NORMAL DISTRIBUTIONS & PROBABILITY

DAY 1

Exploring Normal Distributions with the

EMPIRICAL RULE

Normal Distribution

• A normal distribution is a continuous, symmetrical, bell-shaped distribution of a variable.

Characteristics of Normal Distribution

1. A normal distribution curve is bell-shaped. 2. The mean, median, and mode are equal and are

located at the center of the distribution. 3. The curve is unimodal (i.e., it has only one mode) 4. The curve is symmetric about the mean (its

shape is the same on both sides of a vertical line passing through the center)

5. The curve is continuous. There are no gaps or holes. For each value of X, there is a corresponding value of Y.

Characteristics of Normal Distribution

6. The curve never touches the x axis. No matter how far in either direction the curve extends, it never meets the x axis – but it gets increasingly closer.

7. The total area under a normal distribution curve is equal to 1.00, or 100%.

The area under the part of a normal curve that lies within 1 standard deviation of the mean is approximately 0.68, or 68%; within 2 standard deviations, about 0.95, or 95%; and within 3 standard deviations, about 0.997, or 99.7%.

The Empirical Rule (a.k.a. the “68-95-99.7 Rule”)

• In a normal distribution, almost all data will lie within 3 standard deviations of the mean.

o About 68% of all data lies within 1 standard deviation of the mean.

o About 95% of all data lies within 2 standard deviations of the mean.

o About 99.7% of all data lies within 3 standard deviations of the mean.

Why Do We Need It?

• The Empirical Rule is most often used in statistics for forecasting or predicting final outcomes.

• After a standard deviation is calculated, and before exact data can be collected, the Empirical Rule can be used to estimate impending data.

• The probability based on the Empirical Rule can be used if gathering appropriate data may be time consuming, or even impossible to obtain.

Tips for Using the Empirical Rule

• Before applying the Empirical Rule it is a good idea to identify the data being described and the value of the mean and standard deviation.

• Sketch a graph summarizing the information provided by the empirical rule and identify the percentages for each region of the graph (+/- 1, +/- 2, +/- 3)

• Remember that data must be normally distributed for the Empirical Rule to apply.

The Empirical Rule for a Normal Distribution

The amount of mustard dispensed from

a machine at The Hotdog Emporium

is normally distributed with a mean

of 0.9 ounce and a standard

deviation of 0.1 ounce. If the

machine is used 500 times,

approximately how many times will it

be expected to dispense 1 or more

ounces of mustard.

Choose:

5 16 80 100

The mean is 0.9 and the standard deviation is 0.1. If one standard deviation is

added to the mean, the result is 1.0 ounce. Therefore, dispensing 1 or more

ounces falls into the category above one standard deviation to the right of the

mean. Using the Empirical Rule, 16% of data falls at or above 1 standard

deviation.

16% x 500 = 80 times to dispense one or more ounces of mustard.

A machine is used to fill soda bottles. The amount of soda dispensed

into each bottle varies slightly. Suppose the amount of soda dispensed

into the bottles is normally distributed. If at least 99% of the bottles

must have between 585 and 595 milliliters of soda, find the greatest

standard deviation, to the nearest hundredth, that can be allowed.

The 99% implies a distribution within 3 standard deviations of the mean. The

difference from 585 milliliters to 595 milliliters is 10 milliliters. Symmetrically

divided, there are 5 milliliters used to create 3 standard deviations on one

side of the mean. Dividing 5 by 3, we get the standard deviation to be 1.67

milliliters, to the nearest hundredth.

Battery lifetime is normally distributed for large

samples. The mean lifetime is 500 days and

the standard deviation is 61 days. To the

nearest percent, what percent of batteries have

lifetimes less than 439 days?

Subtracting, we see that 1 s.d. below is 439 days, an exact match to

our question and an indication that the Empirical Rule can be used to

find the answer. The question is asking what percent of a distribution is

beyond one standard deviation to the left of the mean.

