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Operational Excellence through Statistical Thinking

Discover Statoscopex, our online statistical analysis software. It is aimed at Six Sigma and statistical analysis practitioners who want to evaluate and fully exploit the decision-making potential of a data set.
Statoscopex helps Six Sigma and statistical practitioners identify influential factors, quantify their effects and assess their practical relevance, enabling better-informed improvement decisions.
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Master Black Belt

Advanced Features for Your Statistical Studies

Statoscopex offers more than 50 analyses och 100 statistical calculations through its capability study and factor analysis functionalities, including:

For the following metrics: continuous, counting, binary.

  1. Determining the statistical accuracy of the defect rate of a process
  2. Determining the minimum sample size required to estimate the defect rate of a process

For the following parameters: mean, standard deviation, rate of occurrence, proportion.

  1. Determining the confidence interval of the parameter
  2. Determining the minimum sample size required to estimate the statistical parameter with a certain accuracy

For the following parameters: mean, standard deviation, rate of occurrence, proportion.

  1. Translation of the objective of reducing the defect rate in terms of the target effect on the statistical parameter.
  2. Determination of the minimum sample size required to detect the effect on the statistical parameter that achieves the objective of reducing the defect rate.

For the following parameters: mean, standard deviation, rate of occurrence, proportion.

  1. Detect the existence of an effect below or above a threshold value
  2. Determine the minimum sample size required to detect an effect at least equal to a certain size
  3. Determine the smallest detectable effect using two samples (resolution)
  4. Determine the detection power of an effect below or above a threshold value, using samples.

Techniques used: two-sample t-test, two-variance test, two-sample Poisson test, two-proportion test

For the following parameters: mean, standard deviation, rate of occurrence, proportion.

  1. Detect if a parameter differs from its reference value
  2. Determine the minimum sample size required to detect if a parameter differs from its reference value
  3. Determine the smallest detectable difference between the value of a parameter and its reference value, using a sample (resolution)
  4. Determine the detection power of a parameter’s deviation from its reference value, using a sample

Techniques used: one-sample t-test, variance test, one-sample Poisson test, proportion test.

Calculators

Every Statoscopex analysis runs right here. Open the one you need — each panel holds its own form, and results appear in place.

Field of application: a continuous quality metric (Y) following a normal distribution. For example: duration, temperature, size, etc.

Here we determine the expected (probabilistic) defect rate from what is observed in a sample. The calculation also gives the margin of error (statistical precision) on that expected defect rate.

Enter the following data:

  • n: sample size
  • Xbar: the sample mean
  • s: the sample standard deviation
  • LSL: the lower specification limit
  • USL: the upper specification limit

Field of application: a continuous quality metric (Y) following a normal distribution. For example: duration, temperature, size, etc.

Here we determine the minimum sample size to take in order to know the expected defect rate within an acceptable margin of error (statistical precision).

Enter the following data:

  • Relative margin of error: the acceptable % error on the defect rate
  • Assumed performance: an order-of-magnitude estimate of operational performance, expressed as a defect rate or as Sigma

Field of application: a count-based quality metric (Y) following a Poisson distribution. For example: defects per PC, holds per telephone call, errors per document, etc.

Here we determine the expected (probabilistic) defect rate from what is observed in a sample. The calculation also gives the margin of error (statistical precision) on that expected defect rate (normal approximation method). Enter the following data:

  • n: sample size
  • λ̂: the occurrence rate in the sample; typically defects per unit (DPU).

Field of application: a count-based quality metric (Y) following a Poisson distribution. For example: defects per PC, holds per telephone call, errors per document, etc.

Here we determine the minimum sample size to take in order to know the expected defect rate within an acceptable margin of error (statistical precision) (normal approximation method).

Enter the following data:

  • Relative margin of error: the acceptable % error on the defect rate
  • Assumed performance: an order-of-magnitude estimate of operational performance, expressed as a defect rate or as Sigma

Field of application: a binary quality metric (Y) following a binomial distribution. For example: abandoned calls, incorrect invoices, defective parts, etc.

Here we determine the expected (probabilistic) defect rate from what is observed in a sample. The calculation also gives the margin of error (statistical precision) on that expected defect rate (normal approximation method). Enter the following data:

  • n: sample size
  • p: the proportion of defective units in the sample

If no defective unit was observed in the sample, enter a value of p equal to 1/(n+2).

Field of application: a binary quality metric (Y) following a binomial distribution. For example: abandoned calls, incorrect invoices, defective parts, etc.

