Maths and Stats Flashcards
What is Statistical Process Control?
Monitoring of a process to identify process control (capability and performance)
Can be control charts or continuous monitoring.
What is the standard for sampling by attributes
ISO 2859
How is ISO 2859 applied?
1) Decide your acceptance criteria
2) Choose inspection plan (normal level II)
3) Sample as per table
4) Inspect the sample
5) Make a decision
6) Apply switching rules for next time?
What are the switching rules?
- Switching to Tightened Inspection:
If 2 out of 5 consecutive batches are rejected during normal inspection, switch to tightened inspection - Switching to Normal Inspection:
If 5 consecutive batches are accepted during tightened inspection, switch back to normal inspection. - Switching to Reduced Inspection:
If the process is under control and 10 consecutive batches are accepted during normal inspection, and the total number of nonconforming items does not exceed a specified limit, switch to reduced inspection. - Discontinuation of Inspection:
If 5 consecutive batches are rejected during tightened inspection, discontinue inspection until the issue is resolved
What is the standard for sampling by variables?
ISO 3951
What is the difference between sampling by attributes and sampling by variables?
Attributes = inspecting for something that is either yes / no
Variables = inspecting for something continuous (eg diameter). Got a mean and sd.
How is ISO 3951 applied?
The same as ISO 2859 only the mean and SD are the acceptance criteria from the standard. Also have switching rules.
What are AQLs
How are AQL’s different to LQLs?
AQL is the producer risk – the risk that a good batch is rejected
LQL is consumer risk – the risk that bad batch has been classified as good.
LQL is statistically representative of the batch
What is the difference between common cause and special cause variation
Common cause variation = natural noise of process
Special cause variation = variation in process from something that’s not normally there.
What is CUSUM and where is it useful?
CUSUM plots difference from target value. If there’s a change in direction then something odd has happened (special cause variation)
What stats are useful?
Shewhart, cusum, mean, variance, TOST, T-Test, histogram, sd,
Where is maths knowledge useful to a QP?
- Statistical Process Control (SPC): QPs use statistical methods to monitor and control manufacturing processes. Understanding SPC helps in identifying trends, variations, and potential issues in the production process.
- Quality Assurance (QA): Mathematics is essential in QA for designing and interpreting sampling plans, calculating Acceptable Quality Limits (AQLs), and ensuring that products meet specified quality standards.
- Data Analysis: QPs often analyze data from various sources, such as stability studies, validation processes, and routine quality checks. Mathematical skills are crucial for interpreting this data accurately and making informed decisions.
- Risk Assessment: Mathematics helps in quantifying risks and determining the probability of different outcomes. This is vital for assessing the potential impact of deviations and making decisions about product release.
- Validation and Calibration: Mathematical knowledge is used in the validation of processes, equipment, and analytical methods. It ensures that these elements perform consistently and accurately.
- Problem-Solving: Strong mathematical skills enhance a QP’s ability to solve complex problems, optimize processes, and improve overall efficiency.
What are the discrete distributions (i.e. those used for discrete data?)
Binomial and Poisson
What’s the difference between binomial and poisson distributions?
You would use a binomial distribution instead of a Poisson distribution when:
* The number of trials is fixed and known in advance.
* The probability of success is constant across all trials.
* The range of values is finite.
For example, if you wanted to model the number of heads obtained from flipping a coin 10 times, you would use a binomial distribution because the number of trials is fixed at 10 and the probability of success (getting heads) is constant at 0.5. On the other hand, if you wanted to model the number of cars passing through a toll booth in an hour, you would use a Poisson distribution because the number of trials (i.e., the number of hours) is not fixed and the probability of success (i.e., the number of cars passing through) is infinite.
What does a binomial distribution need
Independent chance of event happening with only two outcomes (coin toss)
Defined number of trials
What does a poisson distribution need?
A fixed interval (number per hour / number per area)
Independent, infrequent events
A constant mean rate
For example, if you wanted to model the number of cars passing through a toll booth in an hour, you would use a Poisson distribution because the interval of time is fixed at one hour, the events (i.e., cars passing through) occur independently of each other, the mean rate of events is constant, and the events are rare or infinitesimal.
