The one and two-sample proportion hypothesis tests involve one factor with one and two samples, and likewise, these tests may assume a binomial distribution. If more than two samples exist then use the Chi-Square test.
One Sample Z Proportion Hypothesis Test
The One Sample Proportion Test is used to estimate the proportion of a population. It compares the proportion to a target or reference value and also calculates a range of values that is likely to include the population proportion. This is also called the hypothesis of inequality.
Furthermore, use normal approximation or binomial enumeration based on the sample size. If the sample size is large, then normal approximation will give more accurate results. If the sample size is less then binomial enumeration gives much more accurate results.
Assumptions of the one sample Z Proportion test
Firstly, the data are random values from the population
Secondly, the population follows a binomial distribution
Null hypothesis: population proportion is equal to the hypothesized proportion
Alternative hypothesis: population proportion is not equal to or different from the hypothesized proportion (two-tailed)
Population proportion is greater than the hypothesized proportion (one-tailed)
Population proportion is less than the hypothesized proportion (one-tailed)
Test statistic for one sample Z proportions test
Where
z is the test statistic
p̂is
observed proportion
P0 is the hypothesized probability
n is the sample size
Procedure to execute One Sample Z Proportion Hypothesis Test
Firstly, state the null hypothesis and alternative hypothesis
State alpha, in other words, determines the significance level
Thirdly, compute the test statistic
Fourthly, determine the critical value (from the critical value table)
Then, define the rejection criteria
Finally, interpret the result. In the event that the test statistic falls in the critical region, reject the null hypothesis
Example of One Sample Z Proportion Test
A researcher claims that Republican Party will win in the next Senate elections, especially in Florida State. Statistical data reported that 23% voted for Republican Party in the last election. To test the claim a researcher surveyed 80 people and found 22 said they voted for Republican Party in the last election. Is there enough evidence at α=0.05 to support this claim?
P0=0.23
n=80
p̂=22/80=0.275
Then, define the Null and Alternative hypothesis
Null Hypothesis: p= 0.23
Alternative Hypothesis: p≠0.23
State Alpha
α=0.05
State decision rule
The critical value is ±1.96, hence reject the null hypothesis if the calculated value is less than -1.96 or greater than +1.96
Then, calculate the test statistic
= 0.045 / 0.047 = 0.957
Since the calculated value is between -1.96 and 1.96 and it is not in the critical region, hence failed to reject the null hypothesis.
Six Sigma Black Belt Certification One Sample Proportion ZTest Questions:
Question: Which of the following statement is true, the right-tailed test of a single sample proportion test statistic value is +1.12 and the critical value from the table is +2.89.
(A) Reject the null hypothesis (B) Failed to reject the null hypothesis (C) Accept the null alternative hypothesis (D) None of the above
The preferred non-parametric method for unpaired samples is the Mann-Whitney non parametric hypothesis test or Mann-Whitney test (it is also called as Wilcoxon Rank Sum Test or the Mann-Whitney Wilcoxon Test) and thus the non parametric solution to evaluating two independent datasets comparable to the Student’s T-test.
The Mann-Whitney U test is a non-parametric statistical test used to compare two independent samples. It assesses whether the distribution of values in one sample is stochastically greater or less than the distribution of values in another sample. This test does not specifically compare sample means, but rather it evaluates whether the two samples come from the same population or if one tends to have higher values than the other.
How Decision Rules Differ Between Mann-Whitney U and Parametric t-Tests
It’s important to understand how the Mann-Whitney U test differs in decision-making from parametric tests like the independent or paired t-test. While all of these tests aim to determine if two groups differ, their approaches and interpretations are different:
Parametric t-Tests: These tests compare means and assume that the underlying data follows a normal distribution. A small p-value in a t-test typically leads to the conclusion that there is a statistically significant difference in average values between the two groups.