Here we determine the minimum sample size to take in order to know the expected defect rate within an acceptable margin of error (statistical precision) (normal approximation method).

Enter the following data:

  • Relative margin of error: the acceptable % error on the defect rate
  • Assumed performance: an order-of-magnitude estimate of operational performance, expressed as a defect rate or as Sigma

Field of application: a continuous metric (Y) following a normal distribution. For example: duration, temperature, size, etc.

Here we determine the 1-α confidence interval of the mean µ from what is observed in a sample.

Enter the following data:

  • α: the risk that the true value of the mean falls outside the confidence interval. For a 95% confidence interval, enter α = 5%
  • n: sample size
  • Xbar: the sample mean
  • σ or s: the true standard deviation (if known) or the sample standard deviation

Field of application: a continuous metric (Y) following a normal distribution. For example: duration, temperature, size, etc.

Here we determine the minimum sample size to take in order to know the mean µ within a chosen margin of error (statistical precision).

Enter the following data:

  • σ or s: the true standard deviation (if known) or the sample standard deviation
  • Margin of error: the desired half-width of the 1-α confidence interval of the mean
  • α: the risk that the true value of the mean falls outside the confidence interval. It therefore sets the 1-α confidence level of the margin of error. For a 95% confidence interval, enter α = 5%

Field of application: a continuous metric (Y) following a normal distribution. For example: duration, temperature, size, etc.

Here we determine the shift in the mean required, at constant standard deviation, to meet the target reduction in the defect rate.

Enter the following data:

Summary calculation (if you already know the initial defect rate):

  • Di: the initial defect rate
  • Dc: the target defect rate
  • σ: the standard deviation of Y

Detailed calculation:

  • LSL: the lower specification limit
  • USL: the upper specification limit
  • µi: the initial (current) mean
  • σ: the standard deviation of Y
  • τred. (%): the desired reduction in the defect rate

Field of application: a continuous metric (Y) following a normal distribution. For example: duration, temperature, size, etc.

Here we determine the minimum sample size needed to detect the shift in the mean that would meet, at constant standard deviation, the target reduction in the defect rate.

Enter the following data:

  • α: the risk of wrongly concluding that the required effect exists
  • β: the risk of not detecting the required effect
  • Di: the initial defect rate
  • Dc: the target defect rate

Field of application: a continuous metric (Y) following a normal distribution. For example: duration, temperature, size, etc.

Here we determine whether a shift in the mean above or below a given threshold can be concluded, from the differences between the means observed in two samples. This typically tests the effect of two different operating conditions (the effect of an X, of a solution, and so on). The test used here is the two-sample t-test.

Enter the following data:

  • n1 and n2: the size of each sample
  • Xbar1 and Xbar2: the mean of Y in each sample
  • s1 and s2: the standard deviation of Y in each sample
  • α: the risk that the true effect falls outside the calculated confidence interval; enter 5% for a 95% confidence interval
  • Emin: the threshold of the effect sought

Field of application: a continuous metric (Y) following a normal distribution. For example: duration, temperature, size, etc.

Here we determine, for a two-sample t-test:

  • The minimum sample size required to detect a shift in the mean above a given threshold
  • The smallest detectable shift in the mean (resolution) for given samples
  • The power to detect a shift in the mean above or below a given threshold, for given samples

Enter the following data:

1/ For the sample size:

  • α: the risk of wrongly detecting an effect
  • β: the risk of not detecting the effect
  • σ: standard deviation (known or estimated)
  • Emin to detect: the minimum (useful) effect you want to detect

2/ For the minimum detectable effect:

  • α: the risk of wrongly detecting an effect
  • β: the risk of not detecting the effect
  • n1 and n2: the size of each sample
  • σ1 and σ2: the standard deviation of each sample, if not otherwise known

3/ For the power:

  • n1 and n2: the size of each sample
  • σ1 and σ2: the standard deviation of each sample, if not otherwise known
  • Emin to detect: the minimum (useful) effect you want to detect

Field of application: a continuous metric (Y) following a normal distribution. For example: duration, temperature, size, etc.

Here we determine whether a mean differs from its hypothesised value, from the mean observed in a sample. The test used here is the

test t à un échantillon.

Enter the following data:

  • µ0: the hypothesised value of the mean
  • n: the sample size
  • Xbar: the mean of Y calculated in the sample
  • s or σ: the standard deviation of Y calculated in the sample, or the true standard deviation if known
  • α: the risk that the true mean falls outside the calculated confidence interval; enter 5% for a 95% confidence interval

Domaine d’application : métrique (Y) continue suivant une loi normale. Par exemples : durée, température, taille, etc.

