1. Master Black Belt
Statistics

2. Statistical Notation – Cheat Sheet
Summation
The standard deviation of sample data
The standard deviation of population data
The variance of sample data
The variance of population data
The range of data
The average range of data
Multi-purpose notation, i.e. # of subgroups, # of classes
The absolute value of some term
Greater than, less than
Greater than or equal to, less than or equal to
An individual value, an observation
A particular (1st) individual value
For each, all, individual values
The mean, average of sample data
The grand mean, grand average
The mean of population data
A proportion of sample data
A proportion of population data
Sample size
Population size
Use this as a cheat sheet, don’t bother memorizing all of this. Actually most of the notation in Greek is for population data.
Measure Phase

3. Parameters vs. Statistics
Population: All the items that have the “property of interest” under study.
Frame: An identifiable subset of the population.
Sample: A significantly smaller subset of the population used to make an inference.
Population
Sample
The purpose of sampling is:
To get a “sufficiently accurate” inference for considerably less time, money, and other resources, and also to provide a basis for statistical inference; if sampling is done well, and sufficiently, then the inference is that “what we see in the sample is representative of the population”
A population parameter is a numerical value that summarizes the data for an entire population, a sample has a corresponding numerical value called a statistic.
The population is a collection of all the individuals of interest. It must be defined carefully, such as all the trades completed in If for some reason there are unique subsets of trades it may be appropriate to define those as a unique population, such as, “all sub custodial market trades completed in 2001”, or “emerging market trades”.
Sampling frames are complete lists and should be identical to a population with every element listed only once. It sounds very similar to population, and it is. The difference is how it is used. A sampling frame, such as the list of registered voters could be used to represent the population of adult general public. Maybe there are reasons why this wouldn’t be a good sampling frame. Perhaps a sampling frame of licensed drivers would be a better frame to represent the general public.
The sampling frame is the source for a sample to be drawn.
It is important to recognize the difference between a sample and a population because we typically are dealing with a sample of the what the potential population could be in order to make an inference. The formulas for describing samples and populations are slightly different. In most cases we will be dealing with the formulas for samples.
Sample statistics:
Arithmetic descriptions of a sample
X-bar , s, p, s2, n
Population Parameters:
Arithmetic descriptions of a population
µ,  , P, 2, N
Measure Phase

4. Types of Data Attribute Data (Qualitative)
Is always binary, there are only two possible values (0, 1)
Yes, No
Go, No go
Pass/Fail
Variable Data (Quantitative)
Discrete (Count) Data
Can be categorized in a classification and is based on counts.
Number of defects
Number of defective units
Number of customer returns
Continuous Data
Can be measured on a continuum, it has decimal subdivisions that are meaningful
Time, Pressure, Conveyor Speed, Material feed rate
Money
Pressure
Conveyor Speed
Material feed rate
The nature of data of data is important to understand. Based on the type of data you will have the option to utilize different analyses.
In the next phases, we will study many different statistical tests but it is first important to understand what kind of data you have for example continuous data is the most powerful and attribute data is the least powerful.
Measure Phase

5. Possible values for the variable
Discrete Variables
Discrete Variable
Possible values for the variable
The number of defective needles in boxes of 100 diabetic syringes
0,1,2, …, 100
The number of individuals in groups of 30 with a Type A personality
0,1,2, …, 30
The number of surveys returned out of 300 mailed in a customer satisfaction study.
0,1,2, … 300
The number of employees in 100 having finished high school or obtained a GED
0,1,2, … 100
The number of times you need to flip a coin before a head appears for the first time
1,2,3, …
(note, there is no upper limit because you might need to flip forever before the first head appears.
Shown here are additional discrete variables. Can you think of others within your business?
Measure Phase

6. Possible Values for the Variable
Continuous Variables
Continuous Variable
Possible Values for the Variable
The length of prison time served for individuals convicted of first degree murder
All the real numbers between a and b, where a is the smallest amount of time served and b is the largest.
The household income for households with incomes less than or equal to $30,000
All the real numbers between a and $30,000, where a is the smallest household income in the population
The blood glucose reading for those individuals having glucose readings equal to or greater than 200
All real numbers between 200 and b, where b is the largest glucose reading in all such individuals
Shown here are additional continuous variables. Can you think of others within your business?
Measure Phase

7. Definitions of Scaled Data
Understanding the nature of data and how to represent it can affect the types of statistical tests possible.
Nominal Scale – data consists of names, labels, or categories. Cannot be arranged in an ordering scheme. No arithmetic operations are performed for nominal data.
Ordinal Scale – data is arranged in some order, but differences between data values either cannot be determined or are meaningless.
Interval Scale – data can be arranged in some order and for which differences in data values are meaningful. The data can be arranged in an ordering scheme and differences can be interpreted.
Ratio Scale – data that can be ranked and for which all arithmetic operations including division can be performed. (division by zero is of course excluded) Ratio level data has an absolute zero and a value of zero indicates a complete absence of the characteristic of interest.
Shown here are the four types of scales. It is important to understand these scales as they will dictate the type of statistical analysis that can be performed on your data.
Measure Phase

