One thing that is great about mathematics is the way that seemingly unrelated areas of the subject come together in surprising ways. One instance of this is the application of an idea from calculus to the bell curve. A tool in calculus known as the derivative is used to answer the following question. Where are the inflection points on the graph of the probability density function for the normal distribution?

### Inflection Points

Curves have a variety of features that can be classified and categorized. One item pertaining to curves that we can consider is whether the graph of a function is increasing or decreasing. Another feature pertains to something known as concavity. This can roughly be thought of as the direction that a portion of the curve faces. More formally concavity is the direction of curvature.

A portion of a curve is said to be concave up if it is shaped like the letter U. A portion of a curve is concave down if it is shaped like the following ∩. It is easy to remember what this looks like if we think about a cave opening either upward for concave up or downwards for concave down. An inflection point is where a curve changes concavity. In other words it is a point where a curve goes from concave up to concave down, or vice versa.

### Second Derivatives

In calculus the derivative is a tool that is used in a variety of ways. While the most well-known use of the derivative is to determine the slope of a line tangent to a curve at a given point, there are other applications. One of these applications has to do with finding inflection points of the graph of a function.

If the graph of *y = f( x )* has an inflection point at *x = a*, then the second derivative of *f* evaluated at *a* is zero. We write this in mathematical notation as *f’’( a )* = 0. If the second derivative of a function is zero at a point, this does not automatically imply that we have found an inflection point. However, we can look for potential inflection points by seeing where the second derivative is zero. We will use this method to determine the location of the inflection points of the normal distribution.

### Inflection Points of the Bell Curve

A random variable that is normally distributed with mean μ and standard deviation of σ has a probability density function of

*f( x ) =1/ (σ √(2 π) )exp[-(x - μ) ^{2}/(2σ^{2})]*.

Here we use the notation exp[y] = *e ^{y}*, where

*e*is the mathematical constant approximated by 2.71828.

The first derivative of this probability density function is found by knowing the derivative for *e ^{x}* and applying the chain rule.

*f’ (x ) = -(x - μ)/ (σ ^{3} √(2 π) )exp[-(x -μ) ^{2}/(2σ^{2})] = -(x - μ) f( x )/σ^{2}*.

We now calculate the second derivative of this probability density function. We use the product rule to see that:

*f’’( x ) = - f( x )/σ ^{2} - (x - μ) f’( x )/σ^{2}*

Simplifying this expression we have

*f’’( x ) = - f( x )/σ ^{2} + (x - μ)^{2} f( x )/(σ^{4})*

Now set this expression equal to zero and solve for *x*. Since *f( x )* is a nonzero function we may divide both sides of the equation by this function.

*0 = - 1/σ ^{2} + (x - μ)^{2} /σ^{4}*

To eliminate the fractions we may multiply both sides by *σ*^{4}

*0 = - σ ^{2} + (x - μ)^{2}*

We are now nearly at our goal. To solve for *x* we see that

*σ ^{2} = (x - μ)^{2}*

By taking a square root of both sides (and remembering to take both the positive and negative values of the root

±*σ = x - μ*

From this it is easy to see that the inflection points occur where *x = μ ± σ*. In other words the inflection points are located one standard deviation above the mean and one standard deviation below the mean.