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"Optimism is a strategy for making a better future. Because unless you believe that the future can be better, you are unlikely to step up and...

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What does the word “kaizen” mean to you? You likely know the word kaizen to mean “continuous improvement” or just “improvement”. Perhaps you first heard the term associated with doing a “kaizen event” or project. When I moved to Japan in 2015 I was fascinated to learn about the origins of the concepts of the […]

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What is believed to be true is that St. Patrick was captured by Irish raiders and forced into slavery. He escaped back to his home in Britain and later returned to Ireland as a...

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tl;dr summary: Lean isn't just efficiency… it's safety, quality, delivery, cost, and morale. People often misunderstand that — they don't know or they were taught the wrong things I often have the opportunity to teach a group of experienced healthcare professionals, from a wide range of disciplines, about Lean. My session is part of a […]

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In this and subsequent posts I will have a deeper look at the different pillars of Total Productive Maintenance (TPM). This first post looks at the pillar of focused improvement, which is practically identical with continuous improvement. However, it can be argued if this should be its own pillar or if it should be part ... Read more

The post The Pillars of TPM – Focused Improvement first appeared on AllAboutLean.com.Business has changed drastically over the past year due to the pandemic. Seemingly overnight, companies transformed their operations to continue in this new world where virtual is suddenly very necessary. All said and done, companies "accelerated the digitization of their customer and supply-chain interactions and of their internal operations by three to four years" according to a recent McKinsey & Company report.

If you want to run a successful business and have satisfied customers, then you need to have a proper plan for logistics. In every business, there’s a proper sequence of how every operation works. It starts with manufacturing products, distributing them...

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With this article I try to structure some very important concepts for professionals but which are hostile to many professionals who grew up in the field who have not had the opportunity to deepen their knowledge of these topics from a theoretical point of view.

Obviously it will not be my intention to give an academic and detailed description but an overview and I hope exhaustive for a practical use.

Let's start with some definitions that will come in handy during the reading.

**Measurement**: trivializing the concept, it is simply a unique numerical assignment to an observation.

**Definition domains (discrete or continuous):** if the observed characteristic can assume a finite number of states, we say that the observation and its measurement are defined in a **discrete** set (e.g. outcome of a quality control [Acceptable, Unacceptable]) , otherwise they are defined in a **continuous** set (e.g. measure temperature, length, etc.)

*In the case of infinite definition domains, being our measurement tools limited as they may be (they cannot represent an infinite number of values to be attributed to the observation), the value attributed will always be an approximation of the real, the more precise is our measurement tool, the smaller the difference between the measured value and reality, this difference represents an error that we will call “measurement error (e)”.*

The transformation (function) of an observation into a single numerical value is called a "**random variable**", the "random" attribute refers to the fact that the observation is generated by an experiment (or mechanism, or natural phenomenon, etc.) of which we are unable to predict with certainty.

While we cannot predict the outcome with certainty, there are mathematical theorems that help us in this regard, in particular:

**Law of large numbers:** it establishes the convergence of the mean X_{m} value of a certain sample of random variables (x1, x2, x3 …… .xn) towards the expected value E (x). In other words, thanks to the law of large numbers, we can trust that the average we calculate from a sufficient number of samples is sufficiently close to the true average; however, it says nothing about the probability distribution laws of these random variables.

**Central limit theorem:** answers this problem and establishes the conditions for which a random variable tends to the normal Gaussian distribution. Well, in rough terms, the central limit theorem states that the distribution of the sum of a large number of independent and identically distributed random variables tends to be distributed normally, regardless of the distribution of the individual variables.

*if the number n of the observations is sufficiently large, the probabilistic behavior of the sample mean (whatever the probability law of the variables x1, x2, x3 ....... xn) is of the same type and follows the normal law of Gauss with mean value μ≈ X _{m }and variance \( σ ^ 2≈σ ^ 2 means = 〖σx〗 _n ^ 2 \)*

Now suppose that the object of our observation is the result of a process of which we do not know the details (black box). The analytical study of the measurements of the results obtained can give us useful information on the process in order to know its current state, predict future results and, if necessary, act on the process to improve it.

