Conceptualizing “Distribution” Will Help You in Understanding Your Problem in a Much Better Way

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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.

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 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.

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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. 

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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

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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.

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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)

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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.

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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.

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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.

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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

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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.

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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”

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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)

  1. Hypothesis Testing
    1. Null and alternate hypothesis
    2. Decide α
  2. Test statistics
    1. Use appropriate statistical test to estimate p-value like Z-test, t-test, F-test etc.
  3. Compare p-value and α
  4. 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 (H0): Mean mileage from a liter of new gasoline ≤ 20 Km (first distribution)

Alternate Hypothesis (Ha): Mean mileage from a liter of new gasoline > 20 Km (second distribution)

The above two statement can be represented by following two distribution

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Figure-13: Distributions of null and alternate hypothesis

Now, if H0 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)

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Figure-14: Pictorial view of the condition when null hypothesis is true

On the other hand if H0 is false or Ha 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)

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 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.

 

 

 

 

Why Standard Normal Distribution Table is so important?

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Abstract

Since we are entering the technical/statistical part of the subject hence, it would be better for us to understand the concept first

For many business decisions, we need to calculate the likelihood or probability of an event to occur. Histograms along with relative frequency of a dataset can be used to some extent.. But for every problem we come across we need to draw the histogram and relative frequency to find the probability using area under the curve (AUC).

In order to overcome this limitation a standard normal distribution or Z-distribution or Gaussian distribution was developed and the AUC or probability between any two points on this distribution is well documented in the statistical tables or can be easily found by using excel sheet.

But in order to use standard normal distribution table, we need to convert the parent dataset (irrespective of the unit of measurement) into standard normal distribution using Z-transformation. Once it is done, we can look into the standard normal distribution table to calculate the probabilities.

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From my experience, I found the books belonging to category “statistics for business & economics” are much better for understanding the 6sigma concepts rather than a pure statistical book. Try any of these books as a reference guide.


Introduction

Let’s understand by this example

A company is trying to make a job description for the manager level position and most important criterion was the years of experience a person should possess. They collected a sample of ten manager from their company, data is tabulated below along with its histogram.

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As a HR person, I want to know the mean years of experience of a manager and the various probabilities as discussed below

Average experience = 3.9 years

What is the probability that X ≤ 4 years?

What is the probability that X ≤ 5 years?

What is the probability that 3 < X ≤ 5 years?

In order to calculate the above probabilities, we need to calculate the relative frequency and cumulative frequency

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Now we can answer above questions

What is the probability that X ≤ 4 years? = 0.7 (see cumulative frequency)

What is the probability that X ≤ 5 years? = 0.9

What is the probability that 3 < X ≤ 5 years? = (probability X ≤ 5) – (probability X < 3) = 0.9-0.3 = 0.6 i.e. 60% of the managers have experience between 3 to 5 years.

Area under the curve (AUC) as a measure of probability:

Width of a bar in the histogram = 1 unit

Height of the bar = frequency of the class

Area under the curve for a given bar = 1x frequency of the class

Total area under the curve (AUC) = total area under all bars = 1×1+1×2+1×4+1×2+1×1 = 10

Total area under the curve for class 3 < x ≤ 5 = (AUC of 3rd class + AUC of 4th class) /total AUC = (4+2)/10 = 0.6 = probability of finding x between 3 and 5 (excluding 3)

Now, what about the probability of (3.2 < x ≤ 4.3) =? It will be difficult to calculate by this method, as requires the use of calculus.

Yes, we can use calculus for calculating various probabilities or AUC for this problem. Are we going to do this whole exercise again and again for each and every problem we come across?

With God’s grace, our ancestors gave us the solution in the form of Z-distribution or Standard normal distribution or Gaussian distribution, where the AUC between any two points is already documented.

This Standard normal distribution or Gaussian distribution is widely used in the scientific measurements and for drawing statistical inferences. This normal curve is shown by a perfectly symmetrical and bell shaped curve.

The Standard normal probability distribution has following characteristics

  1. The normal curve is defined by two parameters, µ = 0 and σ = 1. They determine the location and shape of the normal distribution.
  2. The highest point on the normal curve is at the mean which is also the median and mode.
  3. The normal distribution is symmetrical and tails of the curve extend to infinity i.e. it never touches the x-axis.
  4. Probabilities of the normal random variable are given by the AUC. The total AUC for normal distribution is 1. The AUC to the right of the mean = AUC to the left of mean = 0.5.
  5. Percentage of observations within a given interval around the mean in a standard normal distribution is shown below

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The AUC for standard normal distribution have been calculated for all given value of p ≥ z and are available in tables that can be used for calculating probabilities.

