Normal Distribution

Quick reference

• 00:05 Hi, I'm Ray Sheen, and I've already used the term normal distribution several
• 00:09 times in other modules, and I'll be using it a lot more throughout the course.
• 00:13 So let's explain what this really means.
• 00:16 The normal distribution is at the heart of statistical process control,
• 00:21 and therefore is of major importance to Lean Six Sigma.
• 00:24 The normal distribution is what you probably refer to as
• 00:28 the bell-shaped curve.
• 00:29 It is a distribution of data values for a process parameter.
• 00:33 The distribution is symmetrical with a peak center.
• 00:36 This is also known as the Gaussian curve, named after Carl Gauss,
• 00:40 who was a prominent mathematician in the early 1800s.
• 00:44 To be a little more precise, since the normal curve is symmetric,
• 00:49 there are an equal number of data points both above and
• 00:53 below the center value or mean, which is peaked.
• 00:56 This would not be true of a skewed distribution.
• 01:00 There's only one peak and is at the center of the curve.
• 01:04 They're not multiple peaks, nor is the top of the curve flat.
• 01:08 And that the ends of the curve approach the value of zero,
• 01:12 although theoretically it never reaches zero from a practical standpoint, it does.
• 01:17 The area under the curve represents all of the data points in the dataset.
• 01:21 This is what we mean by the form of a data distribution.
• 01:25 It is the form of the curve that it covers over all the data points.
• 01:30 The reason this curve is so important is that it represents normal
• 01:35 random variation or uncertainty in a process.
• 01:38 All the special cause of thing effects are removed, and
• 01:41 I'll talk about that more in another lesson.
• 01:43 What's left is the random uncertainty,
• 01:46 which will inevitably take on the shape of this normal curve.
• 01:49 Remember, in Lean Six Sigma, we want to remove variation.
• 01:53 So our goal is to remove the special cause variation and
• 01:57 limit the amount of the remaining normal random variation.
• 02:02 Let's look at some of the very important characteristics of this curve.
• 02:06 For this illustration, I'll set the mean or
• 02:09 center point to 0 value on our horizontal scale.
• 02:12 That horizontal scale will use the units of sigma or standard deviation.
• 02:18 Based upon the shape of a normal curve,
• 02:21 if we were to draw lines at one standard deviation above and below the mean,
• 02:26 68% of all the data points in our data set would fall within those two lines.
• 02:31 So roughly two-thirds of the data values in the standard normal curve,
• 02:36 meaning random variation,
• 02:37 are within one standard deviation of the average value of the data set.
• 02:43 Let's jump out to two standard deviations.
• 02:46 Now, I can see that 95.45% of the data points in the data set will
• 02:51 fall between -2 standard deviations and +2 standard deviations.
• 02:56 The -3 standard deviation the +3 standard deviation is a significant point for
• 03:01 us to consider.
• 03:02 You can see that this is, 99.73% of all the data points are within that range.
• 03:08 That means that from the standpoint of random variation,
• 03:11 we should only see three data points out of a thousand that go beyond these lines.
• 03:15 The reason the three sigma limits is of interest to us is the process
• 03:20 capability ratios and statistical process control strategies.
• 03:24 These were developed by Professor Schuhart, and were based upon
• 03:28 achieving this plus or minus 3 standard deviation level of performance.
• 03:32 More about that in our course on statistical process control.
• 03:36 Moving on out to +4 sigma to -4 sigma level, as you can see,
• 03:41 we're at a point now where 99.9937% of the data values are between those points.
• 03:49 Continuing out to plus or minus 5 sigma,
• 03:52 we go up to 99.999943 of all the data values.
• 03:56 And finally, at plus or minus 6 sigma, 99.9999998% are under the curve.
• 04:03 For all intents and purposes in the real world, that means everything.
• 04:08 We'll come back to these numbers in the Statistical Process Control course, and
• 04:13 focus in on what they mean for process stability and process control.
• 04:17 One last point, if you need the equation for the curve, here it is.
• 04:22 Let me wrap this up with some comments about the characteristics of a normal
• 04:26 distribution.
• 04:27 The normal distribution is an excellent model for random variation in nature.
• 04:32 This could apply to random variation on the inputs to a process,
• 04:36 such as variation in materials or environmental conditions.
• 04:40 It could also represent the random variation in the output of the process.
• 04:44 That could be variation in timing, quality, or costs.
• 04:48 This variation has no clearly assignable cause, it's just part of the system.
• 04:54 Now, if you go back and follow the item through the process, you can find
• 04:58 several factors that lead to particular final value of a process parameter, but
• 05:03 those factors can and do vary slightly.
• 05:05 And if the variation is within the normal range, it's called random variation.
• 05:10 We'll talk a lot more about this in the session on special cause and common cause.
• 05:15 The bottom line is that in order to change the level of variation,
• 05:19 there needs to be a fundamental change in the process, but
• 05:23 a great characteristic of this type of variation is that it is predictable.
• 05:28 We can calculate a mean and a standard deviation.
• 05:31 As I showed on the last slide, we can with confidence predict what percentages of
• 05:36 the process output will fall within the different standard deviations or
• 05:41 sigma levels.
• 05:42 This predictable center, spread, and shape will allow us then to set up
• 05:46 statistical process control charts to track our process performance and
• 05:50 be able to detect when abnormal conditions are occurring.
• 05:54 The normal distribution is used throughout the Lean Six Sigma process for
• 05:59 both analysis and control.
• 06:00 It is the most commonly found distribution,
• 06:06 and we will be referring to it often.

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