# Sample Size Matters: Design and Experiments

Previously, I introduced the idea that samples do not look exactly like the populations that they are drawn from, and had a closer look at what impact sample size has on our ability to estimate population statistics like mean, proportion or Cpk from samples. Here, I will have a closer look at how this uncertainty impacts our engineering process. In the next post, I will tie in the engineering impacts and decisions to the business value and costs.

### Difference to detect

When we are testing, we’re either testing to determine that the new product or process performs better than the old or, for cost reduction projects, that the cheaper product or process is at least as good as the existing one.

This means that we need to detect a difference between old and new values, such as a difference in the mean weight between the new and old parts. The larger the sample size, $n$ the smaller the difference, $\Delta$ that we can detect. The error, $\epsilon$, in our estimate of the differences gets smaller as sample size increases:

$\epsilon_{\Delta} \propto \frac{\sigma}{\sqrt{n}}$

Given the uncertainties in our estimate of $\mu$ and $\sigma$, illustrated above, it should be clear now that with small sample sizes we can only detect large differences of many multiples of the sample standard deviation, $S$.

### Mean

When trying to determine if a new product or process is better than an old one, we are usually interested in shifting the mean. We want a product to be lighter, provide more power, or a process to work faster. In such cases, we need to estimate the difference of the means, $\Delta = \mu_2 - \mu_1$ and ensure that it is different than 0 (or some other pre-determined value). The minimum difference that we can reliable detect is plotted below for different sample sizes.

### Standard deviation

In many Six Sigma projects, and any time we want to shift the mean closer to a specification limit, we need to compare the new population standard deviation with the old. The simplest way of making this comparison is by taking the ratio $F = \sigma_{2}^{2} / \sigma_{1}^{2}$, where $\sigma_{2}^{2}$ is the larger of the two variances. The dependence on sample size is illustrated below.

You can see from the inset plot, which includes sample sizes of 2 and 3, that small sample sizes really hurt comparisons of variance, and that interesting differences in variance can’t be detected until we have more than 10 samples.

### Proportions

Proportions, such as fraction of defective parts between a new and old design, can be compared by looking at the difference between the two proportions, $\Delta = \left| p_1 - p_0 \right|$.

You can see from this that proportions data provides much less information than variable data; we need much larger sample sizes to achieve usefully small $\Delta$.

### Summary and look forward

When designing experiments, the goal is to detect some difference between two populations. The uncertainty in our measurements and the variation in the parts has a big impact on how many parts we need to test, or greatly limits what we can learn from an experiment.

Next time, I’ll show how these calculations of sample size and uncertainty impact the busines.

# Sample Size Matters: Uncertainty in Measurement

In my previous post, I gave a brief introduction to populations and samples, and stated that sample size impacts our ability to know what a population really looks like. In this post, I want to show this relationship in more detail. In future posts, I will look at how sample size considerations impact our engineering process and what impacts this has on the business.

### Mean and sample size

The error in our estimate of the mean, $E$, is proportional to the standard deviation of the sample, $S$, and the sample size, $n$.

$E \propto \frac{S}{\sqrt{n}}$

We can visualize this easily enough by plotting the 95% confidence interval. When we sample and calculate the sample mean ($\overline{X}$), the true population mean, $\mu$, (what we really want to know) is likely to be anywhere in the shaded region of the graph below.

This graph shows the 95% confidence region for the true population mean, $\mu$; there’s a 95% chance that the true population mean is within this band. The “0” line on the y axis is our estimate of the mean, $\overline{X}$. We can’t know what the true population mean is, but it’s clear that if we use more samples, we can be sure that our estimate is closer to the true mean.

### Standard deviation and sample size

Likewise, when we calculate the sample standard deviation, $S$, the true standard deviation, $\sigma$ has a 95% chance of being within the confidence band below. For small sample sizes (roughly less than 10), the measured standard deviation can be off from the true standard deviation by several times. Even for ten samples, the potential error is nearly $\pm 1$ standard deviation.