A very useful tool for carrying out an analytical study of the measurements is the **SPC - Statistical Process Control** and the related Control Charts.

In statistical process control **Mean**, **Range** and **Standard Deviation** are the most used statistics to analyze data measurements.

In the previous article we have indicated the three main characteristics of a statistical distribution: **Position**, **Scatter** and **Pattern**.

We have also defined the term "Measurement" and its possible definition domains "Discrete" and "Continuous".

When talking about Statistical Process Control (SPC), in the case of Continuous Measurements, it is common to speak of Mean as a position statistic, Range and Standard Deviation as scatter statistics.

**Control Charts** are the tools used to monitor these statistics.

There are different types of Control Charts, we will see below the main ones in the case of Continuous type Measurements: **X-Chart** for single Measurements (n = 1), **R-Chart** (2 ≤n ≤ 9), **S-Chart** (9 <n≤25)

The choice between a Control Chart rather than another depends precisely on the number of measurements that make up the sets (subgroups).

Each control chart always analyzes the two statistics, of position (Mean) and scatter (Range or Standard Deviation)

The measurement system to lead to statistically reliable data must be:

**Accurate:**the numerical value attributed to the property must be "close" to reality;**Repeatable:**if the measurement is made on the same object repeatedly, albeit by different individuals, the results must be close to each other;**Linear:**accurate and repeatable measurements must be linearly distributed over the full range of possible results;**Reproducible:**different measurements if made under the same external conditions and by the same individual must lead to the same result;**Stable:**Measurements on the same object produce the same results if done in the future as in the past.

**X-Chart for Single Measurements (n = 1)**

Used when the particular process in question does not allow to carry out consecutive measurements (quality control with destructive test, cycle time too high, etc.)

**Dispersion (Range):** R = largest subgroup - smallest subgroup

¯R = sum of ranges / number of ranges

LCL = D3 x ¯R

UCL = D_{4 }x ¯R

**Position (Average):** ¯X = sum of measurements / number of measurements

LCL = ¯X− E_{2} ¯R

UCL = ¯X + E_{2}¯R

Where by subgroup we mean each consecutive pair of measurements, D_{3}, D_{4}, E_{3} for n = 2 are taken from the table.

**R-Chart for few measurements (2≤n≤9)**

In case the number of measurements of the subgroup are limited (from 2 to 9) it is usual to use the pair of statistics "Average" and "Range" for the control chart.

**Dispersion (Range):** R = largest subgroup - smallest subgroup

¯R = sum of ranges of subgroups / number of subgroups

LCL = D_{3 }x ¯R

UCL = D_{4}x ¯R

**Position (Mean):** ¯X = sum of measurements in the subgroup / subgroup size

X ̿ = sum of subgroup averages / number of subgroups

LCL = X ̿ - A_{2} ¯R

UCL = X ̿ + A_{2} ¯R

D_{3}, D4, A_{2} for 2≤n≤9 are taken from the table.

**S-Chart for more measurements (9 <n <25)**

Used when the number of measurements is greater (9 <n <25). The deviation (Sigma) is used instead of the Range because as the number of measurements increases, the standard deviation is more efficient.

**Dispersion (Sigma): **S = \( \sqrt[2]{ \frac{\sum{1,n} (X_i-¯X)^2}{n-1} } \)

¯S = sum of the sigma of the subgroups / number of subgroups

LCL = B_{3 }x ¯S

UCL = B_{4 }x ¯S

**Position (Mean):** ¯X = sum of measurements in the subgroup / subgroup size

X ̿ = sum of subgroup averages / number of subgroups

LCL = X ̿ - A_{3} ¯S

UCL = X ̿ + A_{3}¯S

Where by subgroup we mean each consecutive pair of measurements, B_{3}, B_{4}, A_{3} for 9 <n≤25 are taken from the table.