The Earth's mantle transition zone could potentially store a large amount of water, as the minerals wadsleyite and ringwoodite incorporate a significant amount of water in their crystal structure. The water content in the transition zone can be estimated from the electrical conductivities of hydrous wadsleyite and ringwoodite, although such estimates depend on accurate knowledge of the two conduction mechanisms in these minerals (small polaron and proton conductions), which early studies have failed to distinguish between. Here we report the electrical conductivity of these two minerals obtained by high-pressure multi-anvil experiments. We found that the small polaron conductions of these minerals are substantially lower than previously estimated. The contributions of proton conduction are small at temperatures corresponding to the mantle transition zone and the conductivity of wadsleyite is considerably lower than that of ringwoodite for both mechanisms. The dry model mantle shows considerable conductivity jumps associated with the olivine-wadsleyite, wadsleyite-ringwoodite and post-spinel transitions. Such a dry model explains well the currently available conductivity-depth profiles obtained from geoelectromagnetic studies. We therefore conclude that there is no need to introduce a significant amount of water in the mantle transition to satisfy electrical conductivity constraints.