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Fundamentals of Statistics 2: The Normal Distribution :: Above, Below and Between Probabilities
The normal curve is very helpful. If we can show that our data is like the normal curve we can use its properties to find probabilities for events to occur. All we need to know is the mean and standard deviation of a population and we're ready to start.



Area Below and Above.

The easiest and most frequent thing we do is find probabilities of events less extreme or more extreme than an event. We do this by using the z-score. Let's take the example of IQ scores. Most IQ tests have means of 100 and standard deviations of 15. Let's say you take an IQ test and get the score of 125. Are you super smart or just mediocre ? (of course you're smart, you're reading this!). 

To find your relative standing we convert your score to a z-score using the formula :

 
 z = (125-100)/15 = 1.6667

The z-score is the number of standard deviations you are from the mean of 0 (recall that by subtracting the mean and dividing the result by the standard deviation you convert your data to the standard normal distribution which has a mean of 0 and standard deviation of 1).

So you're 1.6667 standard deviations above the mean. We look up 1.667 in a table of normal values, use the Excel function =NORMSDIST(1.6667) or use the online calculator and we get .952213. That means you're in the 95th percentile or you have a higher IQ score than about 95% of the population (you go!).



Because the area under the normal curve (or the total probability space) adds up to 1, we can find the percent of the population with a higher IQ score than you. We just subtract the score from 1 = 1-.952213 = .047787. So about 4.7% of the population has a higher IQ score than 125.  You can see how these are the areas under the normal in the figure above. The very small white area on the right is 4.7% of the area and the large green part to the left represents 95.22% of the area.

Area between

It is also often the case that we want to know what percent of the population will score between two points. For example, maybe you think many of your friends might score between say 115 and 125.  What percent of the population would we expect to score between these values? 

To find the area between two points we :
  1. convert each raw score to a z-score
  2. find the area for the two z-scores
  3. subtract the smaller area from the larger area


The z-scores are (115-100)/15 = 1 and we already calculated the z-score for 125 = (125-100)/15 = 1.6667. Looking up the areas we find .9522 and .8413. Subtracting the smaller from larger we get .9522-.8413 = .1109. Or about 11% of the population would score between a 115 and 125.

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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