Nous déterminons ici, dans le cadre de la mise en œuvre d’un test t à un échantillon :

  • The minimum sample size required to detect whether the mean is above or below the hypothesised mean
  • The smallest detectable deviation between the mean and the hypothesised mean, for given samples
  • The power to detect a deviation between the mean and the hypothesised mean, for given samples

Introduire les données suivantes :

1/ Pour la détermination de la taille d’échantillon :

  • α: the risk of wrongly detecting a deviation
  • β: the risk of not detecting the deviation
  • σ: standard deviation (known or estimated)
  • εmin to detect: the minimum (useful) deviation you want to detect

2/ Pour la détermination de l’écart minimal détectable :

  • α: the risk of wrongly detecting a deviation
  • β: the risk of not detecting the deviation
  • n: the sample size
  • σ: the sample standard deviation, if not otherwise known

3/ Pour la détermination de la puissance :

  • n: the sample size
  • σ: the sample standard deviation, if not otherwise known
  • εmin to detect: the minimum (useful) deviation you want to detect

Field of application: a continuous metric (Y) following a normal distribution. For example: duration, temperature, size, etc.

Here we determine the 1-α confidence interval of the standard deviation σ from what is observed in a sample.

Enter the following data:

  • α: the risk that the true value of the standard deviation falls outside the confidence interval. For a 95% confidence interval, enter α = 5%
  • s: the standard deviation of Y in the sample
  • n: sample size

Field of application: a continuous metric (Y) following a normal distribution. For example: duration, temperature, size, etc.

Here we determine the minimum sample size to take in order to know the standard deviation σ within a chosen margin of error (statistical precision).

Enter the following data:

  • s: the standard deviation of Y in the sample
  • Margin of error: the desired half-width of the 1-α confidence interval of the mean
  • α: the risk that the true value of the standard deviation falls outside the confidence interval. It therefore sets the 1-α confidence level of the margin of error. For a 95% confidence interval, enter α = 5%

Field of application: a continuous metric (Y) following a normal distribution. For example: duration, temperature, size, etc.

Here we determine the reduction in the standard deviation required, at constant mean, to meet the target reduction in the defect rate.

Enter the following data:

Summary calculation (if you already know the initial defect rate):

  • Di: the initial defect rate
  • Dc: the target defect rate

Detailed calculation:

  • SL: the specification limit on Y (lower or upper)
  • LSL: the lower specification limit of Y
  • USL: the upper specification limit of Y
  • µ: the mean
  • σi: the initial (current) standard deviation
  • τred. (%): the desired reduction in the defect rate

Field of application: a continuous metric (Y) following a normal distribution. For example: duration, temperature, size, etc.

Here we determine whether a decrease or increase in the standard deviation above or below a given threshold can be concluded, from the differences between the standard deviations observed in two samples. This typically tests the effect of two different operating conditions (the effect of an X, of a solution, and so on). The test used here is the

test de deux variances (test F)

.

Enter the following data:

  • n1 and n2: the size of each sample
  • s1 and s2: the standard deviation of Y in each sample
  • α: the risk that the true effect falls outside the calculated confidence interval; enter 5% for a 95% confidence interval
  • Emin: the threshold of the effect sought

Field of application: a continuous metric (Y) following a normal distribution. For example: duration, temperature, size, etc.

Here we determine, for a two-variance test (F-test):

  • The minimum sample size required to detect a decrease or increase in the standard deviation above or below a given threshold
  • The smallest detectable reduction in the standard deviation (resolution), for given samples

The calculations assume a power of 79.16% and α = 5%.

Enter the following data:

1/ For the sample size: E

min

to detect: the minimum (useful) effect you want to detect

2/ For the minimum detectable effect: n1 and n2: the size of each sample

Field of application: a continuous metric (Y) following a normal distribution. For example: duration, temperature, size, etc.

Here we determine whether a standard deviation differs from its hypothesised value, from the standard deviation observed in a sample. The test used here is the one-variance test (F-test).

Enter the following data:

  • σ0: the hypothesised value of the standard deviation
  • n: the sample size
  • Xbar: the mean of Y calculated in the sample
  • s: the standard deviation of Y calculated in the sample
  • α: the risk that the true standard deviation falls outside the calculated confidence interval; enter 5% for a 95% confidence interval

Field of application: a count-based quality metric (Y) following a Poisson distribution. For example: defects per PC, holds per telephone call, errors per document, etc.

