Wiki Page, Math 13 - Perez Sandbox : Module 12 - Introduction to Normal Probability Distributions (2a of 5)

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2026-01-17T21:15:13-08:00

Here is what we know so far about normal probability curves:

The shape is a bell-shape.

The mean is the center value associated with the peak.

The standard deviation measures the spread about the mean. The standard deviation affects the shape of the normal curve. The normal curve is always bell-shaped, but it can be short and squat or tall and skinny depending on the value of the standard deviation.

The area under the normal probability curve is 1.

Normal probability curves also have another important feature: the area under the curve associated with values within one standard deviation of the mean is always 0.68. This is true for every normal probability curve, regardless of the value of the mean or standard deviation.

We examine this important feature in the next example.

EXAMPLE One standard deviation on each side of the mean

Let’s start with a quantitative variable X that has a normal probability distribution with mean = 10 and standard deviation = 2. Let’s practice our new notation: and .

The normal probability curve for X is shown below.

As expected, the mean is located at the center of the normal curve. The other two arrows point to values a distance of 1 standard deviation on each side of the mean.

The point 1 standard deviation less than the mean is represented by . Since and , this point is located at 10 − 2 = 8, as shown.

The point 1 standard deviation more than the mean is represented by . Since and , this point is located at 10 + 2 = 12, as shown.

You will notice we have indicated that the area of the green region is 0.68. So we can say that the probability of X being between 8 and 12 equals 0.68.

Or, using our probability notation, we could write: P(8<X<10) = 0.68.

EXAMPLE

Baby birth weights have a normal probability distribution with mean grams and standard deviation grams.

What is the probability that a baby's birth weight is within one standard deviation of the mean birth weight?

Answer: The probability is 0.68 = 68%. To be more precise, there is a 68% chance that a randomly selected baby will weigh between 2,900 grams (i.e., 3,500 − 600 = 2,900) and 4,100 grams (i.e., 3,500 + 600 = 4,100) at birth.

Let's Summarize

For any normal probability curve, the central area within 1 standard deviation of the mean equals 0.68. This is the same as saying ...

68% of the time we will expect X to have a value within 1 standard deviation of the mean.

Are you surprised by these statements? After all, we have seen that the standard deviation affects the shape of the normal curve. The normal curve is always bell-shaped but sometimes it is short and squat or tall and skinny. Regardless, the area under the normal probability curve is always 1 (100%). In addition, the values within one standard deviation of the mean occur 68% of the time. We examine this idea further in the next example.

EXAMPLE

Look at the red arrows pointing at the normal curve in the following figure. At these points, the curve changes the direction of its bend and goes from bending upward to bending downward, or vice versa. A point like this on a curve is called an inflection point. Every normal curve has inflection points at exactly 1 standard deviation on each side of the mean.

This diagram shows what we mean by concave down (bending down) and concave up (bending up) and the transition between the two.

With the following applet, you can look at a variety of normal curves. Use the slider to change the standard deviation to get different normal curves.

Observe that the two properties we discussed in the examples remain true for any standard deviation you select:

The probability that a value is within 1 standard deviation of the mean is 0.68 = 68%.

The x-values of the inflection points correspond to 1 standard deviation above and below the mean.

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