# 1. Overview

The standard deviation, σ, represents the spread of data from its mean. Taking the examination results, for example, if the standard deviation, σ, of a mathematics examination is 10 marks, and that of a world history examination is 5 marks, it means that the mathematics examination has a greater variation than the world history examination in the spread of marks between students with high scores and those with low scores. As σ only focuses on the variation from the mean, it does not represent the actual mean.

Six Sigma is a term that takes into consideration the fact that if the σ (representing the standard deviation) of a sample fits six times between the mean and the nearest specification limit, there will be practically no items that fail to meet the specifications. To put it another way, Six Sigma primarily focuses on achieving a status with no variation. It specifically defines that defect levels should be below approximately three defects per million opportunities.

## 1-1. Improvement Cycle

To achieve a status with no variation, the concept of Six Sigma is to focus primarily on process improvements, not on products or components. For example, process improvements are ensured by implementing the MAIC cycle, measuring the actual deviation, and identifying the reasons for the deviation. Six Sigma is a top-down process emanating from management, and it initiates activities based on customer feedback.

In business, particularly from the perspective of quality control, the concept of the standard deviation and the number of defects is often considered to be more important than the concept of the mean. That is why Six Sigma has developed as a management theory around the world. Six Sigma is a registered trademark of Motorola in the United States.

## 1-2. The Standard Deviation

Sigma (σ) means the standard deviation that is used to indicate the spread in statistics. When the figures in a sample vary widely, for example, 1000, 500, 200 and 10, the standard deviation σ is large. When the figures do not vary widely, for example, 502, 500, 499 and 499, the standard deviation σ of the sample is small.

The standard deviation commonly used as an indicator for academic achievement clearly shows the position of the specific sample in the statistical population. The figure is converted so that the variation (standard deviation σ) of the entire spread is always 10. If the sample is equal to the mean μ, the variation is 50.

This type of distribution is called **the normal distribution (Gaussian distribution)**, and probability is known based on the distance from the center point in the normal distribution graph. (The graph illustrated below was created using NORMSDIST in Microsoft Excel.)

- Standard deviation of 60 or above (40 or below): 15.87% of the overall statistical population
- Standard deviation of 70 or above (30 or below): 2.275% of the overall statistical population
- Standard deviation of 80 or above (20 or below): 0.135% of the overall statistical population
- Standard deviation of 90 or above (10 or below): 0.003% of the overall statistical population
- Standard deviation of 100 or above (0 or below): 0.00002% of the overall statistical population

Samples with a standard deviation outside the range between 100 and 0 take place only approximately once in a group of 2 million cases. When there is an adequate number of samples, the ratio is as follows. This appears to be mathematically correct in the same way, stating that 50% of the overall group consists of samples that equal or exceed the mean score.

- Standard deviation between 40 and 60 (between +1σ and -1σ): 68.2689492% of the overall group
- Standard deviation between 30 and 70 (between +2σ and -2σ): 95.4499736% of the overall group
- Standard deviation between 20 and 80 (between +3σ and -3σ):
**99.7300204% of the overall group** - Standard deviation between 10 and 90 (between +4σ and -4σ): 99.9936658% of the overall group
- Standard deviation between 0 and 100 (between +5σ and -5σ): 99.9999427% of the overall group
- Standard deviation between -10 and 110 (between +6σ and -6σ): 99.9999998026825% of the overall group

The probability of a sample being 3σ away from μ, the center value, is 0.3%, which is markedly low. This is called **the Three Sigma Rule** or **the 68-95-97.7 Rule**.

Etymologically speaking, Six Sigma aims to eliminate variation in order to make all samples within +6σ to -6σ (110 to -10 in terms of the standard deviation) meet the specifications. To put it another way, Six Sigma’s goal is to contain the occurrence of defects to two per billion or less.

Six Sigma’s actual target concept, however, is to achieve a defect occurrence probability of -4.5σ or more (5 or below in terms of the standard deviation) based on a variation of approximately 1.5σ. This figure was obtained historically in light of the necessity of allowing for various long-term variation factors, while disregarding good results. This produces a figure of 3.4 defects per million. (In either case, an unachievable figure is set as the target value.)

# 2. History

- 1980s: Six Sigma is developed to improve the manufacturing processes of Motorola, a major telecommunication equipment manufacturer.
- 1998: Toshiba adopts Six Sigma to improve its management quality.

Six Sigma has also been adopted by a number of other companies, including General Electric, Texas Instruments, and Ford in the United States, and Sony, Shimano, and Hitachi Maxell in Japan.

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# Y. External Sites

- en.wikipedia.org > Six Sigma