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Fundamentals of Statistics 2: The Normal Distribution :: Converting Percentages to Z-scores
We saw how we can convert raw scores to z-scores then find the area under the curve which gives us the probabilities above, below and between values. There are times where we want to know what values would account for say the top 5% of a population.

Photo By: HeadGeekette

For example, maybe we want to know what we'd need to score to be considered for the Mensa High IQ society. You're in Mensa if you score in the top 2% of IQ scores. So what would we need to score?

To find out we work backwards. We first need to find out what the z-score is for the top 2%. Remember that the total area under the normal curve adds up to 1 (or 100 percent when we're working with percentages). So we want the z-score for .98. To find it we need to use an inverse normal table or the excel function = NORMSINV(.98) and we get the z-score of 2.054.

Ok, so we need a score that is 2.054 standard deviations above the mean. We know our mean of 100 and standard deviation of 15 so we can just plug these values into our z-score equation and solve for the unknown score, which will represent as the variable x. We need to use a little bit of algebra and solve for the unknown x by isolating it in our equation.
  1. 2.054 = (x - 100)/15
  2. 2.054*15 = x-100 
  3. 30.80623 = x -100
  4. 30.80623+100 = x
  5. 130.8  = x
So we'd need to score just about a 131 to get into that oh so desirable Mensa club and hang out with the rest of the geeks.

If you need more practice with finding areas with z-scores, I put together a Crash Course in Z-scores which provides a visual and easy-to-follow guide with plenty of practice understanding the normal curve and z-scores.

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