### Proportion and sample size

For proportions, the situation is similar: there is a 95% chance that the true sample proportion, $p$, is within the shaded band based on the measured sample proportion $\hat{p}$. Since this confidence interval depends on $\hat{p}$ and cannot be standardized the way $\mu$ and $\sigma$ can be, confidence intervals for two different proportions are plotted.

For small $n$, proportions data tells us very little.

### Process capability and production costs

The cost of poor quality in product or process design can be characterized by the Cpk:

$Cpk = \mathrm{minimum} \begin{cases}\frac{USL - \mu}{3\sigma} \\\frac{\mu - LSL}{3\sigma}\end{cases}$

Where USL is the upper specification limit (also called the upper tolerance) and LSL is the lower specification limit (or lower tolerance).

We can estimate the defect rate (defects per opportunity, or DPO) from the Cpk:

$DPO = 1 - \Pr\left(X < 3 \times Cpk - 1.5\right)$

That probability function is calculated in R with pnorm(3 * Cpk - 1.5) and in Excel with NORMSDIST(3 * Cpk - 1.5). The 1.5 is a typical value used to account for uncorrected or undetected process drift.

Since we don’t know $\mu$ and $\sigma$, we have to substitute $\overline{X}$ and $S$. The uncertainty in these estimates of the population $\mu$ and $\sigma$ mean that we have uncertainty in what the true process Cpk (or defect rates) will be once we’re in production. When our sample testing tells us that the Cpk should be 1.67 (the blue line), the true process Cpk will actually turn out to be somewhere in the shaded band:

Below the blue line, our product or process is failing to meet customer expectations, and will result in lost customers or higher warranty costs. Above the blue line, we’ve added more cost to the production of the product than we need to, reducing our gross profit margin. Since that gray band doesn’t completely disappear, even at 100 samples, we can never eliminate these risks; we have to find a way to manage them effectively.

The impact of this may be more evident when we convert from Cpk to defect rates (ppm):

### Summary and a look forward

With a fair sampling process, samples will look similar to—and statistically indistinguishable from—the population that they were drawn from. How much they look like the population depends critically on how many samples are tested. The uncertainties, or errors in our estimates, resulting from sample size decisions have impacts all through our design analysis and production planning.

In the next post, I will explore in more detail how these uncertainties impact our experiment designs.

# Sample Size Matters

I find that Six Sigma and Design for Six Sigma courses are often eye-opening experiences for participants. There is an experience of discovering that there are tools available to answer problems that have vexed them, and learning that good engineering and science decisions can lead directly to good business outcomes through logical steps.

One of the most remarkable such moments is when students realize the importance of sample size. In the best cases, there is a forehead-slapping moment where the student realizes that much of the testing they’ve done in the past has probably been a complete waste of time; that while they thought they were seeing interesting differences and making good decisions, they were in fact only fooling themselves by comparing too-small data sets.

I want to show in the next few blog posts why sample size matters, both from a technical perspective and from a business perspective.

### Design example

Throughout the next few posts, I’ll use the example of a manufactured product which the customer requires weigh at least 100 kg, sells for about $140 and that costs$120 to manufacture and convert to a sale (the cost of goods sold, or COGS, is $120). Amount Sales 140 COGS 120 Material 60 Labor and Overhead 60 Gross Profit 20 We want to develop a new version of the product, using a modified design and a new process that, by design, will reduce the cost of material by 10%. The old cost of material was 50% of COGS, or$60. To achieve the material cost reduction of 10%, we have to remove $6 in material costs, improving gross profit to$26.

We believe that the current design masses 120 kg, so we estimate that our new part mass should be $120 - 0.1 \times 120 = 108$ kg.