Answer: 16%

A shoe manufacturer collected data

regarding men's shoe sizes and found

that the distribution of sizes exactly fits

the normal curve. If the mean shoe size

is 11 and the standard deviation is 1.5,

find:

a. the probability that a man's shoe size

is greater than or equal to 11.

b. the probability that a man's shoe size

is greater than or equal to 14.

a. 50% In a normal distribution, the mean divides the data into two equal

areas. Since 11 is the mean, 50% of the data is above 11 and 50% is below 11.

The probability is 0.5.

b. 14 is exactly two standard deviations above the mean. Using the Empirical Rule

we see that 2.5% will fall above two standard deviations. Probability is 0.025.

Five hundred values are normally distributed with a mean of 125 and a standard

deviation of 10.

a. What percent of the values lies in the interval 115 - 135,

to the nearest percent?

b. What interval about the mean includes 95% of the data?

a. What percent of the values is in the interval 115 - 135?

mean + one standard deviation = 135

mean - one standard deviation = 115

Percent within one standard deviation of the mean = 68%

b. 2 standard deviations about the mean for a total interval size

of 40, with the mean in the center.

mean + 2 standard deviations = 145

mean - 2 standard deviations = 105

Interval: [105,145]

Example: Estimating with the Empirical Rule

• You have purchased fluorescent light bulbs for your home. The average bulb life is 500 hours with a standard deviation of 24. The data is normally distributed.

• One of your bulbs burns out at 450 hours. Would you send the bulb back for a refund?

Problem: You have purchased fluorescent light bulbs for your home. The average bulb life is 500 hours with a standard deviation of 24. The data is normally distributed. One of your bulbs burns out at 450 hours. Would you send the bulb back for a refund?

Solution: • According to the Empirical Rule:

o 68% of the light bulbs should last between 500 ± 24 or between 476 to 524 hours.

o 95% of the light bulbs should last between 500 ± 2(24) or between 452 to 548 hours.

o 99.7% of the light bulbs should last between 500 ± 3(24) or between 428 to 572 hours.

• If the light bulb only lasted 450 hours, I would consider it a defective bulb. Less than 2.5% should last less than 452 hours.

Your Turn • The scores for all high school seniors taking the verbal

section of the Scholastic Aptitude Test (SAT) in a particular year had a mean of 490 and a standard deviation of 100. The distribution of SAT scores is normal.

1. What percentage of seniors scored between 390 and 590 on this SAT test?

2. One student scored 795 on this test. How did this student do compared to the rest of the scores?

3. A rather exclusive university only admits students who were among the highest 16% of the scores on this test. What score would a student need on this test to be qualified for admittance to this university?

1. What percentage of seniors scored between 390 and 590 on this SAT test? • The mean = 490 and the standard deviation = 100. So, we can

draw the Normal Curve to depict the data.

Calculate the difference

between the mean and

the test scores:

|390-490| = 100 1

s.d.

|590-490| = 100 1

s.d. So, about 68% of

seniors would score

between 390 and 590

on the SAT test.

2. One student scored 795 on this test. How did this student do compared to the other scores? • Calculate the difference between the mean and the student’s

test score:795-390 = 405

• This student’s score is more than 3 std. deviations from the mean (3 standard deviations = 3 * 100)

Since about 99.7% of seniors

would score between 190 and

790 on the SAT test, this is an

exceptionally high score.

Only 0.15% of students would

have scores above 790. This

is an example of using the

Empirical Rule to estimate an

answer.

3 . A rather exclusive university only admits students who were among the highest 16% of the scores on this test. What score would a student need on this test to be qualified for admittance to this university?

• About 68% of the scores are between 390 and 590, so this leaves 32% of the scores outside this interval.

• Since a bell-shaped curve is symmetrical, one-half of the scores (16%) are on each end of the distribution.

• 16% of the students scored above 590 on this SAT test.

So, to qualify for admittance to this university, a student would need to score 590 or above on the SAT test.