Here we determine the reduction in the occurrence rate required to meet the target reduction in the defect rate.

Enter the following data:

Summary calculation (if you already know the initial defect rate):

  • Di: the initial defect rate
  • Dc: the target defect rate

Detailed calculation:

  • λi: the initial (current) occurrence rate
  • τred. (%): the desired reduction in the defect rate

Field of application: a count-based quality metric (Y) following a Poisson distribution. For example: defects per PC, holds per telephone call, errors per document, etc.

Here we determine the minimum sample size needed to detect the reduction in the occurrence rate that would meet the target reduction in the defect rate.

Enter the following data:

  • α: the risk of wrongly concluding that the required effect exists
  • β: the risk of not detecting the required effect
  • Di: the initial defect rate
  • Dc: the target defect rate

Field of application: a count-based quality metric (Y) following a Poisson distribution. For example: defects per PC, holds per telephone call, errors per document, etc.

Here we determine whether a decrease or increase in the occurrence rate above or below a given threshold can be concluded, from the differences between the occurrence rates observed in two samples. This typically tests the effect of two different operating conditions (the effect of an X, of a solution, and so on). The test used here is the two-sample Poisson test.

Enter the following data:

  • n1 and n2: the size of each sample
  • d1 and d2: the number of defects counted in each sample
  • α: the risk that the true effect falls outside the calculated confidence interval; enter 5% for a 95% confidence interval
  • Emin: the threshold of the effect sought, λ2 – λ1

Domaine d’application: métrique de qualité (Y) de comptage suivant une loi de Poisson. Par exemples : nombre de défauts par PC, nombre de mises en attente par appel téléphonique, nombre d’erreurs par document, etc.

Nous déterminons ici, dans le cadre de la mise en œuvre d’un test de Poisson à deux échantillons :

  • The minimum sample size required to detect a decrease or increase in the occurrence rate above a given threshold
  • The smallest detectable decrease or increase in the occurrence rate, for given samples
  • The power to detect a decrease or increase in the occurrence rate above or below a given threshold, for given samples

Introduire les données suivantes :

1/ Pour la détermination de la taille d’échantillon :

  • α: the risk of wrongly detecting an effect
  • β: the risk of not detecting the effect
  • λ1: the reference occurrence rate
  • λ2: the comparison occurrence rate corresponding to the effect threshold to detect, Emin = λ2 – λ1

2/ Pour la détermination de l’effet minimal détectable :

  • α: the risk of wrongly detecting an effect
  • β: the risk of not detecting the effect
  • n1 and n2: the size of each sample
  • λ1: the reference occurrence rate

3/ Pour la détermination de la puissance :

  • n1 and n2: the size of each sample
  • λ1: the reference occurrence rate
  • λ2: the comparison occurrence rate corresponding to the effect threshold to detect, Emin = λ2 – λ1

Domaine d’application : métrique (Y) de comptage suivant une loi de Poisson. Par exemples : nombre de défauts par PC, nombre de mises en attente par appel téléphonique, nombre d’erreurs par document, etc.

Nous déterminons ici si un taux d’occurrences diffère de sa valeur hypothétisée, à partir du taux d’occurrences observé dans un échantillon. Le test ici mis en oeuvre est le test de Poisson à un échantillon.

Introduire les données suivantes :

  • λ0: the hypothesised value of the occurrence rate
  • n: the sample size
  • d: the number of defects counted in the sample
  • α: the risk that the true mean falls outside the calculated confidence interval; enter 5% for a 95% confidence interval

Domaine d’application : métrique (Y) de comptage suivant une loi de Poisson. Par exemples : nombre de défauts par PC, nombre de mises en attente par appel téléphonique, nombre d’erreurs par document, etc.

Nous déterminons ici, dans le cadre de la mise en œuvre d’un test de Poisson à un échantillon :

  • The minimum sample size required to detect whether the occurrence rate is above or below its hypothesised value
  • The smallest detectable deviation between the occurrence rate and its hypothesised value, for given samples
  • The power to detect a deviation between the occurrence rate and its hypothesised value, for given samples

Introduire les données suivantes :

1/ Pour la détermination de la taille d’échantillon :

  • α: the risk of wrongly detecting a deviation
  • β: the risk of not detecting the deviation
  • λ0: the hypothesised occurrence rate
  • λ: the comparison occurrence rate such that εmin = λ – λ0

2/ Pour la détermination de l’écart minimal détectable :

  • α: the risk of wrongly detecting a deviation
  • β: the risk of not detecting the deviation
  • n: the sample size
  • λ0: the hypothesised occurrence rate

3/ Pour la détermination de la puissance :

  • n: the sample size
  • λ0: the hypothesised occurrence rate
  • λ: the comparison occurrence rate such that εmin = λ – λ0

Domaine d’application: métrique (Y) binaire suivant une loi binomiale. Par exemples : appels abandonnés, factures erronées, pièces défectueuses, etc.

