Basic QA Statistics Series(Part 4)- Interquartile Range-IQR
REFLECTION: FOR STUDENTS: A good rule in organizational analysis is that no meeting of the minds is really reached until we talk of specific actions or decisions. We can talk of who is responsible for budgets, or inventory, or quality, but little is settled. It is only when we get down to the action words-measure, compute, prepare, check, endorse, recommend, approve-that we can make clear who is to do what. -Joseph M. Juran
FOR ACADEMICS: Without a standard there is no logical basis for making a decision or taking action. -Joseph M. Juran
FOR PROFESSIONALS/PRACTITIONERS: Both pure and applied science have gradually pushed further and further the requirements for accuracy and precision. However, applied science, particularly in the mass production of interchangeable parts, is even more exacting than pure science in certain matters of accuracy and precision. -Walter A. Shewhart
Foundation
When we left this small series on basic QA statistics, we had just discussed basic measures of Dispersion- Range, Variance, and Standard Deviation. As promised, we are now covering the basics of Interquartile Range (IQR for short). IQR is also a measure of dispersion, but as I’m sure you will be exposed to IQR in the future, I thought it best to give it a separate post.
The IQR range, like the other measures of dispersion, is used to measure the spread of the data points in a data set. IQR is best used with different measurements like median and total range to build a complete picture of a data set’s tendency to cluster around its mean. IQR is also a very useful tool to use to identify outliers (values abnormally far from the mean of a data set), but do not worry about the more in-depth math.
First, to Define all of the aspects of IQR
-First Quartile (Q1)- The value at which 25% of the data are less than or equal to this value (does not have to be a value in the data set).
-Second Quartile (Q2)- The value at which 50% of the data are less than or equal to this value. It is also known as the median. The second quartile or median does not have to be a value in the data set.
-Third Quartile (Q3)- This is the point at which 75% of the data are less than or equal to this value. It also does not have to be in the data set.
-Fourth Quartile (Q4)- This value is the maximum value in the data set (100% of the data are less than or equal to this value).
-Interquartile Range (IQR)- IQR is the Third Quartile minus the First Quartile and considered a measure of dispersion.
(Kubiak, 2017)
Calculating Quartiles
There are several methods for calculating quartiles, so the technique I am going to use is just what I consider the most basic without delving into any more in-depth math.
Steps:
- Order the data set from smallest to largest.
- Determine the median (reference my post: Basic QA Statistics Series(Part 2)- Basic Measures of Central Tendency and Measurement Scales).
- This determination separates the data into two sets (an upper half and lower half). This Median is Q2
- The First Quartile (Q1) is found by determining the median of the lower half of the data (not including the Median from the previous step when calculating the lower half data set median).
- Q3 is the median of the upper half of the data set, not including the value for Q1 in the top half median determination
- Q4 is the maximum in the data set.
(Kubiak, 2017)
Data Set: 22,26,24,29,25,24, 23,26,28,30,35,40,56,56,65,57,57,75,76,77,74,74,76,75,72,71,70,79,78, 1000,10,12,13,15,16,12,11,64, 65,35, 25,28, 21,44,46,55,77, 79,85,84,86,15,25,35, 101,12,25,35,65,75
Conclusion
As you can see, I stacked the data deck with a massive outlier in the data set. 1000 is far from the mean, but the IQR is not affected by this enormous outlier, as it only takes into account Q1 and Q3.
This property of IQR helps prevent outliers from convincing you the mean is just fine, when in fact, the entire system may be out of whack but compensated for by outliers in your data. The little chart you see is called a Box and Whisker plot, and we will give it a separate post later after we discuss Histograms in the nest post.
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Kubiak, T. a. (2017). The Certified Six Sigma Black Belt Handbook Third Edition. Milwaukee: ASQ Quality Press.
Basic QA Statistics Series(Part 3)- Basic Measures of Dispersion and Statistical Notation
REFLECTION: FOR STUDENTS: “It is not possible to know what you need to learn.” -Philip Crosby
FOR ACADEMICS: “Quality is the result of a carefully constructed cultural environment. It has to be the fabric of the organization, not part of the fabric.”-Philip Crosby
FOR PROFESSIONALS/PRACTITIONERS: “Quality has to be caused, not controlled.”-Philip Crosby
Foundation
Before we go further, this post will give you the basic notation for simple statistics so we can communicate more efficiently. It will also make understanding instructions from textbooks much less challenging. Please don’t give up here. These notations are just a secret code mathematicians use. If you learn it, you will begin to see that statistics is quite accessible. After the code is passed on, we will move on to the Measures of Dispersion.
