2.  Deviation is a measure of difference for interval and ratio variables between the observed value and the mean.  The sign of deviation (positive or negative), reports the direction of that difference (it is larger when the sign is positive, and smaller if it is negative).  The magnitude of the value indicates the size of the difference.

3.  Measures of Deviation: 1-Standard Deviation - is the frequently used measure of dispersion: it uses squared deviations, and has desirable properties, but is not robust. 2-Average Absolute Deviation - sometimes called the "average deviation" is calculated using the absolute value of deviation – it is the sum of absolute values of the deviations divided by the number of observations. 3-Median absolute deviation - is a robust statistic which uses the median, not the mean, of absolute deviations. 4-Maximum absolute deviation is a highly non- robust measure, which uses the maximum absolute deviation.

4.  Consider a population consisting of the following eight values: 2, 4, 4, 4, 5, 5, 7, 9 The mean is: The deviation from the mean: d1= 2 – 5 = -3d3 = 5 – 5 = 0 d2= 4 – 5 = -1d4 = 7 – 5 = 2

5.  Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" exists from the average (mean, or expected value). A low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data points are spread out over a large range of values.  How to Calculate the Standard Deviation ?

6. Let ‘X’ be a random Variable with a mean value ‘µ’ Then the Standard Deviation noted by ‘σ’ o To calculate the population standard deviation, first compute the difference of each data point from the mean, and square the result of each Next compute the average of these values, and take the square root:

7. Example of two samples with the same mean and different standard deviations. Each sample has 1000 values drawn at random from a gaussian distribution with the specified parameters.

9.  The choice of measure of central tendency, m(X), has a marked effect on the value of the average deviation. For example, for the data set {2, 2, 3, 4, 14}:

10.  The median absolute deviation is a measure of statistical dispersion. It is a more robust estimator of scale than the sample variance or standard deviation, being more resilient to outliers in a data set than the standard deviation.  The median absolute deviation is the median of the absolute deviation from the median.  For a univariate data set X 1, X 2,..., X n, the MAD is defined as the median of the absolute deviations from the data's median:

11.  For the example {2, 2, 3, 4, 14} 3 is the median,  so the absolute deviations from the median are {1, 1, 0, 1, 11} reordered as {0, 1, 1, 1, 11}: with median of 1 So the median absolute deviation MAD =1 in this case unaffected by the value of the outlier 14

12.  Consider the data (1, 1, 2, 2, 4, 6, 9) It has a median value of 2 The absolute deviations about 2(the median) are (1, 1, 0, 0, 2, 4, 7) Reordered: (0, 0, 1, 1, 2, 4, 7) So the median absolute deviation for this data is 1.

13. the variance is a measure of how far a set of numbers is spread out; The variance is computed as the average squared deviation of each number from its mean. Let ‘X’ be a random Variable with a mean value ‘µ’ The Variance is given by

14.  For example, for the numbers 1, 2, and 3, the mean is 2 and the variance is:

15.  Consider a perfect die when thrown has the expected value Its expected absolute deviation – the mean of the absolute deviations from the mean – is But its variance (expected squared deviation )

17. Issa Jawabreh, Amjad Sabri