As the number of measurements increases the mean approaches the true value asymptotically, but for the mean to equal the true value, you would have to take an infinite number of measurements. The second reason is that the mean approaches the true value if and only if the only sources of error are random. In practice, we just assume that the mean equals the true value, and for most work, that's just fine. The mean of a set of measurements indicates the center of the normal distribution. The width of the normal distribution is given by the standard deviation.
Standard deviation is a measure of how widely a series of measurements is spread around the mean.
Measurements that are widely spread apart have a large standard deviation. That is, if the standard deviation is very small, it might take only a handful of measurements before the mean gives a very good estimate of the true value. When the standard deviation is large, many more measurements must be taken before the mean gives as good an estimate of the true value. Two related concepts, defined below, are precision and accuracy. In the figures which follow, imagine that the center of the target represents the true value of some observable that interests us, and that the crosses are individual measurements of that observable.