Question

What is a normal distribution? Draw and label a graph of a normal distribution and explain the relevant terms. How can the mean and standard deviation be used to predict outcomes according to this distribution?

Answer 1

The concept of the normal distribution curve is the most important continuous distribution in statistics. The normal distribution curve plays a key role in statistical methodology and applications.

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 the standard deviation we 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 Properties of a normal distribution

• 
  The mean, mode and median are all equal.
• The curve is symmetric at the center (i.e. around the mean, μ).
• Exactly half of the values are to the left of center and exactly half the values are to the right.
• The total area under the curve is 1.

As with any probability distribution, the parameters for the normal distribution define its shape and probabilities entirely. The normal distribution has two parameters, the mean and standard deviation. The normal distribution does not have just one form. Instead, the shape changes based on the parameter values, as shown in the graphs below.

Mean

The mean is the central tendency of the distribution. It defines the location of the peak for normal distributions. Most values cluster around the mean. On a graph, changing the mean shifts the entire curve left or right on the X-axis.

Standard deviation

The standard deviation is a measure of variability. It defines the width of the normal distribution. The standard deviation determines how far away from the mean the values tend to fall. It represents the typical distance between the observations and the average.

On a graph, changing the standard deviation either tightens or spreads out the width of the distribution along the X-axis. Larger standard deviations produce distributions that are more spread out.

When you have narrow distributions, the probabilities are higher that values won’t fall far from the mean. As you increase the spread of the distribution, the likelihood that observations will be further away from the mean also increases

The normal distribution is a probability function that describes how the values of a variable are distributed. It is a symmetric distribution where most of the observations cluster around the central peak and the probabilities for values further away from the mean taper off equally in both directions. Extreme values in both tails of the distribution are similarly unlikely.