Statistical Analysis

Statistical Deviation

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.

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.

Deviation Consider a population consisting of the following

eight values: 2, 4, 4, 4, 5, 5, 7, 9The mean is:

The deviation from the mean:d1= 2 – 5 = -3 d3 = 5 – 5 = 0d2= 4 – 5 = -1 d4 = 7 – 5 = 2

1-The Standard Deviation

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 ?

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:

Example

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.

2-Average Absolute Deviation In statistics, the absolute deviation of an element

of a data set is the absolute difference between that element and a given point. Typically the point from which the deviation is measured is a measure of central tendency, most often the median or sometimes the mean of the data set.

The Average (Mean) Absolute Deviation is given by

Di = | xi − m(X) |

Di is the absolute deviation

xi is the data element

m(X) is the chosen measure of central tendency of the data set—sometimes the mean ’µ’, but most often the median.

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}:

3-Median Absolute Deviation 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 X1, X2, ..., Xn, the MAD is defined as the median of the absolute deviations from the data's median:

Example 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 1So the median absolute deviation MAD =1  in this case unaffected by the value of the outlier 14

Example 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.

Variance

• 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

Example

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

Example 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 )

Representation of Variance in the Gaussian Probability Distribution Funtion.

Thank You!

Issa Jawabreh, Amjad Sabri