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Statistics Definitions

 
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John



Joined: Jun 13 2005
Posts: 23

PostPosted: Fri May 18, 2007 7:34 pm    Post subject: Statistics Definitions Reply with quote



Statistics Definitions, Generalized

Note: SIMS 2000 uses many of these statistics, and others, in the generation of the various reports available thru out the SIMS 2000 system. P-Value and Mean Separation are the generally most important statistics for our Sensory clients. This list of definitions is in no particular order.

Note: These are general definitions. For further information on these terms it is recommended to search the vast amount on information on the Internet. Such as www.support.sas.com/onlinedoc/913/docMainpage.jsp, www.ats.ucla.edu/stat/sas/output/sas_glm_output.htm, or www.ats.ucla.edu/stat/sas/whatstat/whatstat.htm.


Sample Standard Deviation:
The standard deviation is the most common measure of statistical dispersion,
measuring how widely spread the values in a data set are.
If the data points are close to the mean, then the standard deviation is small.
If many data points are far from the mean, then the standard deviation is large.
If all the data values are equal, then the standard deviation is zero.

Standard Error:
The standard error of a method of measurement or estimation is the estimated standard deviation
of the error in that method. Namely, it is the standard deviation of the difference between the
measured or estimated values and the true values. Notice that the true value is, by definition,
unknown and this implies that the standard error of an estimate is itself an estimated value.
In particular, the standard error of a sample statistic (such as sample mean) is the estimated
standard deviation of the error in the process by which it was generated. In other words, it
is the standard deviation of the sampling distribution of the sample statistic. The notation
for standard error can be any one of SE, SEM (for standard error of measurement or mean).

Confidence Interval (CI):
A confidence interval gives an estimated range of values which is likely to include an unknown
population parameter, the estimated range being calculated from a given set of sample data.
If independent samples are taken repeatedly from the same population, and a confidence interval calculated
for each sample, then a certain percentage (confidence level) of the intervals will include the
unknown population parameter. Confidence intervals are usually calculated so that this percentage is 95%.
SIMS 2000 calculates CI in Excel function CONFIDENCE(alpha, standard_dev, size), Ex. CONFIDENCE(0.05, 2.5, 50).
Alpha is the significance level used to compute the confidence level. The confidence level
equals 100*(1 - alpha)%, or in other words, an alpha of 0.05 indicates a 95 percent confidence level.

Variance:
The variance of a random variable is a measure of statistical dispersion, indicating how its
possible values are spread around the expected value. Where the expected value shows the location
of the distribution, the variance indicates the scale of the values. A more understandable
measure is the square root of the variance, called the standard deviation.

Analysis of Variance:
In statistics, analysis of variance (ANOVA) is a collection of statistical models,
and their associated procedures, in which the observed variance is partitioned into
components due to different explanatory variables. In its simplest form ANOVA gives
a statistical test of whether the means of several groups are all equal or different.
In SIMS 2000 the most common model utilized is the Two-Way AVOVA, aka Factorial ANOVA,
using two independent, variables, Sample and Judge. See SAS Proc GLM Model statements.

F-Value:
Calculation: Sample Mean Square / Error Mean Square, where Sample is the 'Source Variable'.
Tests the null hypothesis that source variable in the model does not explain a significant
portion of variance of the dependent variable, where the dependent variable is the Attribute.
F-distribution, symmetrical for Two-Way AVOVA, for a given significance level alpha, the null
hypothesis is rejected if the observed F is outside the distribution variance tail points.

P-value:
The p-value measures the probability (ranging from zero to one) that the result obtained
in a statistical test is due to chance rather than a true relationship between measures.
Small p-values indicate that it is very unlikely that the results were due to chance.
Therefore, if the p-value is small, statisticians would be confident that the result
obtained is 'real.' When p is less than .05 (P<.05)- meaning that there is a
less than 5% chance that the relationship is due to chance - statisticians usually
conclude that the relationship is strong enough that it is probably not just due to chance.
A p-value of .05 or less is the commonly used standard to determine that a relationship
between variables is significant. Then a multiple comparison procedure, a.k.a.
a mean separation procedure, is used to identify differences between pairs of means.

