##### Analytics

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Once we have the mean we could easily see the difference in actual value with each member and mean value. But, as difference could be positive or negative. This is also referred as measure of spread “like range , variance, standard deviation IQR etc..”. for better understanding E.g suppose mean is 5 , one member has value 2 and other member as value 8. Both are having 3 point away from the mean. While performing sum operations on the difference the result will be a zero and making many further calculation difficult or even impossible.

The benefits of squaring include: (source: stack overflow)

- Squaring always gives a positive value, so the sum will not be zero.
- Squaring emphasizes larger differences – a feature that turns out to be both good and bad (think of the effect outliers have).

Squaring however does have a problem as a measure of spread and that is that the units are all squared, where as we’d might prefer the spread to be in the same units as the original data. Hence the square root allows us to return to the original units.

Thus to understand the deviation from the mean value we made a new term which is called standard deviation.

This is the most commonly used measure of the average spread or dispersion or deviation of data around the mean. The standard deviation is defined as the square root of the variance (V). The variance is defined as the sum of the squared deviations from the mean, divided by n-1.

Operationally, there are several ways of calculation:

Let’s understand n values xi. 1<i<n

Mean: SUM(xi)/n

Stdv.= sqrt( (xi -mean)^2/(n-1) )

n-1 is the degree of freedom for the member. (Probable place of value other than his own place)

The calculation of the mean and the standard deviation can easily be done on a calculator but most conveniently on a PC/laptop with computer programs which have simple ready-to-use functions. (some programs use n rather than n- 1!).

Variability Vs diversity: Std. Deviation is more about variability.

Diversity Set1 > Set 2

Variability Set1 < Set 2

Now the question is which value to be considered while exploring the data. First step is to remove outliers or consider the statistics which are independent from effects of such values. Such measures are known as robust statistics.

Robust | Non-robust | |

Center | Median (in some cases mode) | Mean |

Spread | IQR | SD , Range |

Robust statistics are used in skewed, with extreme observations, in case of symmetric non-robust are more important.