Abstract:
You will be surprised that we all are aware of this concept of distribution and are using it intuitively, all the time! Don’t believe me? Let me ask you a simple question, to which income class do you belong? Let’s assume that your answer is middle income class. On what basis did you made this statement? Probably in your mind you have following distribution of income groups and based on this image in your mind, you are telling your position is towards the left side or towards the middle income group on this distribution. Figure-1: How we are making use of distribution in our daily life, intuitively |
Note:
This article gives a conceptual view of the tools that we use in inferential statistics. Here we are not explanting the concept of sampling or the sampling distribution. Instead we are using distribution of individual values and assuming them to be normally distributed (which is not always the case) in order to explain the concept and also using it for the illustration purpose. We advise readers to read something on “sampling and sampling distribution” immediately after reading this article for better clarity as we are giving oversimplified version of the same in the present article. Don’t miss the “Central limit theorem”. |
Introduction to the Concept of Distribution
When we say that my child is not good at studies, you are drawing a distribution of all students in your mind and implicitly trying to tell the position of your child towards the left of that distribution. Whenever we talk of adjectives like rich, poor, tall, handsome, beautiful, intelligent, cost of living etc., we subconsciously, associate a distribution to those adjective and we just try to pinpoint the position of a given subject onto this distribution.
What we are dealing here is called as inferential statistics because, it helps in drawing inferences about the population based on a sample data. This is just opposite of probability as shown below.
Figure-2: Difference between probability & statistics
This inferential statistics empower us to take a decision based on the small sample drawn from a population.
Why, it is so difficult to take decisions or what causes this difficulty?
This is because we are dealing with samples instead of population. Let’s assume, we are making a batch of one million tablets (population) of Lipitor and before releasing this batch in to the market, we want to make sure that each tablet must be having Lipitor content of 98-102%. Can we analyze all one million tablets? Absolutely not! What we actually do is to analyze, say 100 tablets (Sample) selected at random from one million tablets and based on the results, we accept or reject the whole lot of one million tablets (we usually use z-test or t-test for taking decisions)
BUT, there is a catch. Since we are working with small samples, there is always a chances of taking a wrong decision because the sample thus selected may not be homogeneous enough to represent the entire population (sampling error). This error is denoted by alpha (α) and is decided by the management prior to performing any study i.e. we are accepting an error of α. It means that there is a probability of α that we are accepting a failed batch of Lipitor. Since α is a theoretical threshold limit then, it must be vetted by some experimental probability value. This experimental or the observed probability value is called as p-value (see blog on p-value).
Another aspect of the above discussion arises if we draw two or more samples (of 100 tablets each) and try to analyze them. Let me make it more complicated for you. You are the analyst and I come to you with three samples and want to know from you, whether all these three samples are coming from a single batch (or belong to the same parent population) or not? Point I want to emphasize is that, even though multiple samples are withdrawn from the same population but they would seldom be exactly the same because of the sampling error. The concept is described in following figure-3. This type of decision where sample size ≥ 3, is taken by ANOVA.
Figure-3: The distribution overlap and the decision making (or inferential statistics)
We have seen earlier that α is the theoretical probability or a threshold limit beyond which we assume that the process is no longer the same. This theoretical limit is then tested by collecting a dataset followed by performing some statistical tests (t-test, z-test etc.) to obtain an experimental or observed probability value or the p-value and if, this p-value is found to be less than α, we say that samples are coming from two different populations. This concept is represented below
Figure-4: The relationship between p-value and the alpha value for taking statistical decisions.
Let’s remember the above diagram and try to visualize some more situations that we face every day, where we are supposed to take decisions. But before we do that, one important point, we must identify the target population correctly otherwise whole exercise would be a futile one.
For example
As a high end apparel store, I am interested in the monthly expenditure of females, but wait a second, shouldn’t we specify what kind of females? Yes, we require to study the females of following two categories
The employed and the self-employed females (great! at least we have identified the population categories to be compared). Now next dilemma is whether to consider the females of all age groups or the females below certain age? As my store is more interested in young professionals hence I would compare the above two groups of females but with an age restriction of less than or equal to thirty years.
Figure-5: Identifying the right population for study is important
Another important point, in order to compare two (using z-test or t-test) or more samples (using ANOVA), we also require information about the mean and standard deviation of the samples, before we can tell whether they are coming from same or different parent population.
For example, the mean monthly expenditure on apparel by a sample of 30 employed females is $1500 and the mean expenditure by 30 self-employed females be $1510. Immediately we will try to compare these two means and conclude that two means are almost the same. In back of our mind we are assuming that even though means are different but there will some variation in the data and if, we consider this variation then this difference is not significant. Remember! We have made some kind of distribution in our mind before making this statement. (statistically we do it by two sample t-test)
Figure-6: Significant Overlap between distributions indication no difference between them
What if, the mean expenditure by self-employed females be $1525, then we can say it’s not a big difference to be significant (again we are assuming that there will be a variability in the data). What if, the mean expenditure by self-employed females is $1600, in this case we are certain that the difference is significance. In all three cases discussed above, it is assumed that variance remained constant.
