6.3: Normal Distribution

Empirical Rule

Before looking at the process for finding the probabilities under a normal curve, recall the Empirical Rule that gives approximate values for areas under a bell-shaped distribution. The Empirical Rule, shown in Figure 6-10, is just an approximation for probability under any bell-shaped distribution and will only be used in this section to give you an idea of the size of the probability for different shaded areas. A more precise method for finding probabilities will be demonstrated using technology. Please do not use the empirical rule in the homework questions except for rough estimates.

The Empirical Rule (or 68-95-99.7 Rule)

In a bell-shaped distribution with mean μ and standard deviation σ,

• Approximately 68% of the observations fall within one standard deviation (σ) of the mean μ.
• Approximately 95% of the observations fall within two standard deviations (2σ) of the mean μ.
• Approximately 99.7% of the observations fall within three standard deviations (3σ) of the mean μ.

Gauss

Figure 6-10

For now, we will be working with the most common bell-shaped probability distribution known as the normal distribution, also called the Gaussian distribution, named after the German mathematician Johann Carl Friedrich Gauss. See Figure 6-11.

Figure 6-11

A normal distribution is a special type of distribution for a continuous random variable. Normal distributions are important in statistics because many situations in the real world have normal distributions.

Properties of the normal density curve:

1. Symmetric bell-shaped.
2. Unimodal (one mode).
3. Centered at the mean μ= median = mode.
4. The total area under the curve is equal to 1 or 100%.
5. The spread of a normal distribution is determined by the standard deviation σ. The larger σ is, the more spread out the normal curve is from the mean.
6. Follows the Empirical Rule.

If a continuous random variable X has a Normal distribution with mean μ and standard deviation σ then the distribution is denoted as X~N(μ, σ). Any x values from a Normal distribution can be transformed or standardized into a standard Normal distribution by taking the z-score of x.

The formula for the normal probability density function is: \(f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}\right)}\). We will not be using this formula.

The probability is found by using integral calculus to find the area under the PDF curve. Prior to the handheld calculators and personal computers, there were probability tables made to look up these areas. This text does not use probability tables and will instead rely on technology to compute the area under the curve.

Every time the mean or standard deviation changes the shape of the normal distribution changes. The center of the normal curve will be the mean and the spread of the normal curve gets wider as the standard deviation gets larger.

Figure 6-12 compares two normal distributions N(0, 1) in green on the left and N(7, 6) in blue on the right.

Figure 6-12

“‘So, what's odd about it?’

‘Nothing, it's Perfectly Normal.’”

(Adams, 2002)

6.4.1 Standard Normal Distribution

A normal distribution with mean μ = 0 and standard deviation σ = 1 is called the standard normal distribution.

The letter Z is used exclusively to denote a variable that has a standard normal distribution and is written Z ~ N(0, 1). A particular value of Z is denoted z (lower-case) and is referred to as a z-score.

Recall that a z-score is the number of standard deviations x is from the mean. Anytime you are asked to find a probability of Z use the standard normal distribution.

Standardizing and z-scores:

A z-score is the number of standard deviations an observation x is above or below the mean μ. If the z-score is negative, x is below the mean. If the z-score is positive, x is above the mean.

If x is an observation from a distribution that has mean μ and standard deviation σ, the standardized value of x (or z-score) is \(\mathrm{z}=\frac{x-\mu}{\sigma}\).

To find the area under the probability density curve involves calculus so we will need to rely on a table to find the area.

Compute the area under the standard normal distribution to the left of z = 1.39.

Solution

First, draw a bell-shaped distribution with 0 in the middle as shown in Figure 6-13. Mark 1.39 on the number line and shade to the left of z = 1.39.

Figure 6-13

Note that the lower value of the shaded region is -∞, which the TI-84 does not have. Instead we use a really small number in scientific notation -1E99 or -1*1099 (make sure you use the negative sign (-) not the minus – sign.

6.4.4 How to use a z-table

A normal distribution can have any mean (μ) and any standard deviation (σ). If we had to make a separate probability table for each possible mean and standard deviation, we would need infinitely many tables.

To avoid that, we transform every normal distribution into a standard normal distribution—one with mean 0 and standard deviation 1—using the z-score formula:

\[
z = \frac{x - \mu}{\sigma}
\]

This transformation re-expresses the original value xxx in terms of how many standard deviations it is away from the mean. Because every normal curve becomes the same shape after this transformation, we can use a single universal table—the z-table—to find probabilities for any normal distribution.

What the z-table Shows

Most standard z-tables show cumulative probabilities from the far left tail of the distribution up to the z-score you look up:

\[
P(Z \leq z) = \text{Area under the standard normal curve to the left of } z
\]

If z is positive, you are looking at an area to the right of the mean.
If z is negative, you are looking at an area to the left of the mean.

The table is usually arranged so that:

• The row gives the z-score to the first decimal place.
• The column gives the second decimal place.