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Note: be careful whenever you are using this table as some table give area for ≤ z and some gives area between two z-values.

Let’s try to calculate some of the probabilities using above table

Problem-1:

Probability p(z ≥ 1.25). This problem is depicted below

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Look for z = 1.2 in vertical column and then look for z = 0.05 for second decimal place in horizontal row of the z-table, p(z ≤ -1.25) = 0.8944

Note! The z-distribution table given above give the cumulative probability for p(z ≤ 1.25), but here we want p(z ≥ 1.25). Since total probability or AUC = 1, p(z ≥ 1.25) will be given by 1- p(z ≤ 1.25)

Therefore

p(z ≥ 1.25) = 1- p(z ≤ -1.25) = 1-0.8944 = 0.1056

Problem-2:

Probability p(z ≤ -1.25). This problem is depicted below

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Note! Since above z-distribution table doesn’t contain -1.25 but the p(z ≤ -1.25) = p(z ≥ 1.25) as standard normal curve is symmetrical.

Therefore

Probability p(z ≤ -1.25) = 0.1056

Problem-3:

Probability p(-1.25 ≤ z ≤ 1.25). This problem is depicted below

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For the obvious reasons, this can be calculated by subtracting the AUC of yellow region from one.

p(-1.25 ≤ z ≤ 1.25) = 1- {p(z ≤ -1.25) + p(z ≥ 1.25)} = 1 – (2 x 0.1056) = 0.7888

From the above discussion, we learnt that a standard normal distribution table (which is readily available) could be used for calculating the probabilities.

Now comes the real problem! Somehow I have to convert my original dataset into the standard normal distribution, so that calculating any probabilities becomes easy. In simple words, my original dataset has a mean of 3.9 years with σ = 1.37 years and we need to convert it into the standard normal distribution with a mean of 0 and σ = 1.

The formula for converting any normal random variable x with mean µ and standard deviation σ to the standard normal distribution is by z-transformation and the value so obtained is called as z-score.

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Note that the numerator in the above equation = distance of a data point from the mean. The distance so obtained is divided by σ, giving distance of a data point from the mean in terms of σ i.e. now we can say that a particular data is 1.25σ away from the mean. Now the data becomes unit less!

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Let’s do it for the above example discussed earlier

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Note: Z-distribution table is used only in the cases where number of observations ≥ 30. Here we are using it to demonstrate the concept. Actually we should be using t-distribution in this case.

We can say that the managers with 4 years of experience are 0.073σ away from the mean and on the right hand side. Whereas the managers with 3 years of experience are -0.657σ away from the mean on left hand side.

Now, if you look at the distribution of the Z-scores, it resembles the standard normal distribution with mean = 0 and standard deviation =1.

But, still one question need to be answered. What is the advantage of converting a given data set into standard normal distribution?

There are three advantages, first being, it enables us to calculate the probability between any two points instantaneously. Secondly, once you convert your original data into standard normal distribution, you are ending in a unit less distribution (both numerator & denominator in Z-transformation formula has same units)! Hence, it makes possible to compare an orange with an apple. For example, I wish to compare the variation in the salary of the employees with the variation in their years of experience. Since, salary and experience has different unit of measurements, it is not possible to compare them but, once both distributions are converted to standard normal distribution, we can compare them (now both are unit less).

Third advantage is that, while solving problems, we needn’t to convert everything to z-scores as explained by following example

Historical 100 batches from the plant has given a mean yield of 88% with a standard deviation of 2.1. Now I want to know the various probabilities

Probability of batches having yield between 85% and 90%

Step-1: Transform the yield (x) data into z-scores

What we are looking for is the probability of yield between 85 and 90% i.e. p(85 ≤ x ≤ 90)

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Step-2: Always draw rough the standard normal curve and preempt what area one is interested in

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Step-3: Use the Z-distribution table for calculating probabilities.

The Z-distribution table given above can be used in following way to calculate p(-1.43 ≤ z ≤ 0.95)

Diagrammatically, p(-1.43 ≤ z ≤ 0.95) = p(z ≤ 0.95) – p(z ≤ -1.43), is represented below

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p(-1.43 ≤ z ≤ 0.95) = p(z ≤ 0.95) – p(z ≤ -1.43)= 0.83-0.076 = 0.75

75% of the batches or there is a probability of 0.75 that the yield will be between 85 and 90%.

It can also be interpreted as “probability of getting a sample mean between 85 and 90 given that population mean is 88% with standard deviation of 2.1”.

