Let X1, . . . , Xn be independent exponential random variables with respective hazard rates λ1, . . . , λn, and Y1, . . . , Yn be independent and identically distributed random variables from an exponential distribution with hazard rate λ. Then, we prove that X2:n, the second order statistic from X1, . . . , Xn, is larger than Y2:n, the second order statistic from Y1, . . . , Yn, in terms of the dispersive order if and only if
$$\lambda\geq \sqrt{\frac{1}{{n\choose 2}}\sum_{1\leq i < j\leq n}\lambda_i\lambda_j}.$$We also show that X2:n is smaller than Y2:n in terms of the dispersive order if and only if
$$ \lambda\le\frac{\sum^{n}_{i=1} \lambda_i-{\rm max}_{1\leq i\leq n} \lambda_i}{n-1}. $$Moreover, we extend the above two results to the proportional hazard rates model. These two results established here form nice extensions of the corresponding results on hazard rate, likelihood ratio, and MRL orderings established recently by Pǎltǎnea (J Stat Plan Inference 138:1993–1997, 2008), Zhao et al. (J Multivar Anal 100:952–962, 2009), and Zhao and Balakrishnan (J Stat Plan Inference 139:3027–3037, 2009), respectively.