All statistics and graphs for Tolerance Intervals (Normal Distribution)

Find definitions and interpretation guidance for every statistic and graph that is provided with tolerance intervals.

Confidence level

Because data samples are random, it is unlikely that two samples from the same population will yield identical tolerance intervals. But, if you collect many samples, a certain percentage of the resulting tolerance intervals will contain the minimum proportion of the population that you specify.

The confidence level is the likelihood that the tolerance interval actually includes the minimum percentage. For example, an engineer wants to know the range within which 99% of the future product will fall, with 98% confidence. 98% is the confidence level for the tolerance interval.
Method Confidence level 98% Percent of population in interval 99%
Note

You can specify the confidence level for the analysis in the Options dialog box. Minitab displays the target confidence level in the Methods table. By default, the confidence level is 95%. For the nonparametric method, Minitab calculates the achieved confidence level. The achieved confidence level is the exact confidence level that Minitab calculates. The achieved confidence level is usually greater than or equal to the target confidence level, unless your sample size is too small.

Percent of population in the interval

The percent of population in the interval is the minimum percentage of the population that you want the tolerance interval to include. For example, an engineer wants to know the range that will include 99% of the future product, with 98% confidence. 99% is the percent of population in the tolerance interval.
Method Confidence level 98% Percent of population in interval 99%

N

The sample size (N) is the total number of observations in the sample. In this data, the sample size is 400.
Statistics Variable N Mean StDev C1 400 0.604 3.671

Mean

The mean summarizes the sample values with a single value that represents the center of the data. The mean is the average of the data, which is the sum of all the observations divided by the number of observations.

In this data, the mean is 0.604.
Statistics Variable N Mean StDev C1 400 0.604 3.671

StDev

The standard deviation is the most common measure of dispersion, or how spread out the data are from the mean.

A larger standard deviation indicates that your data are spread more widely around the mean and will result in a wider tolerance interval. A smaller standard deviation indicates that your data are distributed more closely around the mean and will result in a narrower tolerance interval.

In this data, the standard deviation is 3.671.
Statistics Variable N Mean StDev C1 400 0.604 3.671

Normal method

Tolerance intervals are a range of values for a specific quality characteristic of a product that likely covers a specified percentage of future product output. Use the normal method tolerance interval if you can safely assume that your sample comes from a normally distributed population.

If your data follow a normal distribution, then the normal method is more precise and economical than the nonparametric method. The normal method allows you to achieve smaller margins of error with fewer observations.

The normal method is not robust to severe departures from normality. If you are unsure of the parent distribution, or you know that the parent distribution is not normal, then use the nonparametric method.

Interpretation

Two-sided
Use a two-sided interval to determine the interval that contains a certain percentage of the population measurements.
In this example, using the normal method, you can be 98% confident that at least 99% of all fill measurements are between –9.604 and 10.813 of the target value.
Method Confidence level 98% Percent of population in interval 99%
98% Tolerance Interval Nonparametric Achieved Variable Normal Method Method Confidence C1 (-9.604, 10.813) (-9.300, 10.700) 91.0% Achieved confidence level applies only to nonparametric method.
Upper bound
Use an upper bound to determine the interval that indicates that a certain percentage of population measurements will not be greater than an upper limit.
In this example, the normal upper bound is 9.043, so you can be 95% confident that 95% of the product will have measurements of 9.043 or less.
Method Confidence level 95% Percent of population in interval 95%
95% Upper Tolerance Bound Normal Nonparametric Achieved Variable Method Method Confidence C1 9.043 12.000 95.1% Achieved confidence level applies only to nonparametric method.
Lower bound
Use a lower bound to determine the interval that indicates that a certain percentage of population measurements will not be less than a lower limit.
In this example, the normal lower bound is 1085.947, so you can be 95% confident that at least 95% of the product will have measurements of 1085.947 or greater.
Method Confidence level 95% Percent of population in interval 95%
95% Lower Tolerance Bound Normal Nonparametric Achieved Variable Method Method Confidence Hours 1085.947 1070.700 96.3% Achieved confidence level applies only to nonparametric method.

Nonparametric method

Tolerance intervals are a range of values for a specific quality characteristic of a product that likely covers a specified proportion of future product output. If you cannot safely assume that your sample comes from a normally distribution population, you must use the nonparametric method tolerance interval.

The nonparametric method requires only that the data are continuous. However, the nonparametric method requires large sample sizes for the results to be accurate. If your sample size is not large enough, the nonparametric interval is a non-informative interval that ranges from negative infinity to infinity. In this case, Minitab displays a finite interval based on the range of your data. As a result, the achieved confidence level is much lower than the target confidence level.

