Applied Business Statistics

SESSION 7 & 8..
Session started with an introduction to PERMAP.

PERMAP is a program that uses multidimensional scaling (MDS) to reduce multiple pair wise relationships to 2-D pictures, commonly called perceptual maps.

The purpose of PERMAP is to provide a particularly convenient method of producing perceptual maps and to do so in a way that helps the researcher avoid a number of common mistakes, as described in following sections.

Uses: A major advantage of perceptual maps is that they deal with problems associated with substantiating and communicating results based on data involving more than two dimensions. They discussed the importance of graphical communications and the role of the eye in interpreting and distinguishing object (factor, stimulus, characteristic) grouping.

Another important aspect of perceptual maps is that they are forgiving of missing or imprecise data points. Whereas some analytical techniques cannot tolerate missing elements in the input matrix.The perceptual map simply presents the data in the two dimensions that best explain the variance.

PERMAP provides an interactive, visual system for the construction of perceptual maps from multidimensional dissimilarity data. It can treat up to 30 objects and can aggregate an unlimited number of matrices (cases) describing the pair wise differences or similarities among the objects. Aggregation can be accomplished using any of three methods, and the use of weighting factors is available.

Than we continued with a new topic named Z score.

An Introduction to Z SCORE -

Z-Score is basically a statistical measurement of a score's relationship to the mean in a group of scores. A Z-score of 0 means the score is the same as the mean. A Z-score can also be positive or negative, indicating whether it is above or below the mean and by how many standard deviations.
Each z-score corresponds to a point in a normal distribution and as such is sometimes called a normal deviate since a z-score will describe how much a point deviates from a mean or specification point.

If a Z-Score….

Has a value of 0, it is equal to the group mean.
Is positive, it is above the group mean.
Is negative, it is below the group mean.
Is equal to +1, it is 1 Standard Deviation above the mean.
Is equal to +2, it is 2 Standard Deviations above the mean.
Is equal to -1, it is 1 Standard Deviation below the mean.
Is equal to -2, it is 2 Standard Deviations below the mean.

This theory helps us to determine and compare individual scores from different bunches of data, standardize scores from different groups of data and than compare raw scores from different bunches of data.

NOW HOW TO CALCULATE Z SCORE-

Collect the samples of the variable of interest

Sample- If we have 25 kids of varying ages i.e 8, 7.5, 9, 8.5, 8, 7, 9.5, 8, 9, 6.5, 8.5, 9,8, 7, 8, 9, 8.5, 7, 6.5, 7, 8.5, 7, 9, 8, 8, 7.

Find the sample mean

total of the samples / no of samples = 199 /25= 7.96

Determine the standard deviation of the sample

Determine the variation of each sample from the mean by noting the difference in values of the 2 numbers. Subtraction can be used to determine these variances, but remember to change all negative values obtained this way into positive values as variance is defined as distance from the mean, regardless of whether the sample is below or above the mean. Square each individual sample variance and add the squared values together. Divide this sum of the squares by the number of samples used. The result is the sample variance. Take the square root of the variance. This square root is the standard deviation.

7.96 - 6.5 = 1.46

7.96 - 9.5 = -1.54

(1.54)^2 = 2.3716

(1.46)^2 = 2.1316

now dividing the sum total with 25.

4.5032 / 25 = 0.1801

So Standard Deviation = 0.4244

Calculate the Z scores

A Z score may be calculated for each sample. Subtract the sample group mean from the value of the individual sample of interest. Divide the result of that subtraction by the standard deviation of the sample group. The result of that division is the Z score of the chosen sample, indicating how many standard deviations away from the mean the chosen sample lies. Negative numbers are permitted, as the Z score not only gives the sample distance from the mean, but also indicates if the chosen sample lies below (negative Z score) or above (positive Z score) the mean.

6.5 - 7.96 = -1.46

-1.46 / 0.42 = -3.4762 i.e Z SCORE.

Than with the today's second session, we came to know about Bubble chart..

A bubble chart is a variation of a scatter chart in which the data points are replaced with bubbles, and an additional dimension of the data is represented in the size of the bubbles. Just like a scatter chart, a bubble chart does not use a category axis — both horizontal and vertical axes are value axes. In addition to the x values and y values that are plotted in a scatter chart, a bubble chart plots x values, y values, and z (size) values.

A graphical representation -