What are examples of continuous distributions? i.e. measurement of variables
Normal (symmetrical bell curve – infinite range)
Log normal (where the log forms a normal bell curve)
Uniform / boxcar – stepped as data broken into ranges
What’s a histogram do?
Plot frequency of a measurement
Highlights repeating values
Can allow an estimate of probabilities
What are boxplots and what can they do?
Allows Comparison of sets of data
Have 50% of sample data in the box
Whiskers are a specified amount of values.
What’s the Central Limit Theorem
That the averages taken of a set of repeat samples on a distribution are normal.
Can make confidence intervals and do t-tests to ask if there’s a difference between the means of two samples or ANOVA to see whether there’s a difference between the means of three or more sample sets.
What do trend and scatter plots show
Trend = show changes in data bot not data distribution
Scatter = shows relationship between two sets of measurements linked by a common property
What are measurements of location?
Mean = estimates location of population
Median = the middle value of an ordered set
Geometric mean = average of the log of the population values (calculate the average of the log and convert it back to the normal scale)
What are measurements of variability?
Standard deviation – the spread of data of a population
Relative standard deviation = sd as a % of sample average (useful for comparing data sets
Variance = sd2 – useful in analysis of variances
Range – difference between highest and lowest value
Standard error – used in significance testing. An estimation of standard deviations of sample averages.
What is a t-test used for?
A t-test is a statistical test used to compare the means of two groups.
What does a one-sample t-test do
Compares the mean of a single group to a known value
What does a two-sample t-test do
Compares the means of two independent groups
What does a paired t-test do
Compares means from the same group at different times (e.g., before and after treatment).
What is two one-sided t-test (TOST)
The Two One-Sided t-Tests (TOST) procedure is used to test for equivalence rather than difference.
How does TOST work?
To test equivalence, to t-tests are performed:
One test checks if the mean of group A is significantly less than the mean of group B
The other test checks if the mean of group B is significantly less than the mean of group B.
If both tests pass then the groups are considered equivalent.
The null hypothesis is that the difference in the means is outside the equivalence margin
The alternative hypothesis is that the difference in the means is within the equivalence margin
What is ANOVA used for
To test if the means between 3 or more groups are equivalent
What are the Shewhart chart rules for identifying special cause variation?
Rule 1 – One point beyond the 3 σ control limit
Rule 2 – Eight or more points on one side of the centerline without crossing
Rule 3 – Four out of five points in zone B or beyond
Rule 4 – Six points or more in a row steadily increasing or decreasing
Rule 5 – Two out of three points in zone A
Rule 6 – 14 points in a row alternating up and down
Rule 7 – Any noticeable/predictable pattern, cycle, or trend
What is a p-value and what is good
p-value tells you if the result could have happened by chance
<0.05 = unlikely to be by chance
what does an f-test do?
An F-test is a statistical test used to compare the variances of two samples or the ratio of variances between multiple samples.
It’s commonly used in the context of ANOVA (Analysis of Variance) and regression analysis
What stats in PQR
Shewhart for Cpk and Ppk
Means and confidence intervals
What stats in validation
Process capability
Regression Analysis
Histograms and pareto analysis
Where is an f-test used
- Regression Analysis: In multiple regression analysis, an F-test can be used to determine if the overall regression model is significant. This means checking if the relationship between the dependent variable and the set of independent variables is statistically significant1.
- ANOVA (Analysis of Variance): When comparing the means of three or more groups, an F-test is used in ANOVA to determine if there are any statistically significant differences between the group means
3 Quality Control in Manufacturing: A factory might use an F-test to compare the variances in the dimensions of products produced by two different machines. This helps in identifying if one machine produces more consistent results than the other
What is a Type 1 error wrt hypothesis testing
The null hypothesis is true but is rejected i.e. a false positive
- the tester states there’s a statistically significant difference even though there isn’t one
What is a Type 2 error wrt hypothesis testing
the failure to reject a null hypothesis that is actually false - a false negative
