Question

What is the relationship between the normal curve and the standard deviation?

Answer 1

The standard deviation is based on the normal distribution curve.

Explanation:

The "Normal Distribution Curve" is the distribution of values around the mean of an evenly-dispersed population. That is, an equal number of value-differences from the Mean lie on each side of the mean at any given value.

The "Standard Deviation" is a calculation of the "width" of that curve based on a sample or population value. It is commonly used in a short-hand form (`1sigma, 2sigma, 3sigma`) to indicate a particular range of the population included. There is no mathematical or statistical reason to restrict oneself to those values. They are simply convenient designations for standardization of statistical reporting.

For details on their derivations and significance see the following references:
https://www.mathsisfun.com/data/standard-normal-distribution.html

https://www.thoughtco.com/bell-curve-normal-distribution-defined-2312350

http://stattrek.com/probability-distributions/normal.aspx