Comparison Tests

Comparison Tests. These tests are used to compare averages to determine if there is a significant difference between two values.
  1. Comparing the sample to the true value. Method #1. The t-test is used to determine if there is a significant difference between an experimental average and the population mean () or "true value". This method is used to compare experimental results to quality control standards and standard reference materials. This comparison is based upon the confidence interval for the sample mean calculated above. If the difference between the measured value and the true value is greater than the uncertainty in the measurement, there is a significant difference between the two values at that confidence level. This may be expressed mathematically that IF:

    Then there is no significant difference at the stated confidence level. This could be stated as "there is no significant difference between the experimental results and the accepted value for the Standard Reference Materials at the 95% confidence interval."


  2. Comparing the sample to the true value, Method #2. This is same test as above, but it is often easier to understand the meaning of the test by calculating an experimental value for t (texperimental). Then the experimental t-score (t experimental ) is compared to t-critical (tc), the value of t found in a table. texperimental is calculated as follows:

    There is a significant difference between the sample average and the true value if texperimental is greater than tc. tc is chosen for N-1 degrees of freedom at the desired percent confidence interval. If the experimental value may be greater or less than the true value, use a two sided t-score. If specifically testing for a significant increase or decrease (but not both) use a single sided value for tc.


  3. Comparing two experimental averages. The t-test may also be used to compare two experimental averages. This is most accurately done by using the pooled standard deviation and calculating texperimental as:

    If texperimental is greater than tcritical then there is a significant difference between the two means. tcritical is determined at the appropriate confidence level from a table of the t-statistic for N1 + N2 - 2 degrees of freedom.


Student's t

P1 sided t.60 t.70 t.80 t.90 t.95 t.975 t.99 t.995
P2 sided t.20 t.40 t.60 t.80 t.90 t.95 t.98 t.99
df
1 0.325 0.727 1.376 3.078 6.314 12.71 31.82 63.66
2 0.289 0.617 1.061 1.886 2.920 4.303 6.965 9.925
3 0.277 0.584 0.978 1.638 2.353 3.182 4.541 5.841
4 0.271 0.569 0.941 1.533 2.132 2.776 3.747 4.607
5 0.267 0.559 0.920 1.476 2.015 2.571 3.365 4.032
6 0.265 0.553 0.920 1.440 1.943 2.447 3.143 3.707
7 0.263 0.549 0.896 1.415 1.895 2.365 2.998 3.499
8 0.262 0.546 0.889 1.397 1.860 2.306 2.896 3.355
9 0.261 0.543 0.883 1.383 1.833 2.262 2.821 3.250
10 0.260 0.542 0.879 1.372 1.812 2.228 2.764 3.169
-
20 0.257 0.533 0.860 1.325 1.725 2.086 2.528 2.845
-
inf 0.283 0.524 0.842 1.282 1.645 1.960 2.326 2.576