# Statistical methods in manufacturing

I recently had a chat about process improvement, 6 sigma, and statistics and I was asked the question: “Have you ever used the T-test?”

I was genuinely puzzled as I haven’t. Never.

I started to think why, as the T-test is a basic statistical method we had learnt at the uni. As I looked back, in most cases we try to improve a specific and continuously monitored parameter (temperature, pressure), or a quality characteristic of the product (weight, size) and the mean of the population and the minimum-maximum are recorded for the whole population, or statistically relevant sample size. So no need to waste time sampling, measuring and doing fancy calculations to see any difference. When I needed to look at the changes in the mean, there were usually complex trials with 2 parameters and multiple settings – where a T-test is not really useful.

When we need to make improvements and use statistics is the real 6sigma work – improving process stability. The outcome of your efforts is the improvement in the variance, so you need to consider using the F-test or my favourite, the ANOVA. The benefit of using ANOVA (I mean the one-way ANOVA) is simple: you spend time setting up your Excel calculation table once, but later, with minor adjustments (if any) you can run a quick analysis on multiple sets of samples. Furthermore, It’s the best tool to investigate process stability over the production run: start-up, steady run, after a material change, or around stoppages.

The two-way ANOVA is when magic happens. You can test the impact of two different parameters (different levels of both parameters) and the interaction of these parameters on the mean of the dependent variable that you would like to improve. Excellent tool for investigating and understanding complex processes.

For example, you would like to test the effect of pressure and temperature on the finished good yield in a process step. The null hypotheses are the following:

– There is no difference in average yield at any temperature setting,

– There is no difference in average yield at any pressure setting,

– and most importantly: There is no interaction between pressure and temperature settings.

## A bit of mathematics behind

Here’s a concise breakdown of each of the basic statistical methods, including their purpose, required data, calculation method, and the null hypothesis:

**T-Test:**

**Purpose:** The T-test is used to determine if there’s a significant difference between the means of two groups. *Imagine you’re testing whether a new manufacturing process improves product quality compared to the old process. The T-test helps you decide if the improvements you’re seeing are real or just due to chance.*

**Data:** You need two groups with continuous data. Sample sizes can be relatively small.

**Calculation:** It calculates the t-statistic, which considers the means, standard deviations, and sample sizes of both groups.

**Outcome: ****Null hypothesis (H0): The means of the two groups are equal.**

**Z-Test:**

**Purpose:** Similar to the T-test, but used for larger sample sizes. *It’s like comparing two ice cream shops to see if one consistently gets higher ratings than the other. The Z-test helps you decide if the difference in ratings is actually important or just a fluke.*

**Data:** Two groups with continuous data, and a larger sample size compared to the T-test.

**Calculation:** Involves calculating the z-score, which compares the means and standard deviations of the groups.

**Outcome: **Null hypothesis (H0): The means of the two groups are equal.

**F-Test:**

**Purpose:** Used to compare variances of two or more groups to determine if they are significantly different. *In a manufacturing setting, you might use the F-test to compare the performance of different machines or processes to see if any of them are significantly better.*

**Data:** Two or more groups with continuous data.

**Calculation:** Compares the variances within each group to the variance between the groups.

**Outcome: **Null hypothesis (H0): The variances of the groups are equal.

**One-WayANOVA:**

**Purpose:** Extends the F-test to compare means of more than two groups. *The One-Way ANOVA is your tool for checking if the different materials lead to different widget quality. It helps you determine if there’s a meaningful difference among the groups.*

**Data:** Multiple groups with continuous data.

**Calculation:** Compares the variation within each group to the variation between groups, giving an F-ratio.

**Outcome:** Null hypothesis (H0): The means of all groups are equal.

**Two-Way ANOVA:**** **

**Purpose:** Examines how two factors (variables) impact a response variable. *The Two-Way ANOVA steps in to help you understand if both the materials and machines have an impact on the final product quality. It’s like being a detective who needs to figure out if different ingredients and cooking methods affect the taste of a dish.*

**Data:** Multiple groups based on two independent factors, with continuous data.

**Calculation:** Computes F-ratios for both factors and their interaction.

**Outcome:** Null hypothesis (H0): The means across groups and interactions are equal.

Remember that in all these tests, the null hypothesis (H0) assumes that there is no significant difference or effect. The tests calculate specific statistics based on your data and compare them to critical values or p-values to determine if you can reject the null hypothesis. If the p-value is very small (typically below a chosen significance level), you have evidence to reject the null hypothesis and conclude that there’s a significant difference or effect. These statistical methods are like special tools in a toolkit that help manufacturers make informed decisions about their processes, materials, and product quality. They help you see patterns and differences in data that might not be obvious at first glance.

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