A Day With New Concepts of Statistics - 19th July , 2013
Today's class began with the introduction of PERMAP.
PERMAP is a free, Windows-based, real-time interactive program for making perceptual maps (also called product maps, strategic maps, sociograms, sociometric maps, psychometric maps, stimulus-response maps, relationship maps, concept maps, etc.). Its fundamental purpose is to uncover any "hidden structure" that might be residing in a complex data set.
PERMAP is a program that uses multidimensional scaling (MDS) to reduce multiple pair wise relationships to 2-D pictures, commonly called perceptual maps. The Churchill data are in the form of correlation coefficients that show the relationships between 10 factors that influence the image of a department store. These correlation coefficients were calculated from responses to semantic differential scale questions given to a random selection of shoppers.
Purpose of PERMAP. The use of MDS for the construction of perceptual maps is well developed and several computer programs are available. In fact, MDS was one of the earliest uses of high-speed computers in psychology and the social sciences. 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.
Usefulness of perceptual maps: A major advantage of MDS and 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.
Although experts may be able to extract the subtle relationships represented in a matrix of numbers, this skill is not widespread. 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, MDS results are often unaffected. This is because it is not uncommon for there to be much redundancy in the information given by a complete matrix of dissimilarities.
Although experts may be able to extract the subtle relationships represented in a matrix of numbers, this skill is not widespread. 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, MDS results are often unaffected. This is because it is not uncommon for there to be much redundancy in the information given by a complete matrix of dissimilarities.
Existing perceptual mapping difficulties: Although the theory behind making perceptual maps is well developed, its application has been controversial There are four major concerns that PERMAP can help alleviate. These include avoiding local minima (i.e., configurations that are optimal with respect to small changes in configuration but not optimal with respect to all possible changes), proving complete convergence, minimizing the influence of outliers, and combining multiple correlation matrices. With care, batch-operated programs can be used in such a manner that all of these difficulties are properly addressed, but moving to a visually interactive program renders these difficulties easier to deal with.
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.
PERMAP was designed to be simple and easy to use by a novice and to offer enough advanced features that it would be of value to the expert. Its major improvement over existing perceptual mapping programs is that it was designed specifically to combat certain common errors associated with multidimensional scaling. For instance, it is particularly effective at showing incomplete convergence, trapping by a local minima, and outlier influence. It is also effective at revealing the importance, or lack of importance, of the choice of the distance metric used in the objective function. Overall, the program provides a means for the researcher to go beyond just finding a solution to developing a feel for the suitability, stability, and variability of the solution.
Then We learned What is Z-Score and how to calculate it.
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.
In addition to showing a score's relationship to the mean, the Z-score shows statisticians whether a score is typical or atypical for a particular data set. Z-scores also allow analysts to convert scores from different data sets into scores that can be accurately compared to each other. One real-life application of z-scores occurs in usability testing.
Z-Score is used for normalization.
If a Z-Score….
ü Has a value of 0, it is equal to the group mean.
ü Is positive, it is above the group mean.
ü Is negative, it is below the group mean.
ü Is equal to +1, it is 1 Standard Deviation above the mean.
ü Is equal to +2, it is 2 Standard Deviations above the mean.
ü Is equal to -1, it is 1 Standard Deviation below the mean.
ü Is equal to -2, it is 2 Standard Deviations below the mean.
If you've encountered the z-score in a statistics book you usually get some formula like:
The above formula is for obtaining a z-score for an entire population. Usability testing obviously samples a very small subset of the population and thus the following formula is used:
Where x-bar and s are used as estimators for the population's true mean and standard deviation. Both formulas essentially calculate the same thing:
ETS publishes the means and standard deviations of a set of test takers on the GRE website.
By plugging in your scores you get the following:
Figure 1: Verbal Score
Figure 2: Quantitative Score
When testing software with users, task times are usually a good metric that will reveal the individual differences in performance. For task times there typically is only an upper spec limit. That is, it usually doesn't matter how fast a user completes a task, but it does matter if a user takes too long. For example, say you and your product team determined that a task should be completed in 120 seconds. 120 seconds becomes your Upper Spec Limit (USL). You sampled 10 users and got these task times:
In the case of task times, a negative process sigma is ideal--as you want more people completing the task below the task time, not above it. You can simply drop the negative when communicating the results in the event it causes confusion. If you were to make radical improvements to the UI and then sampled another set of ten users, here are more results:
In the redesign, the average of the new sample is well below the spec limit and the process sigma is now very high. The corresponding defect area is now only .01% and the quality area is 99.98%
Of course having users perform that much below the spec limit is not very common due to the inherent variability in user performance.