Nous déterminons ici l’objectif de réduction de la proportion requis pour réaliser l’objectif de réduction du taux de défectueux.

Introduire les données suivantes :

  • πi: the initial defect rate
  • τred.: the desired reduction in the defect rate

Field of application: a binary metric (Y) following a binomial distribution. For example: abandoned calls, incorrect invoices, defective parts, etc.

Here we determine the minimum sample size needed to detect the reduction in the proportion that would meet the target reduction in the defect rate.

Enter the following data:

  • α: the risk of wrongly concluding that the required effect exists
  • β: the risk of not detecting the required effect
  • Di: the initial defect rate
  • Dc: the target defect rate

Field of application: a binary metric (Y) following a binomial distribution. For example: abandoned calls, incorrect invoices, defective parts, etc.

Here we determine whether a decrease or increase in the proportion above or below a given threshold can be concluded, from the differences between the proportions observed in two samples. This typically tests the effect of two different operating conditions (the effect of an X, of a solution, and so on). The test used here is the

test de deux proportions

.

Enter the following data:

  • n1 and n2: the size of each sample
  • D1 and D2: the number of defective units counted in each sample
  • α: the risk that the true effect falls outside the calculated confidence interval; enter 5% for a 95% confidence interval
  • Emin: the threshold of the effect sought, π2 – π1

Field of application: a binary metric (Y) following a binomial distribution. For example: abandoned calls, incorrect invoices, defective parts, etc.

Here we determine, for a two-proportion test:

  • The minimum sample size required to detect a decrease or increase in the proportion above a given threshold
  • The smallest detectable decrease or increase in the proportion (resolution), for given samples
  • The power to detect an effect on the proportion above or below a given threshold, for given samples

Introduire les données suivantes :

1/ Pour la détermination de la taille d’échantillon :

  • α: the risk of wrongly detecting an effect
  • β: the risk of not detecting the effect
  • π1: the reference proportion
  • π2: the comparison proportion corresponding to the effect threshold to detect, Emin = π2 – π1

2/ Pour la détermination de l’effet minimal détectable :

  • α: the risk of wrongly detecting an effect
  • β: the risk of not detecting the effect
  • n1 and n2: the size of each sample
  • π1: the reference proportion

3/ Pour la détermination de la puissance :

  • n1 and n2: the size of each sample
  • π1: the reference proportion
  • π2: the comparison proportion corresponding to the effect threshold to detect, Emin = π2 – π1

Domaine d’application: métrique de qualité (Y) binaire suivant une loi binomiale. Par exemples : appels abandonnés, factures erronées, pièces défectueuses, etc.

Nous déterminons ici si une proportion diffère de sa valeur hypothétisée, à partir de la proportion observée dans un échantillon. Le test ici mis en oeuvre est le test d’une proportion.

Introduire les données suivantes :

  • π0: the hypothesised value of the proportion
  • n: the sample size
  • D: the number of defective units counted in the sample
  • α: the risk that the true mean falls outside the calculated confidence interval; enter 5% for a 95% confidence interval

Field of application: a binary quality metric (Y) following a binomial distribution. For example: abandoned calls, incorrect invoices, defective parts, etc.

Here we determine, for a one-proportion test:

  • The minimum sample size required to detect whether the proportion is above or below its hypothesised value
  • The smallest detectable deviation between the proportion and its hypothesised value, for a given sample
  • The power to detect a deviation between the proportion and its hypothesised value, for a given sample

Enter the following data:

1/ For the sample size:

  • α: the risk of wrongly detecting a deviation
  • β: the risk of not detecting the deviation
  • π0: the hypothesised proportion
  • π: the comparison proportion such that εmin = π – π0

2/ For the minimum detectable deviation:

  • α: the risk of wrongly detecting a deviation
  • β: the risk of not detecting the deviation
  • n: the sample size
  • π0: the hypothesised proportion

3/ For the power:

  • n: the sample size
  • π0: the hypothesised proportion
  • π: the comparison proportion such that εmin = π – π0
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