Mann-Whitney U Test: This test does not compare means. Instead, it ranks the data and tests whether values in one group tend to be higher or lower than those in the other group. It makes no assumption about the shape of the distribution, making it a safer option when data is skewed or ordinal. A significant p-value here suggests that one population tends to yield higher (or lower) values than the other, but not necessarily that their medians or means differ.
So, while both tests help you reject or accept a null hypothesis, the t-test focuses on differences in central tendency (mean), and the Mann-Whitney focuses on differences in rank distributions.
A few researchers also interpret the Mann-Whitney test will help to compare the medians of the two populations. Usually, parametric tests will compare the means (Null hypothesis: μ1=μ2) between the independent groups. Whereas, the null hypotheses and the two-sided hypotheses non-parametric tests can be indicated as follows:
Null Hypothesis (H₀): The two populations come from the same distribution.
Alternative Hypothesis (H₁): The two populations come from different distributions.
Note: Unlike t-tests, Mann-Whitney does not assume a difference in means, only a difference in distribution or tendency.
The Mann-Whitney test is regularly performed as a two-sided test, therefore the investigative hypothesis indicates that the two populations are not equal, instead of specifying the directionality.
Assumptions of the Mann-Whitney:
The sample drawn from the population is random.
Independence within the samples and also mutual independence is assumed. That means that an observation is in one group or in the other (it cannot be in both).
The Mann-Whitney U test can be used in many industries, but it is most frequently applied in healthcare, nursing, business, and related fields. In medicine, it is a more efficient method to know the effect of two medicines and whether they are equal or not. It is also used to know whether or not a particular medicine cures an ailment or not.
Procedure to conduct Mann-Whitney test
Combine the two data samples into a single dataset and sort them in ascending order.
Assign ranks to all observations.
Separate the ranks back into their original groups.
Calculate the U statistic using the sum of ranks for each group.
Find the critical U value from the Mann-Whitney table based on your sample sizes and significance level.
Compare the calculated U value to the critical value and draw a conclusion.
Calculation of Mann-Whitney Test
Case 1: For comparing two small sets of observations (samples sizes less than 20)
U1 = n1n2+0.5n1(n1+1)-R1
U2 = n1n2+0.5n2(n2+1)-R2
Where
U1+U2=n1n2
n1 is the observations from the first population
n2 is the observations from the second population
R1 is the sum of observation ranks for the first population
R2 is the sum of observation ranks for the second population
Finally, calculate the U statistic as the smaller U1 and U2. U= min (U1, U2)
Case2: Normal approximation and tie correction
For sample sizes above ~20, the distribution of U rapidly approaches the normal distribution
U mean = μU =0.5 n1n2
Example of Mann-Whitney Test
A researcher, while conducting studies on the Biomass of various trees, wished to determine if there was a difference in the biomass of male and female Juniper trees. So, he randomly selected 6 individuals of each gender from the field. He dries them to constant moisture, chips them, and then weighs them to the nearest kg.
H0: There is no difference between the biomass of male and female Juniper trees– Biomassmale = Biomassfemale (medians are equal)
H1: There is a difference between the biomass of male and female Juniper trees– Biomassmale ≠ Biomassfemale (medians are not equal)
The data are ratings (ordinal data), and hence a non-parametric test is appropriate – the Mann-Whitney U test (the non-parametric counterpart of an independent measures t-test).
Mann-Whitney Table: For Two-Tailed Test at a 5% Significance Level
Ucritical = 5
Calculated U is a value less than the critical value of U for a 0.05 significance level. Ucalculated < Ucritical . Hence, we can reject the null hypothesis.
Therefore, we reject the null hypothesis and conclude that there is a statistically significant difference in the biomass distributions between male and female Juniper trees.
Six Sigma Black Belt Certification Mann-Whitney Test Questions:
Question: Which of the following test is calculated by ranking all the participant’s scores from lowest to highest and adding up the ranks separately for each of the groups?
Six Sigma terminology, sometimes politely called “terminology” but more accurately described as jargon, can overwhelm new learners. If you are studying for a Green Belt or Black Belt, the vocabulary alone can feel like learning a new language.
