You just can’t knock down this Monster “Variance” —- Part-3

 

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If x & y are two variables, then irrespective of whether you add or subtract them the variance will always add up.

A store wants to know the mean and the variance of sales made by male and female customers in a day. He also wants to see the variance in case sales by both gender are added in pair randomly. Lastly he wants to analyze the mean and variance because of the gender effect (i.e. difference of means and variance). Data of sales in hundred dollars is given below

Picture38

Using Excel Sheet Mean is calculated by typing formula  =average(array)

Variance is calculated by typing formula  =var.s(array)

Array = column of data, Var.s = variance of sample

Picture44

But most surprising element is that, irrespective of whether you add or subtract the data, variance always increases. This monster will always raise its head. This is indicated by the resultant variance which is always greater than the individual variances.

Picture45

In general, the variance always gets added irrespective of whether we are adding or subtracting the individual variances.

Picture46

where ρ is the correlation coefficient between two variables.

If two random variables are not correlated or they are independent then, ρ = 0 and above formula will get reduced to

Picture47

 Try to calculate the variance for x+y and x-y, are you getting little bit different answer? use correlation coefficient into the equation!

Calculating correlation coefficient (ρ) in excel

Type formula in a cell  =correl(array1, array2)

array1 = column x, array2 = column y

Understanding the Monster “Variance” part-1

Why it is so Important to Know the Monster “Variance”? — part-2

 

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Why it is so Important to Know the Monster “Variance”? — part-2

 

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Variance occupies the central role in the six-sigma methodology. Any process whether from manufacturing or service industry has many inputs and the variance from each input gets add up in the final product.

Hence variance has an additive property as shown below

Picture32

 Note: you can add two variances but not the standard deviations

Consequence of the variance addition and six sigma

Say if a product/services which is the output of some process, which in turn have many inputs. Then the variance from the input (Picture41) and from the process (Picture42) adds up to give the final variance (Picture43) in the product/services.

DMAIC methodology of 6Sigma try to identify the inputs that contributes maximum towards the variance in the final product and once identified, its effect is studied in detail to minimize the variance from the input. This is done by reducing the variance in the input itself.

Example: if the quality of a input material used to manufacture a product is found to be critical, then steps would be taken to reduce the fluctuation of the quality of that input material from batch to batch either by requesting/threatening the vendor or by performing the rework of the input material at your end.

Related articles:

Understanding the Monster “Variance” part-1

You just can’t knock down this Monster “Variance” —- Part-3

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Understanding the Monster “Variance” part-1

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This is one of the ways of calculating the variability in the data set.  Variance helps us in understanding how the data is arranged around the mean. In order to do so, we need to calculate the deviation of each observation from the mean in the data set .

For example: following is the time taken by me during the week to reach the office. The deviation of each  observation from the mean  time is given below.

Picture3a

Now next step is to calculate the average deviation from the mean using well-known formula

Picture21

Note that the sum of all positive deviations = sum of all negative deviations which indicates that the mean divided the data set in two equal halves. As a result the sum of all deviation becomes zero, hence we need some other way to calculate this average deviation about the mean.

In order to avoid the issue, a very simple idea was used

Negative number → Square of negative number → positive number → square root of this number → parent number

Hence square of all the deviations are calculated and summed-up to give sum of squares (simply SS) [1]. This SS is then divided by total number of observations to give average variance s² around the mean.[2] The square root of this variance gives standard deviation s, the most common measure of variability.

Picture4

What it physically means is that on an average data is deviating 7.42 units or simply one standard deviation (±1s) in either of the directions in a given data set.

Picture5

Above discussion about the sample standard deviation represented by s. For population, variance is represented by σ² and standard deviation by σ.

Picture40

The sample variance s² is the estimator of the population variance σ². The standard deviation is easier to interpret than the variance because the standard deviation is measured in the same units as the data.


 [1] Popularly known as sum of squares, this most widely term used in ANOVA and Regression analysis

[2] SS divided by its degree of freedom → mean sum of squares or MSE, these concepts would appear in ANOVA & Regression analysis.

Related articles:

Why it is so Important to Know the Monster “Variance”? — part-2

You just can’t knock down this Monster “Variance” —- Part-3

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Now it’s important to understand the concept of sigma or the standard deviation

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We have seen that we need to restrict the width of the car for a given width of the garage. This is analogous to the with of the process (voice of customer, VOP) Vs the width of the customer’s specification (voice of the customer or VOC).
The width of the process is measured in terms of standard deviation denoted by σ (sigma).

The target of the 6sigma methodology is to reduce this variance (width of the car) to such an extent that even by mistake it should not cross the customer’s specification (or should not hit the wall of the garage).

Before we work towards reducing the σ, we should know about this monster very well as we will be encountering him at every step during the 6sigma journey.

There are two very important characteristics of any data set

Location and the spread of the data set.

Location represents the point in the data set where there is maximum clustering of the data –> Mean and median.

Spread represents the variability in the data set, there will be some observations that will be above the mean and there will be some that will be below the mean. Standard deviation σ measures the average spread of the data from the mean in either direction of the mean.

Office arrival time for last 5 days with average time are given below, deviation of each observation from the mean is also captured.

Picture3

Let’s calculate the average deviation

Picture21

Note that sum of all positive deviations = sum of all negative deviations which indicates that the mean divided the data in two equal halves.

Sum of all deviation itself becomes zero, hence we need some other way to calculate this average deviation about the mean.

 In order to circumvent the issue, a very simple idea was used

Square of negative number → positive number → square root of this number → ±parent number

Hence square of all the deviations are calculated and summed-up to give sum of squares (simply SS) [1]. This SS is then divided by total number of observations to give average variance around the mean.[2] The square root of this variance gives standard deviation s, the most common measure of variability.

Picture4What it typically means that “on an average data is 7.42 units (= 1 standard deviation ±1σ) in either direction of the mean in the given data set. Mean of the data set is at ZERO standard deviation.

If process a stabilized and normally distributed then following holds true

 Picture16

i.e. 99.7 % of the observation in the data set would be between ±3σ.

Now we can understand whey we have taken 12σ as the width of the garage and 6σ as the width of the car!

The concept of ‘σ’ is the most important concept in understanding 6sigma. If we can understand it, downstream we wouldn’t be having any problem in understanding other topics. At this moment one important point to be noted here is that the calculation of σ depend on the type of data or data distribution we are handling.

Calculation of mean and σ would be different depending on whether we are dealing with normal distribution, binomial distribution, Poisson distribution etc. The importance of this would be realized when we would be studying the various types of control charts. At that time we just have to remember that “we must calculate mean and σ according to the distribution”.


[1] Popularly known as sum of squares, this most widely term used in ANOVA and Regression analysis

[2] SS divided by its degree of freedom → mean sum of squares or MSE, these concepts would appear in ANOVA & Regression analysis.

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