Essentials

204 ESSENTIALS OF STATISTICSthe SD is (NPQ) . Any value for X can be converted to a z score with the followingformula:z X NP (9.1)QNP(For small N—say, less than 40—a correction to the above formula is recommended,because the binomial distribution is not smooth and continuous. Thecontinuity correction involves reducing the absolute value of the numerator of Formula9.1 by .5.) Applying Formula 9.1 (with continuity correction) to our babiesexample, we get⏐X NP⏐ .5 ⏐8 10(.5)⏐ .5 ⏐8 5⏐ .5 2.5z 1.58 NPQ 10(.5)( .5) 2.5 1 .58From Table A.1 we can see that the area beyond 1.58 is .0571, which is very closeto the probability we found earlier by counting (.055). Even with N as small as 10the ND can serve as a pretty good approximation.The Sign TestThe binomial test, whether used in its exact form (adding up the appropriateprobabilities of the binomial distribution) or in terms of the ND approximation,has many applications in the social sciences involving dichotomous events (e.g.,does the gender balance among chief executive officers reflect a selection bias?).A particularly useful application of the binomial test is to evaluate the significanceof an RM or matched-pairs experiment in which the DV cannot be measured precisely.Imagine that you have some treatment for enhancing creativity in children.Children are matched in pairs based on the creativity of drawings they have alreadyproduced in an art class. Then, one member of each pair is selected at randomto get the new treatment, and the other member gets a control condition.Each child produces a drawing after the treatment or control condition is finished.It may not be possible to measure the creativity of each drawing in any preciseway, but it is reasonable to suppose that a panel of artists could decide foreach matched pair of children which drawing expressed the greater creativity(without knowing, of course, which child received the new treatment in eachcase).For, say, 20 pairs of children the data from the experiment would boil down tothe number of pairs (X ) for which the treated child produced the more creativedrawing. Say in this case that X equals 16. We can test the significance of this re-