Table 20.2 Person score and measure distribution

TABLE 20.2

TABLE OF SAMPLE NORMS (500/100) AND FREQUENCIES CORRESPONDING TO COMPLETE TEST

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| SCORE MEASURE S.E.|NORMED S.E. FREQUENCY % CUM.FREQ. % PERCENTILE|

|------------------------+--------------------------------------------------|

| 0 -6.17E 1.83| 147 107 0 .0 0 .0 0 |

| 1 -4.86 1.08| 225 63 0 .0 0 .0 0 |

| 2 -3.94 .85| 278 50 1 2.9 1 2.9 1 |

| 3 -3.27 .79| 318 46 2 5.9 3 8.8 6 |

| 4 -2.64 .78| 355 46 2 5.9 5 14.7 12 |

| 5 -1.97 .83| 394 49 2 5.9 7 20.6 18 |

| 6 -1.19 .92| 440 54 3 8.8 10 29.4 25 |

| 7 -.23 1.00| 496 59 12 35.3 22 64.7 47 |

| 8 .80 .97| 557 57 5 14.7 27 79.4 72 |

| 9 1.72 .92| 610 54 4 11.8 31 91.2 85 |

| 10 2.55 .89| 660 52 1 2.9 32 94.1 93 |

| 11 3.37 .89| 707 52 2 5.9 34 100.0 97 |

| 12 4.21 .93| 756 54 0 .0 34 100.0 100 |

| 13 5.23 1.12| 817 66 0 .0 34 100.0 100 |

| 14 6.60E 1.84| 897 108 0 .0 34 100.0 100 |

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The columns in the Table of Sample Norms and Frequencies are:

Measures on the Complete Test:
SCORE	raw score on a complete test containing all calibrated items
MEASURE	measure corresponding to score
	If a person did not take all items or items are weighted, then that person is stratified with the measure on the complete test nearest the person's estimated measure (as reported in Table 18), regardless of that person's observed score.
S.E.	standard error of the measure (model)
	The statistical information is (USCALE/S.E.)²
Statistics for this sample:
NORMED	measures linearly locally-rescaled so that the mean person measure for this sample is 500 and the sample measure sample standard deviation is 100. Equivalent to UPMEAN=500, USCALE=100/(Person S.D.)
S.E.	standard error of the normed measure (for a score based on all the items)
FREQUENCY	count of sample with measures at or near (for missing data) the complete test measure
%	percentage of sample included in FREQUENCY
CUM.FREQ.	count of sample with measures near or below the test measure, the cumulative frequency.
%	percentage of sample include in CUM. FREQ.
PERCENTILE	mid-range percentage of sample below the test measure, constrained to the range 1-99 for non-zero frequencies.

Logit measures support direct probabilistic inferences about relative performances between persons and absolute performances relative to items. Normed measures support descriptions about the location of subjects within a sample (and maybe a population). Report the measures which are most relevant to your audience.

This Table is easy to paste into Excel. Use Excel's "data", "text to columns" feature to put the scores and measures into columns.

Example: I want to stratify my sample into low, medium, high ability groups.

The separation index is based on the statistical fiction that your data accord exactly with a normal distribution and that the average measurement error (RMSE) precisely summarizes the precision of your data. In practice, these assumptions are only met approximately.

If your data are "complete" (everyone responds to every item), then a convenient places to start is Table 20.2. Starting at the lowest score, look down the scores until you find the score that best characterizes (as a first guess) your "low group". Then mentally multiply its S.E. by 3, and add it to the measure for the low group. This will take you to the measure for the middle group, which will be approximately statistically significantly different (p<.05) from the low group. Do the same again for the middle group, and it will take you to the high group. Same again may take you to an even higher group, or up into the outliers at the top of the test. The cut-points will be half-way between the group centers that you have identified.

Do the same process from the top score downwards for another version of the stratification.

Then synthesize the two stratifications by adjusting the group by moving their center scores further apart (not closer together).