One reader captured this challenge perfectly in a survey response:
“I’m studying for Green Belt and the jargon is mind blowing. I think to make this more accessible and understandable for fresh students who are not aware of LSS, the narrative needs to be diluted and the jargon made simpler. The classic example is something like the explanation of a standard distribution as the mechanism for gaining the benchmark is not very clear.” – Karle
Karle is absolutely right.
Many Six Sigma books and instructors assume you already speak the language. Terms like sigma level, process capability, control limits, and standard deviation are often introduced quickly, sometimes with little connection to real-world applications.
The result is predictable: students memorize vocabulary without understanding how it works.
This article fixes that.
Instead of repeating textbook definitions, we will explain the most common Six Sigma terms in clear language, connect them to real business problems, and show how they fit inside the broader improvement framework. If you want the more formal definitions alongside these, ASQ’s quality glossary is a solid cross-reference.
Why Six Sigma Jargon Exists
Before learning the terminology, it helps to understand why the jargon exists in the first place.
Six Sigma developed at Motorola in the 1980s as a data-driven methodology for reducing defects and improving processes. The approach draws heavily from statistics, quality engineering, and operations management.
The IASSC Lean Six Sigma Green Belt Study Guide is a free, quick-reference list of essential material to prepare for and pass the certification exam. Master the IASSC Six Sigma Greenbelt Body of Knowledge with this Study Guide.
When you’re ready, there are a few ways I can help:
First, join 12,000+ other Six Sigma professionals by subscribing to my email newsletter. A short read every Monday to start your work week off correctly. Always free.
* * *
If you’re looking to pass your Six Sigma Green Belt or Black Belt exams, I’d recommend starting with my affordable study guide:
1)→ 🟢Pass Your Six Sigma Green Belt: a study guide organized by the exam Body of Knowledge, 1,000+ practice questions with detailed answer explanations, and 8 full-length mock exams
2)→ ⚫Pass Your Six Sigma Black Belt: the full Black Belt Body of Knowledge, 2,000+ practice questions with detailed answer explanations, and 8 full-length mock exams
You’ve spent so much effort learning Lean Six Sigma. Why leave passing your certification exam up to chance? This comprehensive study guide offers 1,000+ exam-like questions for Green Belts (2,000+ for Black Belts) with full answer walkthroughs, access to instructors, detailed study material, and more.
Control charts are among the hardest things to understand for those studying Six Sigma. Here’s an easy Control Charts Study Guide for you.
While studying for the Six Sigma Black Belt Exam, I noticed many questions on the control charts. Besides that, I noticed that there were a lot of different types of control charts. You had to use one for certain kinds of data or the number of units in the sample. Then you had to use other control charts in different instances.
Preparing for a Six Sigma exam? Get the I-MR Chart Worked Exercise kit.
A step-by-step Excel workbook (live formulas, Phase I baseline + Phase II monitoring + a blank template) and a printable PDF walkthrough covering the moving-range method, the Rule 1 and Rule 4 signals, and the exam angle. Enter your email and I'll send it now.
You'll also get my weekly Six Sigma newsletter. Unsubscribe anytime.
Beyond that, several of the data sets go with different distribution types. You take this for granted when a question is specifically about data. However, I found it was easy to get tripped up on the exam when it came up in control chart questions.
Very confusing, right????
So, in the spirit of continuous improvement, we made the following control charts study guide.
Control Charts Study Guide Overview
Control Chart: A control chart is one of the primary techniques of statistical process control (SPC). It plots quality characteristics measured or computed from a sample against the sample number or time. Walter Shewhart invented the control chart at Bell Labs in the 1920s.
Statistical Process Control: Statistical Process Control (SPC) is a statistical method to measure, monitor, and control a process. In other words, SPC is a quality control method that employs statistical methods to measure, monitor, and control a process.
Common Cause: A cause of variation in the process is due to chance but not assignable to any factor. It is the variation that is inherent in the process. A process under the influence of a common cause will always be stable and predictable.