8. Possible nominal level data values for the variable
Nominal Scale
Qualitative Variable
Possible nominal level data values for the variable
Blood Types
A, B, AB, O
State of Residence
Alabama, …, Wyoming
Country of Birth
United States, China, other
Listed are some examples of nominal data. The only analysis is whether they are different or not.
Time to weigh in!
Measure Phase

9. Possible Ordinal level data values
Ordinal Scale
Qualitative Variable
Possible Ordinal level data values
Automobile Sizes
Subcompact, compact, intermediate, full size, luxury
Product rating
Poor, good, excellent
Baseball team classification
Class A, Class AA, Class AAA, Major League
These are examples of ordinal data.
Measure Phase

10. Interval Scale Interval Variable Possible Scores
IQ scores of students in BlackBelt Training
100…
(the difference between scores is measurable and has meaning but a difference of 20 points between 100 and 120 does not indicate that one student is 1.2 times more intelligent )
These are examples of interval data.
Measure Phase

11. Ratio Scale Ratio Variable Possible Scores
Grams of fat consumed per adult in the United States
0 …
(If person A consumes 25 grams of fat and person B consumes 50 grams, we can say that person B consumes twice as much fat as person A. If a person C consumes zero grams of fat per day, we can say there is a complete absence of fat consumed on that day. Note that a ratio is interpretable and an absolute zero exists.)
Shown here is an example of ratio data.
Measure Phase

12. Converting Attribute Data to Continuous Data
Continuous data is always more desirable
In many cases attribute data can be converted to continuous
Which is more useful?
15 scratches or Total scratch length of 9.25”
22 foreign materials or 2.5 fm/square inch
200 defects or 25 defects/hour
Continuous data provides us more opportunity for statistical analyses. Attribute data can often be converted to continuous by converting it to a rate.
Measure Phase

13. Descriptive Statistics
Measures of Location (central tendency)
Mean
Median
Mode
Measures of Variation (dispersion)
Range
Interquartile Range
Standard deviation
Variance
We will review the descriptive statistics shown here which are the most commonly used.
1) For each of the measures of location, how alike or different are they?
2) For each measure of variation, how alike or different are they?
3) What do these similarities or differences tell us?
Measure Phase

14. Descriptive Statistics
Open the MINITAB™ Project “Measure Data Sets.mpj” and select the worksheet “basicstatistics.mtw”
We are going to use the MINITAB™ worksheet shown here to create graphs and statistics. Open the worksheet “basicstatistics.mtw”.
Measure Phase

15. Mean is: Measures of Location Population
Commonly referred to as the average.
The arithmetic balance point of a distribution of data.
Stat>Basic Statistics>Display Descriptive Statistics…>Graphs… >Histogram of data, with normal curve
Sample
Population
Mean are the most common measure of location. A “mean”, implies that you are talking about the population, or inferring something about the population. Conversely, average, you are implies something about sample data.
Although the symbol is different, there is no mathematical difference between the mean of a sample and mean of a population.
Descriptive Statistics: Data
Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3
Data
Variable Maximum
Data
Measure Phase

16. Median is: Measures of Location
The mid-point, or 50th percentile, of a distribution of data.
Arrange the data from low to high, or high to low.
It is the single middle value in the ordered list if there is an odd number of observations
It is the average of the two middle values in the ordered list if there are an even number of observations
The physical center of a data set is the median and unaffected by large data values. This is why people use Median when discussing average salary for an American worker, people like Bill gates and Warren Buffet skew the average number.
Descriptive Statistics: Data
Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3
Data
Variable Maximum
Data
Measure Phase

17. Trimmed Mean is a: Measures of Location
Compromise between the mean and median.
The trimmed mean is calculated by eliminating a specified percentage of the smallest and largest observations from the data set and then calculating the average of the remaining observations
Useful for data with potential extreme values.
Stat>Basic Statistics>Display Descriptive Statistics…>Statistics…> Trimmed Mean
Descriptive Statistics: Data
Variable N N* Mean SE Mean TrMean StDev Minimum Q1 Median
Data
Variable Q3 Maximum
Data
The trimmed mean (highlighted above) is less susceptible to the effects of extreme scores.
Measure Phase

18. Mode is: Measures of Location Mode = 5
The most frequently occurring value in a distribution of data.
Mode = 5
It is possible to have multiple modes, when this happens it’s called bi-modal distributions. Here we only have One mode = 5
Measure Phase

19. Interquartile Range is the:
Measures of Variation
Range is the:
Difference between the largest observation and the smallest observation in the data set.
A small range would indicate a small amount of variability and a large range a large amount of variability.
Interquartile Range is the:
Difference between the 75th percentile and the 25th percentile.
Descriptive Statistics: Data
Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3
Data
Variable Maximum
Data
A range is typically used for small data sets which is completely efficient in estimating variation for a sample of 2. As your data increases the standard deviation is a more appropriate measure of variation.
Use Range or Interquartile Range when the data distribution is skewed.
Measure Phase

