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We can summarize these distributions by discussing their
The ViSta summary report presents measures of each of these characteristics of a variable's distribution. For these four variables, the report is:
range = URL(Max) - LRL(Min)
where URL(Max) stands for "Upper Real Limit of the maximum score" and LRL(min)means the "Lower Real Limit of the minimum score".
If the distribution consists of whole numbers (integers), the range is
range (of integers) = Max - Min + 1
Problem: Because the range doesn't consider all of the scores in the distribution, only the extremes, it often does not give an accurate description of the variability.
ViSta's Range: ViSta calculates the range as simply
range = Max - Min
ViSta leaves off the term dealing with real limits. This is another way the range is commonly defined.
Example:For the variables in our class data (see table above), the ranges are:
Questions:
We begin defining the interquartile range by first defining quartiles:
IQR = Q3 - Q1
SIQR = (Q3 - Q1) / 2
Evaluation: The IQR and SIQR are more stable than the Range because they focus on the middle half of the scores and, therefore, can't be influenced by extreme scores. However, the actual value of the scores aren't used, which would be an improvement.
Example:For the variables in our class data (see table above), the IQR values are:
Questions:
Examples
Consider, once again, the four variables in our class survey data. Here are their means and standard deviations:
One interprets the mean as showing that the typical age is about 20 1/4 years (20 years and 3 months). The standard deviation shows that the average person is within 1 3/4 years (1 year and 9 months) of that age. That is, most people in class are between 20.24-1.77= 18.47 (18 1/2 years old) and 20.24+1.77=22.01 years old (between 18 1/2 and 22).
The mean tells us that the average GPA is 3.06, corresponding to just above a B average. The standard deviation tells us that, on the average person has a GPA that is between 2.62 and 3.50. In other words, the typical GPA range is between a B- and B+/A-. Not bad!
The mean tells us that the average SAT score on the Math section is about 590 (which is certainly better than the average score in the whole population). The standard deviation tells us that, the typical person has a SAT score on the Math section that is between 495 and 685, or, roughly, between 500 and 700. This seems like a fairly big variation, suggesting that some students did quite a lot better than others on SAT Math.
Typically, you rated your satisfaction with your experience at UNC averages about 7, which is above the middle of the scale (which was 5). Furthermore, most of you rated your experience in the 5 to 9 range, which is from the middle of the scale to the top of it. Its hard to know what this means, exactly, since we don't have a well defined reference for the scale, as we do for the other variables.
Don't you feel as though you've learned more about the variability of the scores on the four variables discussed above than you did from the range or IQR?
This is because it uses every score in the distribution to come up with a value for the variation in the scores, not just two scores (as for the range) or some of the scores (as for the IQR).
Also, the standard deviation and variance are very much involved in inferential statistics, whereas the other measures are not involved.
For these reasons, we will see these measures repeatedly throughout the book.
Check out the HyperStat site. Pay particular attention to the first two chapters, especially the one on Describing Univariate Data.
Use the Histogram Explorer to get a better understanding of histograms and distributions. Follow the Basic Instructions given there. Use the Practice Guessing.
Also try this interactive demonstration of how to calculate the standard deviation and variance.