The mean of a normal distribution is 25. the standard deviation is 2. what are the six sigma limits?

Are you a one in a million?  This is a tongue in cheek post that might shed a little bit of light on statistics.

Índice Show

• How do you calculate sigma from standard deviation?
• What is the standard deviation for Six Sigma?
• What is the value when it is 2 standard deviations above the mean in a normal probability distribution?
• Is standard deviation 1

The other day I was having an interesting conversation with a successful entrepreneur friend of mine and we were discussing how hard it is to get a startup exit let alone more than one.  I have had three in 35 years.  One within my first 10 years, and then another one not until 23 years later.  This friend of mine has had a few exits as well and at one point he joked and said “I guess we are 6-Sigma’s”.  We both laughed but that joke had me thinking all night.  So when I couldn’t sleep I pulled my lovely R and Rstudio out and tried to figure out what Sigma am I.

First we should start with what a Sigma is. It’s the eighteenth letter in the Greek alphabet, but for the purposes of this discussion it is used in statistics as a representation of Standard Deviation (SD). A “1-Sigma” is one standard deviation from the norm (norm is also called mean or average).  When plotted against a normal distribution, a “bell-shaped curve” (see Figure 2), or on a control chart (see Figure 1), it shows how much variance a particular data item is from the average.

Figure 1 – From Wikipedia on Walter Shewhart

The 68-95-99.7 rule is another way of looking at variation. In a normal distribution, it is postulated that things that are true 68% of the time are considered 1-Sigma events.  Things that are true 95% of the time are considered 2-Sigma events and the three-Sigma rule implies that heuristically nearly all values lie within three standard deviations of the mean (3-Sigma).

Figure 2 – From Wikipedia on 68-95-99.7 Rule

You might have also heard the term Six Sigma. Six Sigma, by pure definition, is just 6 standard deviations from the norm. However, companies like Motorola (1986) and General Electric (1995) used the term as a quality initiative. The initiative is widely known through it’s banner slogan “Defects Per Million Opportunities (DPMO)”.

The Six Sigma quality initiative has a goal of not producing more than 3.4 defects per million opportunities. In common IT parlance, you could loosely consider this an service level of 99.9966. In reality this number is actually closer to a 4.5-Sigma, but Motorola determined that due to long term drift, they need to add something called the 1.5 Sigma Shift.

The bottom line is that Sigma is used often as a term to differentiate data points. Next time your friendly aunt tells you that you are a 1 in a million, you can thank her for putting you in an group with 6,999 other people (based on a world population of 7 billion).  This group would collectedly be known as, at least from your aunt’s perspective, a group of 5-Sigma’s.  So back to my original question, my aunt’s opinion aside, “What Sigma am I?”.  In order to figure this out, I first needed to figure out what other people’s Sigma is (see Figure 3).

Figure 3 – My Table

I created my table chart below based on a chart I used from Wikipedia (see Figure 4) and a constant value of 7 billion of the world population.  I cheated and took the frequency ratios and used them in the following R calculations to estimate total population and percentages.

x = 7000000000       # World Population y = c(3,7,22,81,370,2149,15787,1744278,26330254,506797346)format(x/y, scientific=FALSE)# Get the number per 7 billion print(100 * (x-(x/y)) / ((x-(x/y)) + (x/y)),digits=12)

Figure 4 – From Wikipedia on 68-95-97.5 Rule

Let’s look at some of my data points here…

• Roughly speaking, you are a 1-Sigma if you are still using candles to read books. Some estimates predict that 1.6 billion people in this world still do not have electricity. Ironically, this is 130 years after Tesla (Tesla being a bonafide 6-Sigma) invented distributed use of electricity.
• If you make more than $10 per day (hooray), you are a 1.5-Sigma.  Some estimates claim that 80% of the world population lives in poverty making less than $10 dollars per day.
• You might be at least a 2-Sigma if you are reading this blog post.  In other words, you probably make more than 75k per year (estimates are generalizations).
• If you have HIV, you are hovering around 3-Sigma.  A March 2015 study showed that around 15 million people worldwide are infected with HIV.
• If you get breast cancer this year, you are probably somewhere between a 3.5 and 4.0- Sigma. In March of 2010, it was estimated that 1.5 million people got breast cancer.
• If you fall and can’t get up and you don’t have one of those silly TV advertisement devices – “I’ve fallen and I can’t get up” – and you die, you are basically a 4-Sigma. Death from falling is ranked number 24 on the world ranking for causes of death.
• If you win the lottery and and have enough money to buy the Philadelphia Eagles from Jeffery Lurie, you are in that elite group your aunt put you in … 5-Sigma’s. Jeffery Lurie was near the bottom of the Forbes 2000 list of billionaire in 2015. Ironically, if you die from kissing someone in 2015, you are also a 5-Sigma. Death by oral conditions was listed number 83 on the world ranking for causes of death.  My guess is if you die from kissing Jeffery Lurie, you are most likely a 7 or 8-Sigma.
• Sheryl Sandberg is one of around 200 women billionaires worldwide. That puts her in a near proximity of a 5.5-Sigma.
• Einstein by anyone’s estimation is a top 15 individual and has to be considered a 6- Sigma.
• And nudging out Dr Albert for the top spot has to be Jesus. He was clearly a 6-Sigma regardless of your religious belief. 
• If you you happened to discover a purple polka dotted swan in your life time you are probably what Nassim Taleb would call a Black Swan or a 10-Sigma.  In actuality you wouldn’t really be a 10-Sigma in this case.  A 10-Sigma would be an event that happens once every 5.249e+020 years (that’s half a Septillion).   A 10-Sigma is more like a metaphor for something that can never happen or is so great of a number you might as well use infinity.  However, you could actually be a 7-Sigma if per chance you ascended from a completely new solar system and you spotted our friendly purple polka dotted swan friend due to some deviations in evolution. Since the earth is billions of years old and a 7-Sigma event is likely to happen once every 1.07 billion years. 

So this leaves me with my original question. If my dearly departed Aunt Genie were still alive and if she had studied statistics, she would probably call me a 3-Sigma. On a good day my wife might consider me 6-Sigma and lower than a 2 on my bad days. My 16 year old son used to think I was a 6-Sigma, but now he is too cool so I’m probably just a solid 3 from him. And God bless all 12 year old kids because mine thinks (at least for a couple more years) that I’m a 10-Sigma.  The reason I know this is because every night he says to me “Daddy I love you infinity”.

You can always find me at @botchagalupe on twitter…. 

- About The Authors

John Willis

John Willis has worked in the IT management industry for more than 35 years. Currently he is an Evangelist at Docker Inc. Prior to Docker Willis was the VP of Solutions for Socketplane (sold to Docker) and Enstratius (sold to Dell). Prior to to Socketplane and Enstratius Willis was the VP of Training & Services at Opscode where he formalized the training, evangelism, and professional services functions at the firm. Willis also founded Gulf Breeze Software, an award winning IBM business partner, which specializes in deploying Tivoli technology for the enterprise. Willis has authored six IBM Redbooks for IBM on enterprise systems management and was the founder and chief architect at Chain Bridge Systems.

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How do you calculate sigma from standard deviation?

What is the standard deviation for Six Sigma?

What is the value when it is 2 standard deviations above the mean in a normal probability distribution?

Is standard deviation 1