Normal Distribution Explaned

Normal Distribution

Data can be "distributed" (spread out) in different ways.

It can be spread out more on the left		Or more on the right

Or it can be all jumbled up

But there are many cases where the data tends to be around a central value with no bias left or right, and it gets close to a "Normal Distribution" like this:

A Normal Distribution
The "Bell Curve" is a Normal Distribution.
And the yellow histogram shows some data that
follows it closely, but not perfectly (which is usual).

It is often called a "Bell Curve" because it looks like a bell.

Many things closely follow a Normal Distribution:

• heights of people
• size of things produced by machines
• errors in measurements
• blood pressure
• marks on a test
  

We say the data is "normally distributed":

The Normal Distribution has:

• mean = median = mode
• symmetry about the center
• 50% of values less than the mean
  and 50% greater than the mean
  

Standard Deviations

The Standard deviation is a measure of how spread out numbers are (read that page for details on how to calculate it).
When we calculate standard deviation find that (generally):

68% of values are within 1 standard deviation of the mean 95% of values are within 2 standard deviations of the mean 99.7% of values are within 3 standard deviations of the mean

Example: 95% of students at school are between 1.1m and 1.7m tall.

Assuming this data is normally distributed can you calculate the mean and standard deviation?
The mean is halfway between 1.1m and 1.7m:
Mean = (1.1m + 1.7m) / 2 = 1.4m
95% is 2 standard deviations either side of the mean (a total of 4 standard deviations) so:

1 standard deviation	= (1.7m-1.1m) / 4
	= 0.6m / 4
	= 0.15m

And this is the result:

It is good to know the standard deviation, because we can say that any value is:

• likely to be within 1 standard deviation (68 out of 100 should be)
• very likely to be within 2 standard deviations (95 out of 100 should be)
• almost certainly within 3 standard deviations (997 out of 1000 should be)