Figure-7: Insignificant Overlap between distributions indication that there is a difference between them
In real life, whenever we encounter two samples, we are tempted to compare the mean directly for taking decisions. But, in doing so, we forget to consider the standard deviation (variation) that is there in the data of two samples. If we consider the standard deviation and then if we find that there is no significant overlap between the distribution of the monthly expenditure by employed females and the distribution of the monthly expenditure by self-employed, then we can conclude that the expenditure behavior of the two groups are different (see figure-6 & 7 above).
Some other situations that could be understood by drawing the distribution. It will help us in comprehending the situation in a much better way.
Women workforce are protesting that there is a gender biasness in the pay scale in your company, is it so?
Once again, be careful about selecting the population for the study! We should only compare males/females of same designation or with same work experience. Let’s take the designation (males & females at manager and senior manager level) as a criterion for the comparison. Since, we have identified the population, we can now select some random samples from both genders belonging to manager and senior manager level. We can have two situations, either the two distribution overlaps or do not overlap. If there is a significant overlap (p-value > α) then there is no difference in salary based on the gender. On the other hand, if two distribution are far apart (p-value < α), then there is a gender bias.
Figure-8: Intuitive scenarios for taking decision, based on the degree of distribution overlap
Our new gasoline formula gives a better mileage than the other types of gasoline available in the market, should we start selling it?
This problem can be visualized by following diagram. But be careful! While measuring mileage, make sure you are taking same kind of car and testing them on the same road and running them for the same number of kilometers at a same constant speed! Since number of samples ≥ 3, use ANOVA.
Figure-9: Understanding the gasoline efficiency using distribution
New filament increase the life time of a bulb by 10%, should we commercialize it?
For this problem, let’s produce two sets of bulbs, first set with the old filament and second set with the new filament. This is followed by testing the samples from each group for their lifespan, what we are expecting is represented below
Figure-10: Understanding the filament efficiency using distribution
A new catalysts developed by R&D team can increase the yield of the process by 5%, should we scale-up the process?
Here we need to establish whether the 5% increase in yield is really higher or not. Can this case be represented by case-1 or by case-2 in above diagrams?
The efficacy of a new drug is 30% better than of the existing drug in the market, is it so?
The soap manufacturing plant finds that some of the soap are weighing 55 gm. instead of 50-53 gm t(he target weight)., should he reset the process for corrective actions?
A new production process claims to reduce the manufacturing time by 4 hrs, should we invest in this new process?
The students of ABC management school are offered better salary than that of the XYZ School, is it so? Colleges advertise like that!
Let’s have a look how the data is usually manipulated here. In order to promote a brand, companies usually distort the distribution when they compare their products with the other brands.
Figure-11: Misuse of statistics
ABC College or any other company promoting their brands would take samples from the upper band of their distribution and then they compare it with the distribution of the XYZ College or with other available brands. This gives a feeling that ABC College or a given brand in question is better than others. Alternatively, you can take competitor’s samples from the lower end of their distribution for comparison for getting the feel good factor about your brand!
Yield of a process has decreased from 90% to 87%, should we take it as a six sigma project?
Again, we need to establish whether the decrease of 3% yield is really significant or not. Can this case be represented by case-1 or by case-2 in above diagrams?
If we look at the situations described in points 4-8 above, we are forced to think “what is the minimum separation required between the mean of two sample, to tell whether there is significant overlap or not”
Figure-12: What should be the minimum separation between distribution?
This is usually done in following steps (this will be dealt separately in next blog on hypothesis testing)
- Hypothesis Testing
- Null and alternate hypothesis
- Decide α
- Test statistics
- Use appropriate statistical test to estimate p-value like Z-test, t-test, F-test etc.
- Compare p-value and α
- Take decision based on whether p-value is < or > α
Concept of distribution and the hypothesis testing
Let’s see how the above concept of distribution helps in understanding the hypothesis testing. In hypothesis testing we make two statement about the same population based on the sample. These two statement are known as “null” and “alternate” hypothesis.
Null Hypothesis (H_{0}): Mean mileage from a liter of new gasoline ≤ 20 Km (first distribution)
Alternate Hypothesis (H_{a}): Mean mileage from a liter of new gasoline > 20 Km (second distribution)
The above two statement can be represented by following two distribution
Figure-13: Distributions of null and alternate hypothesis
Now, if H_{0} is true i.e. new gasoline is no better than the existing one then, we would expect two distributions to overlap significantly (p-value > a)
Figure-14: Pictorial view of the condition when null hypothesis is true
On the other hand if H_{0} is false or H_{a} is true (new gasoline is really better than the existing one) then these two distribution will be far from each other or there would no significant overlap of the two distributions (p-value < a)
Figure-15: The pictorial view of the case when null hypothesis is not true
Above discussion can be extended to understand ANOVA, Regression analysis etc.
Summary
This article tries to give a pictorial view to a given statistical problem, we can call it as “The Tale of Two Distributions”.
Any business problem that requires decision making can be visualized in the form of a overlapping or a non-overlapping distributions. This will give a pictorial view of the problem to the management and would be easy for comprehending the problem.
Another point that is important here is the exercise if identifying the right target population i.e. we must make sure that an apple is compared to an apple!
Going forward, this understanding will help you in understanding hypothesis testing in upcoming blog.