STATISTICS LECTURE- 19th July, 2013

We begun the session with the introduction on PERMAP, where we went on to say that: 

The program, PERMAP, uses conventional metric multidimensional scaling techniques. That is, it uses pairwise numerical values (correlations, proximities, dissimilarities, etc.) to construct a map showing the relationship between objects. A unique feature of PERMAP is that it embeds the mapping techniques in an interactive, graphical system that minimizes several difficulties associated with multidimensional scaling practices. It is particularly effective at exposing artifacts due to local minima, incomplete convergence, and the effects of outliers. It can associate various attributes with the resultant groupings and provide line-linking options to help the researcher identify the nature of perceived relationships. Problems involving multiple matrices can be treated using three different aggregation methods. The optional use of weighting factors is available.

 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. 

Then we moved on to discuss on a new phenomenon known as Z-Scores, which was very much in continuation with our previous learnings of mean & standard deviation. 

A Z-Score is 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.

Sometimes we want to do more than summarize a bunch of scores. Sometimes we want to talk about particular scores within the bunch. We may want to tell other people about whether or not a score is above or below average. We may want to tell other people how far away a particular score is from average. We might also want to compare scores from different bunches of data. We will want to know which score is better. Z-scores can help with all of this.

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

How to 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 -

This was followed by a video session by TED TV India, where the potential of India and China was shown, as compared to the Western economy. And the sessions ended on this note. 

WRITTEN BY- Noopur Mandhyan

GROUP MEMBERS-

PRANAV SHARMA
PAYAL SINGH

OMKAR GUJAR
RADHIKA AGARWALL