Assignable Cause (“Special Cause”): You can identify and eliminate the variation in a process that is not due to chance. A process under the influence of a special cause will not be stable and predictable.
Rational Sub-Grouping: Rational sub-grouping is the process of organizing the data into groups produced under the same conditions. Rational subgroups help in the estimation process of short-term variations. Thus, rational subgrouping is the basis for operating control charts in a successful manner. These variations later help us predict the long-term variations and their control limits, depending on the type of causes for the variation (special or common).
Control Charts by Data Type Study Guide
The control chart is a graph used to study how a process changes over time. A control chart always has a central line for the average. It also has an upper line for the upper control limit and a lower line for the lower control limit. The control limits are ±3σ from the centerline.
Selecting an appropriate control chart is very important in control chart mapping. Otherwise, we end up with inaccurate control limits for the data. Not all control charts are the same. Different Data Types require different charts.
Control Charts for Continuous Data
Measure the output on a continuous scale. It is possible to measure the quality characteristics of a product. Once you know you are charting continuous data, determine if your population is normal or not and the sample size (n) you are charting.
No matter what you decide, I think you could start off just making a basic Run chart and seeing where that brings you. It is just a basic graph that displays data values in a time order. It can be useful for identifying trends or shifts in the process. A run chart also allows you to test for randomness in the process.
The Data is Normal
Subgroup size n = 1, then use an X-MR or I-MR chart: Use an Individual moving range (I-MR ) chart when data is continuous and not collected in subgroups. In other words, collect a single observation at a time. An I-MR chart provides process variation over time in a graphical method.
If 1 < n < 10, then use an X bar-R chart. It monitors the process performance of continuous data collected in subgroups at set time periods. Since n is small, use the range to estimate the process variation. It consists of two plots to monitor the process mean and the process variation over time.
Else, If n >= 10, then use anX bar-S chart. It is often used to examine the process mean and standard deviation over time. Use the X bar-S chart when the subgroups have a large sample size. The S chart also gives a better understanding of the spread of subgroup data than the range.
Special Note: If your data is Not Normal
Note that there is some lively conversation in the industry on this. For your application purposes, I suggest you read the following and make your own decisions (if you’re just studying for certification, you can probably skip):
Control Charts for Discrete / Attribute Data Study Guide
Attribute control charts work with attribute data. In other words, the data that counts the number of defective items or the number of defects per unit. For example, the number of tubes failed on a shop floor. Unlike variable charts, attribute charts plot only one chart.
Four types of control charts exist for attribute data. P charts plot the proportion of defective items, and np charts are for the number of defectives. U charts are for the average number of defects per unit, and C charts are for the number of defects.
When there are Multiple Defects (i.e., chances of being defective): It computes control limits based on the Poisson distribution.
C chart-control chart for defects. It generally monitors the number of defects in constant-size units.
U chart-control chart for defects per unit. It generally monitors the count type of data where the sample size is greater than one. Plots the average number of defects per sample unit.
When there are NOT Multiple Defects:
This population sorts defects into two piles (it’s Binomial). Samples are either good or bad, positive or negative, right or wrong.
np chart-control chart for defectives. It generally monitors the number of non-conforming or defective items in the measurement process.
p chart-control chart for proportions. It generally analyzes the proportions of non-conforming or defective items in a process.
Significance of Central Limit Theorem
The Central Limit Theorem is important for inferential statistics because it allows us to safely assume that the sampling distribution of the mean will be normal in most cases. This means that we can take advantage of statistical techniques that assume a normal distribution.
The Central Limit Theorem is one of the most profound and useful statistical and probability results. Large samples (more than 30) from any distribution of the sample means will follow a normal distribution. The central limit theorem is vital in statistics for two main reasons: the normality assumption and the precision of the estimates.
The spread of the sample means is less (narrower) than the spread of the population you’re sampling from. So, it does not matter how the original population is skewed.