20. Standard Deviation is:
Measures of Variation
Standard Deviation is:
Equivalent of the average deviation of values from the mean for a distribution of data.
A “unit of measure” for distances from the mean.
Use when data are symmetrical.
Sample
Population
The standard deviation for a sample and population can be equated with short and long term variation.
Usually a sample is taken over a short period of time making it free from the types of variation that can accumulate over time so be aware.
We will explore this further at a later point in the methodology.
Descriptive Statistics: Data
Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3
Data
Variable Maximum
Data
Cannot calculate population standard deviation because this is sample data.
Measure Phase

21. Variance is the: Measures of Variation Population
Average squared deviation of each individual data point from the mean.
Sample
Population
The Variance is the square of the standard deviation. It is common in statistical tests where it is necessary to add up sources of variation to estimate the total. Standard deviations cannot be added, variances can.
Measure Phase

22. What are the characteristics of a Normal distribution?
The normal distribution is the most recognized distribution in statistics.
What are the characteristics of a Normal distribution?
Only random error is present
Process free of assignable cause
Process free of drifts and shifts
So what is present when the data is Non-Normal?
We can begin to discuss the Normal Curve and its properties once we understand the basic concepts of central tendency and dispersion.
As we begin to assess our distributions know that sometimes it’s actually more difficult to determine what is effecting a process if it is normally distributed. When we have a non-normal distribution there is usually special or more obvious causes of variation that can be readily apparent upon process investigation.
Measure Phase

23. The Normal Curve
The normal curve is a smooth, symmetrical, bell-shaped curve, generated by the density function.
It is the most useful continuous probability model as many naturally occurring measurements such as heights, weights, etc. are approximately normally distributed.
The normal distribution is the most commonly used and abused distribution in statistics and serves as the foundation of many statistical tools which will be taught later in the methodology.
Measure Phase

24. Normal Distribution
Each combination of mean and standard deviation generates a unique normal curve
“Standard” Normal Distribution
Has a μ = 0, and σ = 1
Data from any normal distribution can be made to fit the standard normal by converting raw scores to standard scores.
Z-scores measure how many standard deviations from the mean a particular data-value lies.
The shape of the normal distribution is a function of 2 parameters, (the mean and the standard deviation).
We will convert the normal distribution to the standard normal in order to compare various normal distributions and to estimate tail area proportions.
By normalizing the normal distribution this converts the raw scores into standard Z-scores with a mean of 0 and std dev of 1, this practice allows us to use the Z-table.
Measure Phase

25. Convert any raw score to a Z-score using the formula:
Normal Distribution
The area under the curve between any 2 points represents the proportion of the distribution between those points.
Convert any raw score to a Z-score using the formula:
Refer to a set of Standard Normal Tables to find the proportion between μ and x.
The area between the mean and any other point depends upon the standard deviation m x
The area under the curve between any two points represents the proportion of the distribution. The concept of determining the proportion between 2 points under the standard normal curve is a critical component to estimating process capability and will be covered in detail in that module.
Measure Phase

26. The Empirical Rule… Empirical Rule+6 -1 -3 -4 -5 -6 -2 +4 +3 +2 +1 +5
The Empirical rule allows us to predict or more appropriately make an estimate of how our process is performing. You will gain a great deal of understanding within the Process Capability module. Notice the difference between +/- 1 SD and +/- 6 SD.
68.27 % of the data will fall within +/- 1 standard deviation
95.45 % of the data will fall within +/- 2 standard deviations
99.73 % of the data will fall within +/- 3 standard deviations
% of the data will fall within +/- 4 standard deviations
% of the data will fall within +/- 5 standard deviations
% of the data will fall within +/- 6 standard deviations
Measure Phase

27. The Empirical Rule
No matter what the shape of your distribution is, as you travel 3 standard deviations from the mean, the probability of occurrence beyond that point begins to converge to a very low number.
Measure Phase

28. There are many types of distributions:
Why Assess Normality?
While many processes in nature behave according to the normal distribution, many processes in business, particularly in the areas of service and transactions, do not
There are many types of distributions:
There are many statistical tools that assume normal distribution properties in their calculations.
So understanding just how “normal” the data are will impact how we look at the data.
There is no good and bad, its not always better to have “normal” data, look at it in respect to the intent of your project. Again, there is much informational content in non-normal distributions, for this reason it is useful to know how normal our data are.
Go back to your project, what do you want to do with your distribution, normal or non-normal. Many distributions simply by nature can NOT be normal. Assume that your dealing with a time metric, how do you get negative time, without having a flux capacitor as in the movie “Back to the Future.” If your metric is, by nature bound to some setting.
Measure Phase