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Range

 

Range is the simplest measure of variation in a set of data. It is equal to the largest value minus the smallest value. Range = maximum value - minimum value.

 

Standard Deviation

 

Standard Deviation is another measure of the spread of data. It is derived from the distance of each point in the sample from the sample mean. These distances are called deviations. Standard deviation is calculated by first calculating the sum of the square of the deviations (called sum of squares SS). SS is then divided by one less than the sample size. Now if we take the square root of this value, we arrive at the standard deviation (s). In other words, standard deviation is the positive square root of the variance Standard deviation cannot be less than zero. If it is zero, then all sample values are the same.

 

Variance

 

Variance is another measure of the spread of data. It is derived from the distance of each point in the sample from the sample mean. These distances are called deviations. Standard deviation is calculated by first calculating the sum of the square of the deviations (called sum of squares SS). SS is then divided by one less than the sample size to give us the variance of the data set.

 

Inter Quartile Range

 

Interquartile Range (IQR) is the difference between the third and the first quartile (Q3-Q1). Quartiles divides the sample into four equal parts. First quartile has 25% of the points below it and 75% above it. Similarly, third quartile will have 75% points below it and 25% above it.

IQR = value of point at Q3 - value of point at Q1.

 

 

 

An application oriented question on the topic along with responses can be seen below. The best answer was provided by Mohan P B on 31st August 2018. 

 

 

Question

Q. 88  Is standard deviation a superior measure of dispersion as compared to range and interquartile range? Are there any specific scenarios where you will choose to use range instead of standard deviation?

 

Note for website visitors - Two questions are asked every week on this platform. One on Tuesday and the other on Friday.

 

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Standard Deviation indeed has a completeness when measuring dispersion in that every data point in the set is used in the calculation. Would this feature not definitely make it a generally better measure than others like Range, which touches only the two extremities of the data distribution and the Inter-Quartile Range (IQR) which touches just two other points in the distribution? Standard Deviation is a measure that indicates the spread of data from the mean, It is a measure that uses every value in the data set. Range, on the other hand, is the difference between highest and lowest value in a data set - hence, the measure uses only two values in the data set . Interquartile Range (IQR) also uses only two values from the entire data set, IQR = Quartile 3-Quartile 1.

 

 

Further more, two distributions with widely different variations can have the same range by virtue of having the same minimum and maximum values but it is very unlikely for two such widely different distributions to have the same standard deviation. Standard Deviation is not as hyper-sensitive to extremities and outliers as the Range. But Standard Deviation is not suitable when the sample size is very small.

 

This question can have only an affirmative answer, but that alone need not necessarily make Standard Deviation the best measure of dispersion under all circumstances. For example for non-normal processes, Range may be a more useful measure of dispersion than the Standard deviation, which is at its most relevant best in a Normal Distribution.

 

Additionally, when ordinal data are being dealt with, Range could be better suited.

 

Further, if at either ends of a distribution, there are open ended class intervals, the Range may be more appropriate.

 

IQR works better when there are outliers in the data as it discards the outliers by using the measure as IQR = Q3-Q1

 

Finally, when explaining concepts of variation to a diverse crowd consisting of staff at the lower levels of hierarchy, the Range may be easier to explain and understand.

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Measuring variation in Six Sigma can be done in many ways. Let us discuss about some of them (relevant to this context) here , as what are they and why they are needed, with some examples.

 

Standard Deviation Definition: This is the most accurate way of quantifying variation. It is an indicator of the degree of variation in a set of measurements or a process calculated by measuring the average spread of the data around the mean. This is a nice definition provided by Brue Howes, in his book for Six Sigma Green Belt.

Range Definition: This is easy for calculation as this is just about the spread of values and you calculate the difference between the highest and lowest value (to find the range value).