Calculating a Z-Score Example
For example, lets say you took the GRE a few weeks ago and got scores of 630 Verbal and 700 Quantitative. How good are these scores? Which is better, the Verbal or Quantitative score? Using a z-score can tell you how far you are from the mean and thus how well you performed. If you know the mean and standard deviations for a set of GRE test takers you can compare your scores.| Verbal | Quantitative | |
| Mean | 469 | 591 |
| StDev | 119 | 148 |
By plugging in your scores you get the following:
Verbal z = (630 - 469) ÷ 119 = 1.35σ
Quantitative z = (700 - 591) ÷ 148 = .736σ
To convert these sigma values into a percentage you can look them up in a standard z-table, use the Excel formula =NORMSDIST(1.35) or use the Z-Score to Percentile Calculator (choose 1-sided) and get the percentages : 91% Verbal and 77% Quantitative. You can see where your score falls within the sample of other test takers and also see that the verbal score was better than the quantitative score. Assuming the sample data was normally distributed, here's how the scores would look graphically:Figure 1: Verbal Score
Figure 2: Quantitative Score
Z-Scores and Process Sigma
An interactive Graph of the Standard Normal Curve similar to Figures 1 & 2 is available for you to visualize how the z-scores and the area under the normal curve correspond. The graphs also allow you to see the difference between one and two-sided (also called two-tailed) areas. In Six Sigma the process sigma metric is derived using the same method as a z-score. However, in Six Sigma you are measuring the distance a sample mean is above a specification limit--there can be an upper and lower spec limit that a sample must fall between as well. As in the z-score, you still use the same normal-deviates from the z-table to approximate the area under the curve. The process sigma metric is essentially a Z equivalent.| Sample |
| USL: 120 Mean: 104 StDev: 12 |
To calculate the process sigma you subtract the mean (104) of the sample from the target (120) and divide by the sample standard deviation (12). For Sample 1 the process sigma is -1.32σ. The visual representation of the data can be seen below:
In the case of task times, a negative process sigma is ideal--as you want more people completing the task below the task time, not above it. You can simply drop the negative when communicating the results in the event it causes confusion. If you were to make radical improvements to the UI and then sampled another set of ten users, here are more results:
| Sample 2 |
| 60 75 99 88 65 72 75 72 87 65 |
| USL: 120 Mean: 75.8 StDev: 12.14 |
In the redesign, the average of the new sample is well below the spec limit and the process sigma is now very high. The corresponding defect area is now only .01% and the quality area is 99.98%
Of course having users perform that much below the spec limit is not very common due to the inherent variability in user performance.
Next lecture began with world95 file
This file contained data about countries, their population, birth rate, daeth rate etc.
First, we made the histogram for Population of countries.
Then we transformed the population to ln(population) to get normal graph. It is done so that larger values become small.
Last Thing we learned Was Bubble Graph
A bubble chart is a type of chart that displays three dimensions of data. Each entity with its triplet (v1, v2, v3) of associated data is plotted as a disk that expresses two of the vi values through the disk's xy location and the third through its size. Bubble charts can facilitate the understanding of social, economical, medical, and other scientific relationships.
Bubble charts can be considered a variation of the scatter plot, in which the data points are replaced with bubbles.
When to use a Bubble chart
Bubble charts are often used to present financial data. Use a Bubble chart when you want specific values to be more visually represented in your chart by different bubble sizes. Bubble charts are useful when your worksheet has any of the following types of data:
- Three values per data point Three values are required for each bubble. These values can be in rows or columns on the worksheet, but they must be in the following order: x value, y value, and then size value.
- Negative values Bubble sizes can represent negative values, although negative bubbles do not display in the chart by default. You can choose to display them by formatting that data series. When they are displayed, bubbles with negative values are colored white (which cannot be modified) and the size is based on their absolute value. Even though the size of negative bubbles is based on a positive value, their data labels will show the true negative value.
- Multiple data series Plotting multiple data series in a Bubble chart (multiple bubble series) is similar to plotting multiple data series in a Scatter chart (multiple scatter series). While Scatter charts use a single set of x values and multiple sets of y values, Bubble charts use a single set of x values and multiple sets of both y values and size values.
Then we made bubble chart for world95 file used earlier.
We used this table to make the bubble chart and we made the bubble chart in excel.
Number of people / Birth rate per Death rate per
sq. kilometer 1000 people 1000 people
Mean Mean Mean
Region or OECD 108.0 13.0 10
economic East Europe 76.7 13.4 11
group Pacific/Asia 802.5 26.3 9
Africa 62.2 42.0 15
Middle East 126.7 32.7 6
Latn America 88.3 26.9 7
With this we ended the today's lecture.
Neeraj Garg
Pallavi Gupta
Pallavi Gupta
Piyush
Prerna Bansal
Priya Jain
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