Review: Part 1 and 2 covered the definition of Population, Sample, and how the terms Parameter and Statistic relate to Population and Sample, respectively. Also, we covered the concept of what data is, as well as the different kinds of data that exist, and the measurement scales used to analyze measurement data.
STATISTICAL NOTATION
Typically, capital letters and Greek letters are used to refer to population parameters, and lower-case or Roman letters are used to note sample statistics.
I will be providing information in the table below specifically for this post. As posts are added in the series, more tables will be added to address any other notations referenced in the future. This post will become the notation reference page to allow any who are new to statistical notation an easy reference.
(Kubiak, 2017)
MEASURES OF DISPERSION
There are three primary Measures of Dispersion- Range, Variance, and Standard Deviation. I will address each and explain them plainly. If you are new to statistics, I will avoid mathematics as much as possible, but alas, you will find it inescapable.
First comes RANGE. Range is probably the most well known and most easily understood. Range is simply the difference between the largest (Maximum or MAX) value and the smallest (Minimum or MIN) value in a data set.
Example: 24, 36, 54, 89, 12, 14, 44, 55, 75, 86
Min 12, Max, 89
Range (R)= Max-Min = 77
Though Range is easy to use, it is not always as useful as the other measures of dispersion, because sometimes two separate data sets can have very similar ranges, with the other measures looking nothing alike. On that note, comes something a bit more complicated.
At first, it sounds pretty simple:
VARIANCE- This is the measure of how far off the data values are from the mean over-all. Obtaining this measurement by hand can be painful. You have to find the difference between the mean and each data point in the population or sample, square the differences, and then find the average of those squared differences.
Variance RoadMap
- Calculate the mean of all the data points Calculate the difference between the mean and each data point(Xi – μ or x ̅), Xi being a representation ith value of variable X.
- Square the calculated differences for all data points
- Add these Squared values together
- Divide that number by N if the data set is a population (N), or divide by n-1 if the data is a sample
Follow the underlined statements above, and the formula for Variance below is achieved, but most stat software will calculate Variance with minimal effort.
Sample
Population
Standard Deviation (SD)
A negative of Variance is though you can measure the relative spread of the data, it is not representative of the same scale because it has been squared. For example- data collected in inches or seconds and then checked for variance is effectively square inches or seconds squared.
Standard Deviation is more useful because the units of Standard Deviation end up on the same scale and are directly comparable to the mean of the population or sample. Standard Deviation is the Square Root of the Variance and can be described as the average distance from each data point to the mean. The lower the SD, the less spread out the data is. The larger the spread of data, the higher the SD. Once again, most Stats programs and calculators will provide SD with no problem. The SD helps you understand how much your data is varying from the mean.
Two Examples: (using sample sets)
Set 1: 35, 61, 15, 14, 1
Mean(Set 1): 25.2
Set 2: 45, 48, 50, 43, 40
Mean(Set 2): 45.2
S=√(((45-45.2)²+(48-45.2)²+(50-45.2)²+(43-45.2)²+(40-45.2)²)/4)=3.96
When you first glance at the small sample of data, set one looks like it has a much larger spread from the average than set two. When you run the numbers, the SD results back up your “gut feeling.” An analysis is always better than a “gut feeling,” no matter how intuitive you are. The larger the sample set you are looking at, the more the initial appearance of the data can mislead you, so always run those numbers!
Conclusion
To recap, Range is the most well-known and straightforward Measure of Dispersion, but only describes the dispersion of the extremes of the data, and therefore may not always provide much new information. Range is usually most useful with smaller data sets. I should also mention a term known as the Interquartile Range (IQR). I will be dedicating a separate post to IQR next post.
Variance is an overall measure of the variation occurring around the mean using the Sum of Squares methodology. Remember, variance does not relate directly to the mean, so you cannot evaluate a variance number directly, so you should use variance to see how individual numbers relate to each other within a data set. Outliers (data points far from the mean) gain added significance with variance as well. Standard Deviation tends to be the most useful Measure of Dispersion, as it relates directly to the mean, and can be used to compare the spreads of various data sets. Remember, your stats programs will help you, and many online resources will walk you through any calculation. If you have any questions, shoot me a comment, and I will answer it for you. See you next time as we dig a bit deeper into IQR. 😊
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Kubiak, T. a. (2017). The Certified Six Sigma Black Belt Handbook Third Edition. Milwaukee: ASQ Quality Press.
