# Summing z values and summing t values

Following Winer (1971) and Rosenthal (1987) the sum of z values divided by the square root of the number of terms being summed follows a Normal distribution. This approach known as *Stouffer's method* may be computed using this spreadsheet. For, 1 $$\leq$$ k $$\leq$$ K and k-th z value, $$z_text{k}$$

$$ \mbox{Combined z} = \frac{ \sum_text{k} z_text{k}}{\sqrt{K}}$$

Also, note that since chi-square(1) = $$z^text{2}$$ a sum of chi-squares *each having one degree of freedom (df)* may be obtained by square rooting each chi-square(1) and entering as a z-value.

Mosteller and Bush(1954) and Rosenthal (1987) also show how t statistics (each with degrees of freedom over 2) may be summed to obtain a combined t value. These computations may be performed using this spreadsheet. For 1 $$\leq$$ k $$\leq$$ K and k-th t ratio, $$t_text{k}$$ with degree of freedom $$df_text{k}$$

$$ \mbox{Combined t} = \frac{\sum t_text{k}}{\sqrt{ \frac{ \sum_text{k} df_text{k} }{df_text{k}-2} }}$$

Also, note that since $$F(1,df) = t^text{2}(df)$$ a sum of F statistics whose *first degree of freedom is one* may be evaluated by square rooting and treating as a t statistic shose degrees of freedom equal the second F statistic degree of freedom.

t statistics with more than 30 degrees of freedom closely resemble z values so there will be little difference in results summing t statistics with over 30 degrees of freedom and summing z values.

References

Mosteller F, Bush RR (1954) Selected quantitative techniques. In: Lindzey G, ed. Handbook of Social Psychology, Cambridge, Mass: Addison-Wesley, pp 289-334.

Rosenthal R (1987) Judgement studies design analysis and meta-analysis. CUP:Cambridge. The requisite chapter may be viewed here

Winer, B.J. (1971) Statistical principles in experimental design (2nd ed.). New York: McGraw-Hill. (1971 and 1991 editions of this book are available in the CBU library)