Mean Separation Procedure:
See P=Value above.
A mean separation procedure is used to identify differences between pairs of means.

Analysis of Variance (ANOVA):
Analysis of variance is a statistical method used for comparing several Sample means.
The ANOVA is used to test the null hypothesis that the population means are all equal.
With the ANOVA we can assess whether the observed differences among sample means are
statistically significant by examining the F probability (similar to P-value).
This is done by comparing the variation among the means of several groups with the
variation within groups. If we reject the null hypothesis that all sample means
are equal, we need to perform further analysis to draw conclusions about which
population means differ from which others. A multiple comparison procedure is used
to identify differences between pairs of means. A multiple comparison procedure
protects you from calling differences significant when they really aren't.
This is accomplished by adjusting the observed significance level for the
number of comparisons that you are making.

Two-Tailed test:
The two-tailed test is a statistical test used in inference, in which a given statistical
hypothesis will be rejected when the value of the statistic is either sufficiently small
or sufficiently large. The test is named after the 'tail' of data under the far left and
far right of a bell-shaped normal data distribution, or bell curve

One-Tailed test:
The one-tailed test is similar to the two-tailed test, but only tests in one direction.

Duncan's Multiple Range Test:
This test indicates significance of differences between samples by associated letters (a, b, c, etc.).
Most commonly referred to as 'mean separation analysis'.
Where any two sample means that do not share a common letter will differ at some significance level (eg P<.05)
Example Attribute Sweetness: S1 ab S2 a S3 bc S4 c
Samples: S2 and S3 differ, but S1 and S2 do not.

Tukey's Studentized Range Test:
Honestly Significant Difference (HSD)
This test compares each sample mean separately with each other sample mean.
The Tukey test can be selected for Complete Block tests only.

Dunnett's T Test:
The Dunnett test is similar to the Tukey test, but only compares the control sample (must be Sample 1)
with each of the other test samples, i.e., a Difference from Control test.

Binomial Analysis:
Binomial Analysis of test data will be performed for discrimination attributes--duo trio, triangle,
or paired comparison. Related terms include 'Null Hypothesis,' Type I (Alpha) and Type II (Beta) Error,
and One and Two-Tailed Analysis.

Friedman T-test Statistic:
Friedman Statistics generates a nonparametric analysis, and is primarily used for Sample Ranking Analysis SIMS 2000
Defined as follows: T = ( (12/bt(t+1)) * (R) ) - 3b(t+1)
where b = Number of Panelists, t = Number of Samples Presented to Each Panelist, and (R) = sum of squared rank sums
The Friedman statistic is applicable for complete block experimental designs only.

Permutations and Combinations:
The permutation of a number of samples is the number of different ways they can be ordered;
i.e. which is first, second, third, etc. If you wish to choose some samples from a larger
number of samples, the way you position the chosen samples is important.
With combinations, on the other hand, one does not consider the order in which objects were
chosen or placed, just which objects were chosen.
You could summarise permutations and combinations (very simplistically) as
Permutations - position important (although choice may also be important)
Combinations - chosen important.
Note from Sensory Computers Systems, generally:
Permutations nicely randomize Sample Presention by default.
Combinations do not randomize Sample Presention by default.

Latin Square Design Randomization
SIMS 2000 utilizes latin square randomization techniques for many randomization needs in SIMS 2000.
See SIMS 2000 Experimental Design notes for more information and examples on Latin Square Designs.
Latin Square definition: A Latin square is an n × n table filled with n different treatments
in such a way that each treatment occurs exactly once in each row and exactly once in each column.
Here is an example: 1 2 3 / 2 3 1 / 3 1 2


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