Current Design New Design Target
Part Weight 120 108

Seems like we might be done at this point, and I’ve seen plenty of engineering projects that stop here. Unfortunately, this isn’t the whole story. Manufacturing will be unable to produce parts of exactly 108 kg, so they’ll need a tolerance range to check parts against. We have that customer requirement for at least 100 kg, so any variation has to stay above that. We also want to save money relative to the current design, so we don’t want many parts to weigh much more than this, especially since the customer isn’t really willing to pay us for the “extra” material beyond 100 kg.

### Population versus sample statistics

Most of process or product improvement is concerned with reducing the standard deviation, $\sigma$, shifting the mean (a.k.a. average), $\mu$, or reducing a proportion, $p$, of a process or product characteristic. These summary statistics refer to the population characteristics—the mean, standard deviation or proportion of all parts of a certain design that will ever be produced, or all times that a production step will ever be completed in the intended manner.

Since we can’t measure the whole population up front—we will be producing parts for a long time—we have to draw a sample from the population, and use the statistics of that sample to gain insight into the total population. We can visualize this, somewhat crudely, with the following:

We can imagine that the blue circles are conforming parts, and the orange octagons are non-conforming parts. If the sampling process is fair, then the sample proportion $\hat{p}$ will be close to—and statistically indistinguishable from—the true population proportion $p$. In the population we have 44 parts total, 8 defective parts and 36 conforming parts. In the sample that we drew, we have 10 parts total, 9 conforming and 1 defective. While $(p = 8/36 = 1/4 \ne \hat{p} = 1/9$, statistically we have

matrix(c(1, 8, 10-1, 44-8), ncol=2) %>%
chisq.test(simulate.p.value = TRUE)

##
##  Pearson's Chi-squared test with simulated p-value (based on 2000
##  replicates)
##
## data:  matrix(c(1, 8, 10 - 1, 44 - 8), ncol = 2)
## X-squared = 0.3927, df = NA, p-value = 0.6692


With such a high p-value (0.67), we fail to reject the null hypothesis that $\hat{p} = p$; in more colloquial terms, we conclude that the apparent difference between 8/36 and 1/9 is only due to random errors in sampling. (For larger counts of successes and failures, prop.test() would also work and would be more informative.)

From our perspective, of course, we don’t know what the population looks like. We don’t have any way of knowing with certainty—or accessing data about—future performance, so there is no way for us to know what the total population looks like. In lieu of population data, we develop a sampling process that allows us to fairly draw a sample from that population.

While we want to know the true population mean, $\mu$, the true population standard deviation, $\sigma$, or the true population proportion $p$, we can only calculate the sample mean, $\overline{X}$, the sample standard deviation, $S$, or the sample proportion $\hat{p}$.

From the known sample, we then reason backward to what the true population looks like. This is where statistics comes into play; statistics allows us to place rigorous boundaries on what the population may look like, without fooling ourselves. Sample size is critical to controlling the uncertainty in these boundaries.

### Summary and a look forward

Testing in product development—and usually in production—involves sampling a product or process. Samples never look exactly like the population that we are concerned about, but if the sampling process is fair then the samples will be statistically indistinguishable from the population. With due awareness of the statistical uncertainties, we can use samples to make decisions about the population.

In the next post, I will look at how sample size impacts the uncertainty in our estimation of population statistics like the mean and standard deviation. In a later post, I will look at how this uncertainty impacts the business.

### A short aside on statistical tests for proportions

The usual way to compare two proportions would be a proportions test (prop.test() in R), but because we have so few samples to compare, the results may be unreliable and prop.test() generates an appropriate warning. fisher.test() provides an exact estimate of the p-value, but the assumptions are violated with data like this, where we are sampling a fixed number of parts (i.e. row sums are fixed, but column sums are not controlled). This leaves us with using a chi-squared test (chisq.test() in R) which is less informative but does the job. Either the Barnard test or Bayesian estimation based on Monte Carlo simulation would be more informative and possibly more robust.