Woodcock's Test Design Nomograph (RM 6:3 p.244) is a handy graph for obtaining the expected measurement precision of a uniform test. For those designing tests by computer, I suggest a simple formula that performs the same function.
For a uniform test of L items, with logit range W = (Dh -Dl ) between the hardest and easiest items, the average item spacing is given by D = W/(L-1). An estimated person logit standard error at test center is
SE = sqrt( D / Tanh (LD/4))
where tanh() is the hyperbolic tangent. If tanh() is not available to you, compute
tanh x = (exp(x) - exp(-x)) / (exp(x) + exp(-x))
For a 75 item test of range 3 logits, this estimates a central precision of 0.25 logits as does Woodcock's Nomograph.
This formula can be applied iteratively to find the number of items required for a particular precision. How many items are required for a test of 4 logits range to measure with a precision of 0.3 logits? Start with test length at 100 items, then increase or decrease the number of items till a match to the desired precision is found. Following this procedure, the formula estimates a test length of 60 items. This result is also in agreement with Woodcock's Nomograph.
(For further discussion of the design and use of uniform tests, see Wright & Stone's Best Test Design, 1978, p. 137-140, 144-146, 212, 214.)
Computerizing Woodcock's Nomograph, P Pedler Rasch Measurement Transactions, 1993, 6:4 p. 255
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