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53. Mark S.  Aug 13, 5:40 pm     show options
Newsgroups: sci.math,, alt.feminism, sci.stat.math
From: "Mark S." <> - Find messages by this author
Date: Sat, 13 Aug 2005 17:40:40 -0700
Local: Sat, Aug 13 2005 5:40 pm
Subject: Re: Standard Deviation of PISA
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I believe that you have misinterpreted the data reported on page 5 of that
report.  The "standard deviations" reported on page 5 appear to "standard
deviations of the mean", not the standard deviation of the individual scores
that went into computing the mean.

The standard deviation of the mean is equal to SD/sqrt(n), and is a measure
of how accurately the mean has been determined, not the spread in the data.

If you read further in the report, you will find that the spreads in the
data for individual countries are much larger, as would be expected by
anyone familiar with test data.


Dr. Mark H. Shapiro
Editor and Publisher
The Irascible Professor



How do I use statistics?

The main screen’s Stats menu contains the Sample Size and Margin of Error options. Other statistics are available in analysis.

Sample Size lets you determine the right sample size for your work and Hypothesis Testing lets you test whether two groups are statistically the same or whether chance can explain the observed differences.

Sample Size / Margin of Error:

If you want to know the minimum sample size required to supply a specified level of confidence and accuracy, use the Sample Size option. For the hypothesis you are testing you will need to estimate the proportion of the population giving a response. For example, if you expect somewhere between 20 to 30 percent of the population to give a specific response to your question, then Sample Size will show you the minimum sample size if 20 percent respond, or 30 percent. The larger of the two sample sizes will be the one that would assure you of the desired precision.

Supply first the level of confidence and the margin of error that you want your sample size to attain. Then for each value of response in favor that you supply, Sample Size will show you the minimum sample required to achieve that level of accuracy.

Basically, you will supply three pieces of information (responses in favor, margin of error, and level of confidence). The Sample Size command then provides the minimum sample size required to meet those three conditions.

Sample Size is arranged in convenient fashion to let you change any one of your input parameters, with the result displayed on the same screen. You are able to quickly change parameters to study different scenarios. This command very efficiently lets you explore the implications of different parameters. Below is the Sample Size screen.

You may display multiple scenarios at the same time by selecting Edit | Add column from the menu. You may have as many scenarios as fit on the screen.

Sample size / Margin of Error Definitions:

Population Size:The number of people about whom you are gathering information. This could be the number of people who work for your company, or the number of customers your company has. There is very little difference between a sample of twenty thousand and a sample of two billion, so if you are working with a large group, having an exact number doesn’t matter.
Sample:The set of all responses.
Sample Size:The number of responses you have collected (or will collect) in your database. This depends on population size, response in favor, confidence level, and margin of error.
Margin of Error:The amount by which your results are likely to be wrong. If the number of positive responses is 72%, then the real number of responses (which you are likely to get if you ask every single person) ranges from 67% to 77% (which is 72 � 5).
Response in Favor:The percentage of "positive" responses. If you’re looking at people who eat corn chips, and 72% of your sample claims to eat corn chips, then the response in favor is 72.
Standard Deviation:Standard deviation is defined as: sqrt(avg(sqr(x)) - sqr(avg(x))). A small standard deviation that most of the responses in the sample are similar. A large standard deviation means that the responses are not similar.
Confidence:For a confidence level of 90%, if a survey were repeated 100 times, then the response in favor will vary by less than the margin of error 90 of those 100 times. This is a measure of how accurate your measurement is.
Standard Error:Standard error is similar to standard deviation, except that the avg(x) function is replaced by f(x)=sum(x)/[count(x) + 1]. Thus, the standard error is almost the same as standard deviation when the sample size is large.

Hypothesis Testing:

A hypothesis test compares the means of two groups to determine whether they come from the same population, at some given level of confidence.

The summary results are displayed in the screen template. Initially, they reflect the results from the preset values of the parameters. You may change the Confidence Level, Hypothesis Value, Sample Size, and Response Rate parameters as appropriate to your test.

Hypothesis Value is the difference between the two population means (Group A minus Group B). Its preset value is zero, implying that the two populations are the same. As an example, if the hypothesis were that Group A is larger than Group B by a value of 10, then the number 10 would be supplied for this value.

The Confidence Level is default set at a value of 90 percent. You may want to change this value to 95 or 99 percent if you want a higher level of confidence.

The Type of Test is set to two-sided test. If you want to change this to either a left-sided or right-sided test, depending upon the hypothesis, then select the appropriate levels of confidence to match your hypothesis. (Remember that the hypothesized value is testing for Group A minus Group B when you select for left-sided or right-sided test.)

For number values, you will need to supply additional values for each of the two groups, Group A and Group B. These will be Sample Mean and Standard Deviation/Standard Error.




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