distribution, and the chi-square distribution. Within the last 20 years, computers have made it easy to calculate exact probabilities for the various statistics. Prior to that, researchers made extensive use of books containing probability tables.

The Distributions menu has four selections:

Normal distribution

The normal distribution is the most well-known distribution and is often referred to as the z distribution or the bell shaped curve. It is used when the sample size is greater than 30. When the sample size is less than 30, the t distribution is used instead of the normal distribution.

The menu offers three choices: 1) probability of a z value, 2) critical z for a given probability, and 3) probability of a defined range.

Probability of a z value

When you have a z (standardized) value for a variable, you can determine the probability of that value. The software is the electronic equivalent of a normal distribution probability table. When you enter a z value, the area under the normal curve will be calculated. The area not under the curve is referred to as the rejection region. It is also called a two-tailed probability because both tails of the distribution are excluded. The Statistics Calculator reports the two-tailed probability for the z value. A one-tailed probability is used when your research question is concerned with only half of the distribution. Its value is exactly half the two-tailed probability.

Example

z-value: 1.96

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Two-tailed probability = .05

Critical z for a given probability

This menu selection is used to determine the critical z value for a given probability.

Example

A large company designed a pre-employment survey to be administered to prospective employees. Baseline data was established by administering the survey to all current employees. They now want to use the instrument to identify job applicants who have very high or very low scores. Management has decided they want to identify people who score in the upper and lower 3% when compared to the norm. How many standard deviations away from the mean is required to define the upper and lower 3% of the scores?

The total area of rejection is 6%. This includes 3% who scored very high and 3% who scored very low. Thus, the two-tailed probability is .06. The z value required to reject 6% of the area under the curve is 1.881. Thus, new applicants who score higher or lower than 1.881 standard deviations away from the mean are the people to be identified.

Two tailed probability: .06

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z-value = 1.881

Probability of a defined range

Knowing the mean and standard deviation of a sample allows you to establish the area under the curve for any given range. This menu selection will calculate the probability that the mean of a new sample would fall between two specified values (i.e., between the limits of a defined range).

Example

A manufacturer may find that the emission level from a device is 25.9 units with a standard deviation of 2.7. The law limits the maximum emission level to 28.0 units. The manufacturer may want to know what percent of the new devices coming off the assembly line will need to be rejected because they exceed the legal limit.

Sample mean = 25.9
Unbiased standard deviation = 2.7
Lower limit of the range = 0
Upper limit of the range = 28.0

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Probability of a value falling within the range = .7817
Probability of a value falling outside the range = .2183

The area under the curve is the sum of the area defined by the lower limit plus the area defined by the upper limit.

The area under the normal curve is the probability that additional samples would fall between the lower and upper limits. In this case, the area above the upper limit is the rejection area (21.83% of the product would be rejected).

T distribution

Mathematicians used to think that all distributions followed the bell shaped curve. In the early 1900's, an Irish chemist named Gosset, discovered that distributions were much flatter than the bell shaped curve when working with small sample sizes. In fact, the smaller the sample, the flatter the distribution. The t distribution is used instead of the normal distribution when the sample size is small. As the sample size approaches thirty, the t distribution approximates the normal distribution. Thus, the t distribution is generally used instead of the z distribution, because it is correct for both large and small sample sizes, where the z distribution is only correct for large samples.

The menu offers three choices: 1) probability of a t value, 2) critical t value for a given probability, and 3) probability of a defined range.

Probability of a t-value

If you have a t value and the degrees of freedom associated with the value, you can use this program to calculate the two-tailed probability of t. It is the equivalent of computerized table of t values.

Example

t-value: 2.228
df: 10

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Two-tailed probability = .050

Critical t value for a given probability

This program is the opposite of the previous program. It is used if you want to know what critical t value is required to achieve a given probability.

Example

Two-tailed probability: .050
Degrees of freedom: 10

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t-value = 2.228

Probability of a defined range

Knowing the mean and standard deviation of a sample allows you to establish the area under the curve for any given range. You can use this program to calculate the probability that the mean of a new sample would fall between two values.

Example

A company did a survey of 20 people who used its product. The mean average age of the sample was 22.4 years and the unbiased standard deviation was 3.1 years. The company now wants to advertise in a magazine that has a primary readership of people who are between 18 and 24, so they need to know what percent of its potential customers are between 18 and 24 years of age?

Sample mean: 22.4
Unbiased standard deviation: 3.1
Sample size = 20
Lower limit of the range = 18
Upper limit of the range = 24

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Probability of a value falling within the range = .608
Probability of a value falling outside the range = .392

Because of the small sample size, the t distribution is used instead of the z distribution. The area under the curve represents the proportion of customers in the population expected to be between 18 and 24 years of age. In this example, we would predict that 60.8% of the its customers would be expected to be between 18 and 24 years of age, and 39.2% would be outside of the range. The company decided not to advertise.

F distribution

The F-ratio is used to compare variances of two or more samples or populations. Since it is a ratio (i.e., a fraction), there are degrees of freedom for the numerator and denominator. This menu selection may be use to calculate the probability of an F -ratio or to determine the critical value of F for a given probability. These menu selections are the computer equivalent of an F table.

Probability of a F-ratio

If you have a F-ratio and the degrees of freedom associated with the numerator and denominator, you can use this program to calculate the probability.

Example

F-ratio: 2.774
Numerator degrees of freedom: 20
Denominator degrees of freedom: 10

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Two-tailed probability = .0500

Critical F for a given probability

If you know the critical alpha level and the degrees of freedom associated with the numerator and denominator, you can use this program to calculate the F-ratio.

Example

Two-tailed probability = .0500
Numerator degrees of freedom: 20
Denominator degrees of freedom: 10

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F-ratio: 2.774

Chi-square distribution

The chi-square statistic is used to compare the observed frequencies in a table to the expected frequencies. This menu selection may be use to calculate the probability of a chi-square statistic or to determine the critical value of chi-square for a given probability. This menu selection is the computer equivalent of an chi-square table.

Probability of a chi-square statistic

If you have a chi-square value and the degrees of freedom associated with the value, you can use this program to calculate the probability of the chi-square statistic. It is the equivalent of computerized table of chi-square values.

Example

Chi-square value: 18.307
Degrees of freedom: 10

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Probability = .050

Critical chi-square for a given probability

If you have the critical alpha level and the degrees of freedom, you can use this program to calculate the probability of the chi-square statistic. It is the equivalent of computerized table of chi-square values.

Example

Probability = .0500
Degrees of freedom: 10

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Chi-square value: 18.307

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