Interpret all statistics for a histogram with a Weibull distribution

Find definitions and interpretation guidance for every statistic that is provided with a histogram with a fitted Weibull distribution.

N

The sample size (N) is the number of nonmissing observations for a Y variable or a group.

Shape and scale

A Weibull distribution is defined by two parameters: the shape and the scale. When you fit a Weibull distribution, Minitab estimates these parameters from your sample.
Shape
The shape parameter describes how the data are distributed. A shape of 3 approximates a normal curve. Lower shape values result in a right-skewed distribution, higher values result in a left-skewed distribution. For example, the following Weibull distributions all have a scale parameter of 5. The shape parameters for the distributions are 2, 3, and 10.
Shape=2
Shape=3
Shape=10
Scale
The scale parameter describes how spread out the data are. Generally, a larger scale results in a distribution that is more spread out.

Minimum

The minimum is the smallest data value.

In these data, the minimum is 7.

13 17 18 19 12 10 7 9 14

Interpretation

One of the simplest ways to assess the spread of your data is to compare the minimum and maximum.

Maximum

The maximum is the largest data value.

In these data, the maximum is 19.

13 17 18 19 12 10 7 9 14

Interpretation

One of the simplest ways to assess the spread of your data is to compare the minimum and maximum.

Null hypothesis and alternative hypothesis

The null and alternative hypotheses are two mutually exclusive statements about the distribution of the data. The Anderson-Darling test uses sample data to determine whether to reject the null hypothesis.
Null Hypothesis
The null hypothesis states that the data follow a Weibull distribution.
Alternative Hypothesis
The alternative hypothesis states that the data do not follow a Weibull distribution.

AD-value

The Anderson-Darling goodness-of-fit statistic (AD-Value) measures the area between the fitted line (which is based on a Weibull distribution) and the empirical distribution function (which is based on the data points). The Anderson-Darling statistic is a squared distance that is weighted more heavily in the tails of the distribution.

Interpretation

Minitab uses the Anderson-Darling statistic to calculate the p-value. The p-value is a probability that measures the evidence against the null hypothesis. Smaller p-values provide stronger evidence against the null hypothesis. Larger values for the Anderson-Darling statistic indicate that the data do not follow a Weibull distribution.

P-value

The p-value is a probability that measures the evidence against the null hypothesis. Smaller p-values provide stronger evidence against the null hypothesis.

Interpretation

Use the p-value to determine whether the data do not follow a Weibull distribution.

To determine whether the data do not follow a Weibull distribution, compare the p-value to the significance level. Usually, a significance level (denoted as α or alpha) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of concluding that the data do not follow a Weibull distribution when they actually do follow a Weibull distribution.
P-value ≤ α: The data do not follow a Weibull distribution (Reject H0)
If the p-value is less than or equal to the significance level, the decision is to reject the null hypothesis and conclude that your data do not follow a Weibull distribution.
P-value > α : Cannot conclude the data do not follow a Weibull distribution (Fail to reject H0)
If the p-value is larger than the significance level, the decision is to fail to reject the null hypothesis because there is not enough evidence to conclude that your data do not follow a Weibull distribution. However, you cannot conclude that the data do follow a Weibull distribution.
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