The means of the sampling distribution of the mean is equal to the population mean µx̅ =µX
The standard deviation of the sample means equals the standard deviation of the population divided by the square root of the sample size: σ(x̅) = σ(x) / √(n)
Central Limit Theorem allows using confidence intervals, hypothesis testing, DOE, regression analysis, and other analytical techniques. Many statistics have approximately normal distributions for large sample sizes, even when we are sampling from a non-normal distribution.
Control charts and the central limit theorem
The analysis in the above graph uses subgroup sizes of 2, 8, 16, and 32. We can see the impact of the subgrouping. In figure 2 (n=8), the histogram is not as wide and looks more “normally” distributed than Figure 1. Figure 3 shows the histogram for subgroup averages when n = 16. It is even more narrow, and it looks more normally distributed. Figures 1 through 4 show that as n increases, the distribution becomes narrower and more bell-shaped, just as the central limit theorem says.
This means that we can often use well-developed statistical inference procedures and probability calculations based on a normal distribution. That holds even if we are sampling from a population that is not normal, provided we have a large sample size.
Control Chart Analysis Study Guide
The main purpose of a control chart is to improve the process. It provides a “voice of the process” that helps teams to identify special causes of variation in the process. By removing the special cause from the process, the process becomes more stable and consistent.
Out-of-control: A process is “out-of-control,” meaning special causes exist in either the average chart, range chart, or both. Teams must identify and eliminate special causes to achieve a stable process. We can say the process is out-of-control if any point in the control chart is out of control limits or has abnormal patterns of variability.
Control limits are three standard deviations above and below the mean. If the process is in control, 99.73% of the averages will fall within these limits. The same is true for the range control limits because there are two components to every control chart: the average and the range.
Four possible conditions may occur in any process.
Average stable, Variation changing (Example)
Five Common Rules for Control Chart Interpretation
Practical Notes on Using Control Charts and Control Limits to Make Process Improvements
Practical Uses and Questions on Control Charts. Photo by runran
Here are some Practical Notes on Using Control Charts and Control Limits to Make Process Improvements. I recently got into a great discussion on LinkedIn. Before replying to a post, I wanted to clarify my thoughts. Well, 1300 words later and a lot of digging around on forums, this is what I discovered. Hope this helps bring clarity on practical uses of Control Charts – it was a fun Sunday morning exercise!
Why should you use Control Charts to Map your Process?
Charting the process will give you the tools to verify the performance: see if a process is under statistical control. It also helps you monitor the effects of improvement efforts.
Why do you want your Process to be under Statistical Control?
Once you reach an acceptable level of stability, you can feel confident that the process will perform consistently. It is meaningless to say that the process IS 4.5 sigma if its behavior IS NOT consistent over time or is unstable.
Also, once a Process is under statistical control, you can begin to apply some of the other tools to improve performance. Trying to improve performance before a process is under control can actually make things worse. See the Quincunx demonstration for a great visual on this.
Why you would want Control Limits less than 3 Sigma?
Control charts aim to reflect process stability, or an equivalent behavior, over time. Given a normal distribution, 99.73% of your distribution would lay inside six standard deviations or sigmas. On a control chart, half would be above and half below. Thus, positive three sigmas for the UCL – Upper Control Limit and negative three sigmas for the LCL – Lower Control Limit.
The name of the game here is to use control limits to decide if something is broken in your process. You also use them to determine if it is a special event or a common event, because that dictates your next actions. (See below).
A point outside of that Six Sigma band (above the UCL or below the LCL) means that the process is out of control. The reasoning is that a process behaving as usual is very unlikely to deliver such a point. So you could reasonably suspect that the process is no longer under control.
Too Few Sigmas
If you use control limits at too few sigmas, you are reducing the percent likelihood that a point will land in the band. Thus, you will detect any small shift in the process behavior. However, you will also have too many points outside the limits just by chance when there is no special cause to address, giving you many false alarms.
Too Many Sigmas
If you use control limits at too many sigmas, you are increasing the chances of a point landing in the bands. Virtually the whole population will fall within the limits, so there will be no false alarms. But you’ll miss most shifts in the process. That leads to missed opportunities for improvement – in which case, why are you charting the process in the first place?
