a symmetric graph is called normal or a bell curve We will focus on a normal distribution, also referred to as ‘the bell curve’ because of its smooth, bell-like shape. This means the graph is neither skewed left nor skewed right.

The normal distribution is symmetrical. This tells us that the mean is in the center of the normal distribution. A normal distribution is used when dealing with entire populations. For this reason, rather than using x-bar (used for the mean of a sample), we use μ (the Greek letter mu*) to represent the mean of the whole population.

Another key component of the normal distribution is the standard deviation. The standard deviation is the measure of how far a point is from the mean. When dealing with the normal distribution (a population), the symbol σ (the Greek letter sigma**) is used. The larger the standard deviation is, the greater the spread is.

*The greek letter μ is pronounced something like "mew" not like when a cow says, "moo".

**Pronounce the greek letter σ like the word "cigar" but the "ma" interrupts you before you can say the whole word...σ is pronounced "cig" "ma". You may already be familiar with sigma, because you may have seen it written as a capital letter like this: $Σ$ .

Each distribution below shows the mean (μ) and one standard deviation above the mean (μ + σ).


Smaller Spread, smaller σ This produces a narrow curve.	Bigger Spread, bigger σ This produces a shorter, but wider curve.

a bell curve showing three standard deviations above and below the mean A shorthand used in statistics is "N(μ, σ)".

For example, if you see N(5, 1.2) this means you are looking at a normal distribution with a mean of x = 5 (the center) and with a standard deviation of 1.2.

It is custom to label 3 standard deviations above the mean, and 3 standard deviations below the mean (as shown to the right). Each standard deviation above the mean is added to μ and each standard deviation below the mean is subtracted from μ.

a bell curve showing a mean of 30 and 3 standard deviations above and below the mean Consider another example.

N(30,5) is a normal curve with a mean of 30, and a standard deviation of 5. The mean goes in the middle, then count by 5s in both directions. Sketch and label this in your notebook and be sure to write "N(30,5)" next to it.
In your notebook, practice by sketching the following distributions. Label the horizontal axis for 3 standard deviations above and below the mean as shown in the example to the right.
- 1) N(100,20)
- 2) N(9, 1.5)
- 3) N(–4, 0.5)
- Check the sample solution and compare it to yours.

This images on this page were adapted from OpenStax College. Download for free at http://cnx.org/contents/30189442-6998-4686-ac05-ed152b91b9de@18.13