Normal Distribution

MATH 102Contemporary Math

S. Rook

Overview

• Section 15.4 in the textbook:– Normal distribution– Z-scores– Converting raw scores to z-scores– Applications

Normal Distribution

Normal Distribution

• When a sufficient number of data are collected, the resulting histogram becomes nearly symmetric– i.e. split into two equal halves

• By connecting the tops of the bars of the histogram, of the data we obtain a bell- shaped curve more commonly known as the Normal Curve

• If a set of data can be modeled by a normal curve, we can calculate the proportion of the data in ANY interval

Properties of the Normal Distribution

• Given that a set of data follows a normal distribution, the following properties apply to the resulting normal curve:– It is bell-shaped– Its highest point is the mean– It is symmetric with respect to the mean– The total area under it is 1– Approximately:

• 68% of the data lies within 1 standard deviation of the mean

• 95% of the data lies within 2 standard deviations of the mean

• 99.7% of the data lies within 3 standard deviations of the mean

Normal Distribution (Example)

Ex 1: Assume we have a normal distribution with a mean of 10 and a standard deviation of 2. Use the 68-95-99.7 rule to find the percentage of values in the desired interval:

a) Between 10 and 12b) Above 14c) Below 10

Normal Distribution (Example)

Ex 2: Assume we have a normal distribution with a mean of 12 and a standard deviation of 3. Use the 68-95-99.7 rule to find the percentage of values in the desired interval:

a) Above 6b) Below 9c) Between 15 and 18

Z-scores

Z-scores

• Examples 1 and 2 allowed us to determine the percentage of data values that lay within 1, 2, or 3 standard deviations of the mean

• Z-scores allow us to determine the percentage of data that lie within ANY number of standard deviations (sds) of the mean of a standard normal distribution– e.g. z = 2.3 refers to 2.3 sds to the RIGHT of the mean

and z = -1.7 refers to 1.7 sds to the LEFT of the mean

• The standard normal distribution has a mean of 0 and a standard deviation of 1

Calculating Percentage/Proportion of Area

• To calculate the percentage or proportion of area under a standard normal curve, we use the standard normal table– A z-score such as z = -1.92 is looked up by finding the

first two digits (-1.9) in the rows and then the value of the hundredths (0.02) in the columns• e.g. What is the value associated by looking up z = -

1.92?

– The area obtained represents the proportion of data values that lie to the LEFT of (below) the given z-score

Calculating Percentage/Proportion of Area (Continued)

• To find the percentage or proportion of area:– Below a z-score – simply look up the z-score in the

table– Above a z-score – look up the z-score and subtract

it from 1 • Recall that the area underneath the entire normal curve

is 1

– Between two z-scores – look up both z-scores and subtract the smaller from the larger

• It may help to draw the normal curve

Z-scores (Example)

Ex 3: Assume a data set follows the normal distribution and use the standard normal table to find the specified percentage of data values under the standard normal curve:

a) Below z = -1.35 d) Above z = -2.73b) Above z = 2.00 e) Below z = 0.08c) Between z = -0.98 f) Between z = 1.01

and z = 1.72 and z = 1.99

Converting Raw Scores to Z-scores

Converting Raw Scores to Z-scores

• Finding the proportion of data values using z-scores works ONLY with data adhering to a standard normal distribution

• However, given that the data comes from a normal distribution and the distribution’s mean & sd, we can convert a value into a z-score using the formula where:

μ is the mean of the dataσ is the sd of the datax is any value

x

z

Converting Raw Scores to Z-scores (Example)

Ex 4: Given a mean, a standard deviation, and a raw score from a data set that comes from a normal distribution, find the corresponding z-score:

a) μ = 80, σ = 5, x = 87b) μ = 21, σ = 4, x = 14c) μ = 38, σ = 10.3, x = 48

Applications

Applications (Example)

Ex 5: A machine fills bags of candy, but due to slight irregularities in the operation of the machine, not every bag gets exactly the same number of pieces. Assume that the number of pieces per bag has a mean of 200 and a standard deviation of 2. Assuming a normal distribution, what proportion of the bags will have:

a) Less than 197 pieces of candy?b) More than 204 pieces of candy?c) Between 199 and 201 pieces of candy?

Applications (Example)

Ex 6: A supervisor observes and records the amount of time his employees take for lunch. Suppose the length of the average lunch is 42.5 minutes with a standard deviation of 5.8 minutes. Assuming a normal distribution, what proportion of the supervisor’s employees have a lunch break:

a) Less than 50 minutes?b) More than 40 minutes?c) Between 43 to 45 minutes?

Summary

• After studying these slides, you should know how to do the following:– Use the 68-95-99.7 rule to find the percentage of values in a desired

interval of a normal distribution– Compute the proportion of the standard normal curve using z-scores– Compute a z-score given the mean, standard deviation, and a raw

score from a normal distribution– Solve application problems involving normal distributions

• Additional Practice:– See problems in Section 15.4

• Next Lesson:– Study for the Final Exam!