The Perfect Balance
The key here is cost/risk management. We are trying to find the economic trade-off between charting the process and investigating vs. the cost of missing an instance of variation. That missed variation would cause financial issues: the cost of poor quality. Practitioners found (and widely accept) that control limits at three sigmas are a good balance.
Control limits come from historical data and process variation. As your process capability improves, you must recalculate the control limits.
When Should I Recalculate My Control Limits?
The short answer is once the control charts cease to have meaningful business applications. This could be either that you have improved or altered the original practice so much that you have, in effect, a new and different process than what you started with.
What does this look like in practice? After your process has reached a point of statistical control, you should have a good data history. Then you can begin using trend analysis.
Basic trend analysis:
Lack of plotted points near the control limits
Points avoiding the outer zones and various other out-of-control conditions.
As a result of the reduced variation in your process, the new sigma is less than the original sigma. That is, the spread of the new +/- 3 sigma limits will be less than the previous limits.
Can Control Charts tell me the Sigma Level of my Process?
Control charts do NOT measure the sigma level of an overall process. They measure whether a process is in statistical control, i.e., does the process generally follow a normal distribution?
“3 sigma control limits” refers to stability-or equality of behavior over time. “Operating at n sigma” refers to performance.
Since +/- 3 sigma encapsulates 99.73% of the data in a normal distribution, if your process falls within that limit, you have a process that is in statistical control.
The sigma calculation is used to measure the performance or effectiveness of a process at the end of the project. Today, many industries aim for much stricter quality standards, again driven by business.
Performance is how well your process meets the specifications. Note that “specification” is not used in control charts because it has nothing to do with stability. A process operating at six sigma means that the process average is six sigmas away from the closest specification limit. That is twice as far out as the three-sigma control limits (UCL and LCL) on an individuals chart. Thus, virtually no output is out of specification.
Another way to look at this is that the process is what it is regardless of whether you chart it or not, what your control limits are, or what your graph looks like. Changing the control limits does not change your process performance and hence does not change the “sigmas” at which your process operates.
The control chart and sigma calculations are separate measurements, so always treat them that way.
What do you do if a Point falls outside the Control Limits? Ex Above the Upper Control Limit or Below the Lower Control Limit?
This depends on the history of your process.
If you are just getting started and there are points outside the limits, you must get your process under control first. Lower the center and spread of your process values by eliminating or reducing variation.
If a process was under statistical control and then you find a single point outside of the control limits, there is a special cause of variation. First, be sure this is a special, one-time-only variation. Then try to set controls in place to eliminate or prevent it from coming back. Using a prioritization tool like FMEA helps. Note that you may change your standardized process to account for the special cause variation. In that case, you’ll need to re-calculate your control limits, because you now have a new process in place.
If you find multiple points outside the control limits, there is probably a common cause variation. This is when you should apply six sigma methodologies like DMAIC to reduce the variation in your process.
When you’re ready, there are a few ways I can help:
First, join 12,000+ other Six Sigma professionals by subscribing to my email newsletter. A short read every Monday to start your work week off correctly. Always free.
* * *
If you’re looking to pass your Six Sigma Green Belt or Black Belt exams, I’d recommend starting with my affordable study guide:
1)→ 🟢Pass Your Six Sigma Green Belt: a study guide organized by the exam Body of Knowledge, 1,000+ practice questions with detailed answer explanations, and 8 full-length mock exams
2)→ ⚫Pass Your Six Sigma Black Belt: the full Black Belt Body of Knowledge, 2,000+ practice questions with detailed answer explanations, and 8 full-length mock exams
You’ve spent so much effort learning Lean Six Sigma. Why leave passing your certification exam up to chance? This comprehensive study guide offers 1,000+ exam-like questions for Green Belts (2,000+ for Black Belts) with full answer walkthroughs, access to instructors, detailed study material, and more.