• The cell where they intersect gives \[
  P(Z \leq z)
  \]

For example: In a z-table, the row 1.2 and column 0.03 together represent z=1.23

Steps for Using the z-table to Find Probabilities

1. Standardize the value

If X is from a normal distribution with mean μ and standard deviation σ:

\[
z = \frac{x - \mu}{\sigma}
\]

2. Locate the z-score in the table

Use the row for the first decimal place of z, and the column for the second decimal place. The intersection gives \[
P(Z \leq z)
\].

3. Interpret or adjust depending on the question

You're already finished if you want \[
P(Z \leq z)
\]

If you want the probability in the upper-tail:

\[
P(Z > z) = 1 - P(Z \leq z)
\]

If you want the probability between two values​:

\[
P(z_1 \leq Z \leq z_2) = P(Z \leq z_2) - P(Z \leq z_1)
\]

6.4.2 Applications of the Normal Distribution

Many variables are nearly normal, but none are exactly normal. Thus, the normal distribution, while not perfect for any single problem, is very useful for a variety of problems. Variables such as SAT scores and heights of United States adults closely follow the normal distribution. Although we will not be using Excel, it may be valuable to note that the Excel function NORM.S.DIST is for a standard normal when µ = 0 and σ = 1

Using Excel or TI-Calculator to Find Normal Distribution Probabilities

Figure 6-18

Problem 1 - HR analyst and wages

An HR analyst models starting wages for entry-level analysts as normally distributed with mean $52,000 and standard deviation $6,000.

(a) What is the probability that a randomly selected new analyst earns exactly $55,000?
(b) What is the probability that a randomly selected new analyst earns between $54,500 and $55,500?

Solution (a):

\[
P(X = 55{,}000) = 0.
\]

Solution (b):

\[
z_1=\frac{54{,}500-52{,}000}{6{,}000}=\frac{2{,}500}{6{,}000}\approx 0.4167,
\qquad
z_2=\frac{55{,}500-52{,}000}{6{,}000}=\frac{3{,}500}{6{,}000}\approx 0.5833.
\]
\[
P(54{,}500\le X \le 55{,}500)=P(z_1\le Z\le z_2)
= \Phi(0.5833) - \Phi(0.4167).
\]
\[
\Phi(0.5833)\approx 0.720,\quad \Phi(0.4167)\approx 0.661
\;\Rightarrow\;
P(54{,}500\le X \le 55{,}500)\approx 0.720-0.661=0.059.
\]

Problem 2 — Predicting Delivery Times

A logistics manager models delivery times as X∼N(48 minutes, 12 minutes)X\sim N(48\text{ minutes},\,12\text{ minutes})X∼N(48 minutes,12 minutes). What delivery time threshold captures the 90th percentile?

Solution:

\[
\Phi(z_{0.90})=0.90 \;\Rightarrow\; z_{0.90}\approx 1.2816.
\]
\[
x=\mu+z\sigma = 48 + (1.2816)(12) = 48 + 15.3792 \approx 63.38\ \text{minutes}.
\]

---

Problem 3 — Credit Scores and Small Business Loans

A bank’s credit score for small-business loans is approximately normal with mean 680 and standard deviation 50.

(a) What is the probability that a randomly selected applicant scores above 750?
(b) The bank wants to pre-approve the top 5% of applicants. What cutoff should it use?

Solution (a):

\[
z=\frac{750-680}{50}=\frac{70}{50}=1.4,
\qquad
P(X>750)=P(Z>1.4)=1-\Phi(1.4).
\]
\[
\Phi(1.4)\approx 0.9192 \;\Rightarrow\; P(X>750)=1-0.9192=0.0808.
\]

Solution (b):

\[
P(X\le c)=0.95 \;\Rightarrow\; z_{0.95}\approx 1.6449,
\qquad
c=\mu+z\sigma = 680 + (1.6449)(50) = 680 + 82.245 \approx 762.25.
\]

6.4.3 Normal Probability Plot

A normal quantile plot, also called a normal probability plot, is a graph that is useful in assessing normality. A normal quantile plot plots the variable x against each of the x values corresponding z-score. It is not practical to make a normal quantile plot by hand.

Interpreting a normal quantile plot to see if a distribution is approximately normally distributed.

1. All of the points should lie roughly on a straight line y = x.
2. There should be no S pattern present.
3. Outliers appear as points that are far away from the overall pattern of the plot.

Here are two examples of histograms with their corresponding quantile plots. Note that as the distribution becomes closer to a normal distribution the dots on the quantile plot will be in a straighter line. Figures 6-21 is the histogram and figure 6-22 is the corresponding normal probability plot. Note the histogram is skewed to the left and dots do not line up on the y = x line. Figures 6-23 and Figure 6-24 represent a sample that is approximately normally distributed. Note that the dots still do not line up perfectly on the line y = x, but they are close to the line.

Figure 6-21

Figure 6-22

Figure 6-23

Figure 6-24