What does a high standard of living entail? This judgement is relatively subjective, but there are a number of factors that seems to be common to most economists' ideals. These include physical possessions, nutrition, health care, and life expectancy. The more prosperous an economy, the better off the citizens of that economy are in terms of material possessions and health. Thus, prosperity is attainable when wages are high and countries are highly productive.
This is not to say that prosperity is static. Instead, over time different countries becomes more and less prosperous. An economic boom in one country may bring temporary prosperity to that country. Similarly, a depression may wipe out some hard won gains in prosperity. Overall, prosperity is a relatively subjective judgement once the basic necessities of life are in place.
There is a scientific way of measuring prosperity that, while not fully descriptive, is useful in comparing the standard of living across countries. This is called the GDP per capita measure. This is simply calculated by dividing the nominal GDP in a common currency, say US dollars, by the total number of people in the country. This gives the average amount of income that each member of the population potentially has access to. In other words, the more money each individual is able to access the higher the potential standard of living.
This is a useful means of comparing economic wellbeing--that is, prosperity-- across countries. For instance, the GDP per capita in the US is around $25,000 while in Mexico it is around $7000. It stands to reason that by and large, the standard of living in the US is higher than the standard of living in Mexico. This same logic can be used to compare the standard of living between any countries.
As mentioned earlier, the GDP per capita measure is the nominal GDP divided by the population. Thus, for a give amount of output, a country with a smaller population will have a higher standard of living than a country with a larger population. This is a problem often encountered in countries with very low GDP per capita measures of the standard of living. When GDP grows slowly and the population increases rapidly, the GDP per capita and thus the standard of living tends to decline over time. Thus, a major way of increasing the standard of living in a country is to control the population growth rate and thus increase the GDP per capita.