Probability of yield ≥ 90%

What we are looking for is the probability of yield ≥ 90% i.e. p(x ≥ 90)

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= p(z ≥ 0.95)

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Diagrammatically, p(-1.43 ≤ z ≤ 0.95) = p(z ≤ 0.95) – p(z ≤ -1.43), is represented below

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p(x ≥ 90) = p(z ≥ 0.95) = 1-p(z ≤ 0.95) = 1- 0.076 = 0.17, there is only 17% probability of getting yield ≥ 90%

Probability of yield between ≤ 90%

This is very easy, just subtract p(x ≥ 90) from 1

Therefore,

p(x ≤ 90) = 1- p(x ≥ 90) = 1- 0.17 = 0.83 or 83% of the batches would be having yield ≤ 90%.

Now let’s work the problem in reverse way, I want to know the yield corresponding to the probability of ≥ 0.85.

Graphically it can be represented as

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Since the table that we are using gives the probability value ≤ z value hence, first we need to find the z-value corresponding to the probability of 0.85. Let’s look into the z-distribution table and find the probability close to 0.85

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The probability of 0.8508 correspond to the z-value of 1.04

Now we have z-value of 1.04 and we need find corresponding x-value (i.e. yield) using the Z-transformation formula

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Solving for x

x = 90.18

Therefore, there is 0.85 probability of getting yield ≤ 90.18% (as z-distribution table we are using give probability for ≤ z) hence, there is only 0.15 probability that yield would be greater than 90.18%.

Above problem can be represented by following diagram

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Exercise:

The historical data shows that the average time taken to complete the BB exam is 135 minutes with a standard deviation of 15 minutes.

Fins the probability that

  1. Exam is completed in less than 140 minutes
  2. Exam is completed between 135 and 145 minutes
  3. Exam takes more than 150 minutes

Summary:

This articles shows the limitations of histogram and relative frequency methods in calculating probabilities, as for every problem we need to draw them. To overcome this challenge, a standardized method of using standard normal distribution is adopted where, the AUC between any two points on the curve gives the corresponding probability can easily be calculated using excel sheet or by using z-distribution table. The only thing we need to do is to convert the given data into standard normal distribution using Z-transformation. This also enables us to compare two unrelated things as the Z-distribution is a unit less with mean = 0 and standard deviation = 1. If the population standard deviation is known, we can use z-distribution otherwise we have to work with sample’s standard deviation and we have to use Student’s t-distribution.

How our life gets affected by statistics? Are we aware of it?

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Statistics is always following you whether you are aware of it or not.

Statistics is in action even in your kitchen, while cooking rice we will press 3-4 grains of rice between our figures to take decision about the whole pot of rice. There we can’t test each and every grain in the cooking pot.

Even it is working when you are shopping, we will break one or two walnuts before buying few kilograms of it. We can’t break open all walnuts we want to buy.

It even decides the fate of our country, have you ever wonder how opinion polls predicts the winner of the election? It is just not possible to interview the entire population hence, a large random sample of people are selected and interviewed. Based on the analysis of the sample data, opinion predicts the winner. Everywhere you would encounter statistics whether you like it or not. Our starting salary, bonus, increment etc. all is influenced by statistics. Bank’s interest rate, inflation, stock market etc. works on statistics. Finally you will die rich or poor based on the average wealth you accumulated with respect to others. Even after death people will remember that you lived longer than the average life span of a human.

Statistics helps us in taking confident decisions when we have limited information or when can’t get all the information required.

All of us must have experienced the theory of probability in high school. Say we have a bucket full of twelve red and thirteen blue balls, now we draw a sample of three balls one after another and now I want to know the probability of withdrawing three consecutive red balls[1]. Here we have information about the population’s (bucket) composition and based on that we are trying to guess the characteristic of the sample.

Picture1In real world, the information about the population is difficult to obtained due to resource constrains[2]. Let’s try to figure out the average height of all the people in the country. Is it possible to approach each and every individual in the country with measuring tape? It’s just not feasible. Instead, the height of a large sample of people across the country are collected and based on the average height of the sample thus collected, we try to estimate the average height of the whole population, obviously with some margin of error.

This is statistics where we take sample and analyze it to predict the characteristics of the population but with some margin of error. It’s just works in opposite way to probability.

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Now we can understand the manufacturing atmosphere where we can take decision on accepting a production lot based on the analysis of a sample.

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[1] Probability of drawing three consecutive red balls = 12/25*11/24*10/23

[2] That’s why population census is conducted every 10 years, not every year as is involves reaching out to every individual in the country.

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