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Home Introducing the Normal Distribution The Normal Distribution and Z-scores The Standard Normal Distribution and Z-scores Finding Probabilities Associated with a Z-score Why do you need a table to find probabilities with z-scores? Above, Below and Between Probabilities Converting Percentages to Z-scoresView All Tutorials
 The Normal Distribution and Z-scores
Fundamentals of Statistics 2: The Normal Distribution :: Introducing the Normal Distribution
The normal distribution is the most important distribution in statistics. It describes a symmetric bell-shaped distribution. People's heights, weights and IQ scores are all roughly bell-shaped and symmetrical around a mean. This bell-shaped pattern is seen a lot and is why it gets the name normal. Most statistical tests in some way assume data to be roughly normally distributed (even when they're not).

The normal distribution is actually a family of many different bell-shaped distributions. Each can be described by two parameters: the mean μ and standard deviation σ (recall that these are the most common ways of measuring the center and variability of a distribution).


For example, adult male heights are on average 70 inches  (5'10) with a standard deviation of 4 inches. Adult women are on average a bit shorter and less variable in height with a mean height of 65  inches (5'5) and standard deviation of 3.5 inches. If we took a large sample of men and women's heights and graphed the frequency of the heights we'd see something like the following:



When we remove the histogram we see just the different bell-shaped normal distributions.



You're probably never going to need the formula which will generate these graphs, but if you did they are all using the normal distribution probability density function:



μ is the mean,σ is the standard deviation of the population, π is approximately 3.1415 and e is approximately 2.17. The x would be a single value in the graph. For example, for an adult male of 72 inches is an x. Plugging in the values gets you a value of .088016, which is just one small sliver of the bell curve on the right side just past the center.  If you did this for say 100 or so heights then you'd get something that looked like a bell-curve. The density function is rarely used. What's really used a lot is the cumulative probability function which will be explored in future lessons.

 
Home Introducing the Normal Distribution The Normal Distribution and Z-scores The Standard Normal Distribution and Z-scores Finding Probabilities Associated with a Z-score Why do you need a table to find probabilities with z-scores? Above, Below and Between Probabilities Converting Percentages to Z-scoresView All Tutorials
 The Normal Distribution and Z-scores

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December 30, 2019 | Andrew Knighton wrote:
The normal curve makes sense, however the derivation of the formula or even some explanation for why that formula describes the normal curve is not given.
 
July 11, 2019 | Shane Koorbanally wrote:
not easy to comprehend
 
May 24, 2013 | Roger wrote:
The vertical scale is not specified. A frequency of 70 is meaningless! does it mean 70 per 1000 or what?
 
September 16, 2010 | stats.student wrote:
You say that this is "normal," but its difficult to understand what is meant by this without answering the question, as opposed to what? Im a little confused what non-normal graphs would look like...? Other than that, good explanation - thanks!