Interpretation

Two-sided
Use a two-sided interval to determine the interval that contains a certain percentage of the population measurements.
In this example, using the nonparametric method, you can be 91.0% confident that at least 99% of all measurements are between –9.300 and 10.700 of the target value. The achieved confidence is smaller than the target value of 98%.
Method Confidence level 98% Percent of population in interval 99%
98% Tolerance Interval Nonparametric Achieved Variable Normal Method Method Confidence C1 (-9.604, 10.813) (-9.300, 10.700) 91.0% Achieved confidence level applies only to nonparametric method.
Upper bound
Use an upper bound to determine the interval that indicates that a certain percentage of population measurements will not be greater than an upper limit.
In this example, the nonparametric upper bound is 12.000, so you can be 95.1% confident that 95% of the product will measurements of 12.000 inches or less. The achieved confidence is approximately the same as the target value of 95%.
Method Confidence level 95% Percent of population in interval 95%
95% Upper Tolerance Bound Normal Nonparametric Achieved Variable Method Method Confidence C1 9.043 12.000 95.1% Achieved confidence level applies only to nonparametric method.
Lower bound
Use a lower bound to determine the interval that indicates that a certain percentage of population measurements will not be less than a lower limit.
In this example, the nonparametric lower bound of 1070.700, so you can be 96.3% confident that at least 95% of all measurements will be 1070.700 or greater. The achieved confidence is slightly greater than the target value of 95%.
Method Confidence level 95% Percent of population in interval 95%
95% Lower Tolerance Bound Normal Nonparametric Achieved Variable Method Method Confidence Hours 1085.947 1070.700 96.3% Achieved confidence level applies only to nonparametric method.

Achieved confidence

For the nonparametric method, Minitab calculates the achieved confidence level. This is the exact confidence level obtained from your sample. It will generally be greater than or equal to the target confidence level, unless your sample size is too small.

Interpretation

Two-sided
Use a two-sided interval to determine the interval that contains a certain percentage of the population measurements.
In this example, the achieved confidence is 91.0%, which is smaller than the target value of 98%.
98% Tolerance Interval Nonparametric Achieved Variable Normal Method Method Confidence C1 (-9.604, 10.813) (-9.300, 10.700) 91.0% Achieved confidence level applies only to nonparametric method.
Upper bound
Use an upper bound to determine the interval that indicates that a certain percentage of population measurements will not be greater than an upper limit.
In this example, the achieved confidence is 95.1%, which is close to the target value of 95%.
95% Upper Tolerance Bound Normal Nonparametric Achieved Variable Method Method Confidence C1 9.043 12.000 95.1% Achieved confidence level applies only to nonparametric method.
Lower bound
Use a lower bound to determine the interval that indicates that a certain percentage of population measurements will not be less than a lower limit.
In this example, the achieved confidence is 96.3%, which is greater than the target value of 95%.
95% Lower Tolerance Bound Normal Nonparametric Achieved Variable Method Method Confidence Hours 1085.947 1070.700 96.3% Achieved confidence level applies only to nonparametric method.

Tolerance interval plot

Tolerance interval plots contain the following:
  • Histogram: Shows the distribution of your sample data. Each bar on the histogram represents the frequency of data within an interval.
  • Interval plots: Show the mean and the upper and/or lower bounds of the tolerance interval for each method. A vertical line at the end of the interval represents a bound, and an arrow indicates that no bound for that side of the interval exists.
  • Normal probability plot: Shows how well your data fit the normal distribution. If your data are normally distributed, then the data points on the probability plot form a straight line.
  • Statistics table: Displays the sample size, mean, and standard deviation.
  • Normal table: Displays the upper and/or lower bounds of the normal method tolerance interval.
  • Nonparametric table: Displays the upper and/or lower bounds of the nonparametric method tolerance interval, and the achieved confidence level.
  • Normality Test table: Shows the p-value and the Anderson-Darling normality test value. To determine whether you can assume that the data follow a normal distribution, compare the p-value from the normality test to the significance level (α). If the p-value is less than or equal to the significance level, you can conclude that your data do not follow a normal distribution. In this case, you must use the nonparametric method tolerance interval.

Interpretation

The normal probability plot shows that the plotted points form an approximate straight line, which indicates that the data follow a normal distribution. Also, the p-value for the normality test is 0.340, which is greater than the significance level (α = 0.05). Thus, there is not enough evidence to conclude that the data do not follow a normal distribution. The engineer can use the normal method results.

The normal lower bound is 1085.947, so the engineer is 95% confident that at least 95% of all the light bulbs exceed approximately 1086 hours of burn time. For all the light bulbs, the mean burn time is approximately 1248 hours and the standard deviation is approximately 84.1.

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