 

Inter Quartile Range(IQR): This portrays the values which lie in the middle 50% of your data . In other  words, it portrays the difference between 75th percentile(Q3) and the 25th Percentile(Q1). IQR=Q3-Q1. IQR works on the principle by dividing data set into Quartiles (Q1,Q2,Q3) .This uses a box plot as a graphing tool summarising

with 5 points that portrays the centring, spread and distribution of of a set of continuous data. The plot typically has a box, whiskers, Outliers. The 5 points that it shows are - maximum value, minimum value, median, Q3(75th percentile) and Q1(25th percentile). Whiskers are the lines drawn from the end of the box(on either side)

to the maximum and minimum values respectively.

 

Comparison of Standard Deviation, IQR, Range as which is better. Is Standard Deviation better than IQR and Range ?
Let us discuss as which is better than the other.


Standard Deviation : Generally, data distributions are of Normal distribution(Bell Shaped). Eg: Finding the average height of men and women in a college, finding the average marks in a classroom. But there may be cases where the data distribution can be skewed. Eg: when you want to measure current (power) supplied (in watts) in a

power fluctuation happening location and you have that data distribution, or the errors induced in a broadband service provider and so on .... So for skewed data and also for addressing outliers, Standard deviation is not robust.  

 

Inter Quartile Range: This is more robust than Standard Deviation for addressing outliers and also for efficiently addressing ordinal data,non-normal distribution data. Because we are able to calculate the highest and lowest value with the help of the IQR value(Q3 minus Q1), we are able to find the outliers with relative ease.
With respect to Skewdness, by comparing the quartiles values with median, we are able to decipher whether data is skewed or not.

 

Range: Range may not fair better in comparison with Standard Deviation. However, there are cases where Range is needed to be shown.

 

Eg:1: In Health sector (Hospitals), if the visiting Doctor needs the BP range(say as 80-110, 70-100,... ) of his patients.

 

Eg2: Providing Grade/Ranking Class in mark system. Distinction Class- Above 85%, First Class - 60-80%, Second Class: 50-60%; Third class-40-50%. This may not be in practice, these days across the Globe, but still similar grading is prevalent in some parts of the world.

 

Conclusion:

Standard Deviation, IQR, Range are ways of measuring variation. While Standard Deviation is better to use for normally distributed data, IQR is robust enough for working on skewed data and also for ordinal data and also very useful for addressing outliers. Also, Range can be effectively made used of in health sectors which are quite useful and other sectors as well.

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Range, no doubt is the simplest measure for dispersion. Range, however can mislead us when there are outlier in the sample, since only 2 extreme values are used for calculating range. We need not go into the advantages of using standard deviation, since most of us would know it.

 

However, in situations where we deal with small and equal sample sizes, the range will be a very ideal measure. One of the best examples that we have is the usage of range in an Xbar - R chart. Here, the samples are taken in the form of rational sub-groups. Each sub-group consists of a small, say around 4 nos,, but equal sample size. Such sample sizes will be too small for computing standard deviations. The concept of rational sub-grouping and very less time gap between the samples, reduces the possibility of outliers. However, even if we have outliers, those range values will stand out in the control chart and they will be removed during the 'homogenization' exercise. Hence range as a measure of variation can be used for such cases.

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Standard deviation is certainly more accurate measure of dispersion however the application of Standard Deviation is preferred in scenarios when the data are normally distributed as it is extremely sensitive to outlier observations (due to squaring of 'difference from the mean' in numerator). This is the reason that Stdv is used  along with simple mean. Another yet effective way of measuring dispersion is Mean absolute deviation which is sometimes used in studies as standard deviation is slightly more complex to comprehend. Range and Interquatile range are more applicable when the data are non normal.

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It is good to see some of the Excellence Ambassadors attempting all questions, and also great to see some new names on every question. The question asked for explanation on when is Standard Deviation Superior than Range and IQR and conversely, when is Range a better measure than Standard Deviation. The chosen best answer for this question is that of Mohan PB, because of completeness and brief explanation, too. Congratulations!

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