# Relative Standard Deviation

Relative standard deviation is also called as coefficient of variance. The standard deviation discussed over is an absolute measure of dispersion.Karl Pearson developed this measure which is most commonly used for measure of relative variation.The coefficient of standard deviation is defined as ratio of standard deviation the mean of the data.CV is used to express the standard deviation as a percentage of a mean.It is used where we want to compare the variability of two or more series. The coefficient of variation is denoted by C.V. and is obtained as follows:

C.V.=S.D/Mean*100

Where,

S.D. =standard deviation

Note: lesser the CV,higher the consistency.

higher the CV,lesser the consistency.

 Worker 1 WORKER 2 S.D 30 MINUTES 25 MINUTES MEAN 6 MINUTES 4 MINUTES

According to table:fastest worker is worker 2 because he complete job in 25 minutes,and consistent worker is worker 2 because his CV is lesser than worker 1’s CV.

When the relative dispersion stated in the terms of standard deviation and arithmetic mean,the resulting percentage is known as coefficient of variation or coefficient of variability.

Comparison to standard deviation:

Advantage:The coefficient of variation or variability is being useful because the standard deviation of data must always be understood in context of arithmetic mean of the data.The real value of CV is independent of the unit in which measurement of data has been taken, so it is a dimensionless number.For comparison between data set with different mean and different units, a person would use CV instead of standard deviation.

Unlike standard deviation,CV cannot be directly used to construct confidence intervals for arithmetic mean.

When the value of arithmetic mean is close to zero, the CV will approach to infinity and is being sensitive to small changes in arithmetic mean.This is being a case in which value do not originate from ratio scales.

Applications:The coefficient of variance is commonly used in the field of probability such as renewal theory,reliability theory,queueing theory.In such fields,the exponential distribution is being more important than normal distribution. The standard deviation of an exponential distribution is equal to its arithmetic mean, so CV=1.Distribution in which CV<1,are considered low-variance and when CV>1,are considered high-variance.In these fields some formulas are expressed using squared coefficient of variation(SCV).While many natural processes show a correlation between amount of variation and average value around it, there is a need to design a accurate sensor device in such a way that coefficient of variation is close to zero.

What does relative standard deviation tells us?

It is a measure of precision. Relative standard deviation is sometime called as coefficient of variation CV,and calculated as a percentage.

S.D.=standard deviation

X=mean

RSD=S.D./X,as a percentage RSD=S.D.*100/X

The RSD allow standard deviation of various measurements to be compared more meaningfully.For ex. If one is measuring the concentration of 2 compounds X and Y,result is 0.5(+/-)0.4 ng/ml for compound X and 10(+/-)2 ng/ml for compound Y,one may look at standard deviation for compound X and say because its lower (0.4 vs. 2) than for Y, the measurement for X is more precise.When %RSD is used the new values for compound X and Y are 0.5(+/-)80% and 10(+/-)20%,therefore measurement of compound Y is more precise.

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