Basic QA Statistics Series(Part 2)- Basic Measures of Central Tendency and Measurement Scales
REFLECTION: FOR STUDENTS: Learning is not compulsory… neither is survival. – W. Edwards Deming
FOR ACADEMICS: Our schools must preserve and nurture the yearning for learning that everyone is born with. -W Edwards Deming
FOR PROFESSIONALS/PRACTITIONERS: Data are not taken for museum purposes; they are taken as a basis for doing something. If nothing is to be done with the data, then there is no use in collecting any. The ultimate purpose of taking data is to provide a basis for action or a recommendation for action. The step intermediate between the collection of data and the action is prediction. -W. Edwards Deming
Foundation
The previous post covered just the definition of Population and Sample and the descriptions of each using Parameters for Population and Statistics for a Sample. We also mentioned data. To be able to communicate about data, we first have to define data. Define should always be the first step for better understanding.
Data are characteristics or information (usually numerical) that are collected through observation. In a more technical sense, data consists of a set of values of qualitative or quantitative variables concerning one or more persons or objects.
The two broadest categories of Data are: Qualitative and Quantitative-
Qualitative data deals with characteristics and descriptors that cannot be easily measured but can be observed in terms of the attributes, properties, and of course, qualities of an object (such as color and shape). Quantitative data are data that can be measured, verified, and manipulated. Numerical data such as length and weight of objects are all Quantitative.
On the next level of Data are Discrete and Continuous Data.
Discrete Data– Pyzdek and Keller defined discrete data as such: “Data are said to be discrete when they take on only a finite number of points that can be represented by the non-negative integers” (Kubiak, 2017). Discrete data is count data and sometimes called categorical or attribute data. A count cannot be made more precise. You cannot have 2.2 fully functional cars.
Continuous Data– Pyzdek and Keller state- “ Data are said to be Continuous when they exist on an interval, or on several intervals.” Another term used is Variable data. Height, weight, and temperature are continuous data because between any two values on the measurement scale, there is an infinite number of other values (Kubiak, 2017).
Measurement Scales
- Nominal
- Classifies data into categories with no order implied
- Ordinal
- Refers to data positions within a set, where the order is essential, but precise differences between the values are not explicitly defined (example: poor, ok, excellent).
- Interval
- An Interval scale has meaningful differences but no absolute zero. (Ex: Temperature, excluding the Kelvin scale)
- Ratio
- Ratio scales have meaningful differences and an absolute zero. (Ex: Length, weight and age)
(Kubiak, 2017)
I know that it seems like a lot to digest, but recording data correctly is critical. Next, we will discuss the Central Limit Theorem: Per the central limit theorem, the mean of a sample of data will be closer to the mean of the overall population in question, as the sample size increases, notwithstanding the actual distribution of the data. In other words, the true form of the distribution does not have to be normally distributed (a bell curve) as long as the sample size is sufficiently large(Kubiak, 2017). There will eventually be a separate post(s) on sampling, distribution, and choosing the ideal sample size, but we are starting at the basics.
Note: Ordinal Data can be confusing. It depends on the how the ordinal scale is arranged. The Likert Scale would be considered quantitative ordinal, while the Movie rating scale would be considered qualitative ordinal.
(Kubiak, 2017)
Measures of Central Tendency
Three Common ways for quantifying the centrality of a population or sample include the
- Mean
- Arithmetic Average of a data set. This is the sum of the values divided by the number of individual values Ex: [1,3,5,10] Average is 4.75
- Median
- This is the middle value of an ordered data set. When the data are made up of an odd number of values, the median value is the central value of the ordered set. [1, 3, 5], so 3 is the median. When there is an even number of data points, the median is the average of the two middle values of the ordered set [1, 3, 5, 10]. In this case, the Median is the average of 3 and 5: (3+5)/2=4
- Mode
- The mode is the most frequently found value in a data set. It is possible for there to be more than one mode. EX: [1,2,3,5,1,6,8,1,8,1,3]- The Mode is 1
(Kubiak, 2017)
Conclusion
Correctly recording Data and using the proper scale to track your Data is the first step to understanding your process outputs.
A next baby step is knowing how to measure your process based upon your data scale. Being able to calculate the Measures of Central Tendency helps you, but Stats software will do much of this for you. Still, you need to know what you are seeing. It is always most helpful to know what those stat software programs are doing with your data so you can more robustly defend your decisions. Next time we will go a little deeper and talk about Measures of Dispersion (and it is precisely what it sounds like!).
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Kubiak, T. a. (2017). The Certified Six Sigma Black Belt Handbook Third Edition. Milwaukee: ASQ Quality Press.
Quality Concepts Matter 