The standard normal curve. Properties of normal distributions.

1. The Normal
Curve
Curve by JasonUnbound

2. z-scores
A z-score may also be described
by the following formula:
z = the z-score (standardized score)
x = a number in the distribution
μ = the population mean
σ = standard deviation for the population

3. Standardizing Two Sets of Scores
Two consumer groups,
Vancouver Halifax
one in Vancouver and Cereal Brand Rating Cereal Brand Rating
one in Halifax, recently A 1 P 25
tested five brands of B 10 Q 35
breakfast cereal for C 15 R 45
taste appeal. Each D 21 S 50
consumer group used E 28 T 70
a different rating
system. HOMEWORK
Use z-scores to determine which cereal has the higher taste
appeal rating.
T has the higher taste appeal rating.

4. The table shows the lengths in millimetres of 52 arrowheads.
16 16 17 17 18 18 18 18 19 20 20 21 21
21 22 22 22 23 23 23 24 24 25 25 25 26
26 26 26 27 27 27 27 27 28 28 28 28 29
30 30 30 30 30 30 31 33 33 34 35 39 40
HOMEWORK
(a) Calculate the mean length and the standard deviation.
(b) Determine the lengths of arrowheads one standard deviation
below and one standard deviation above the mean.
(c) How many arrowheads are within one standard
deviation of the mean?
(d) What percent of the arrowheads are within one standard
deviation of the mean length?

5. HOMEWORK
The table shows the lengths in millimetres of 52 arrowheads.
16 16 17 17 18 18 18 18 19 20 20 21 21
21 22 22 22 23 23 23 24 24 25 25 25 26
26 26 26 27 27 27 27 27 28 28 28 28 29
30 30 30 30 30 30 31 33 33 34 35 39 40
(a) Calculate the mean length (b) Determine the lengths of arrowheads
and the standard deviation. one standard deviation below and one
standard deviation above the mean.
(c) How many arrowheads (d) What percent of the arrowheads
are within one standard are within one standard deviation
deviation of the mean? of the mean length?

6. Using Z-Score, Mean, and Standard Deviation
to Calculate the Real Score
Numerous packages of raisins were weighed. The mean mass was
1600 grams, and the standard deviation was 40 grams. Trudy bought a
package that had a z-score of -1.6. What was the mass of Trudy's
package of raisins?
HOMEWORK

7. HOMEWORK
A survey was conducted at DMCI to determine the number of music
CDs each student owned. The results of the survey showed that the
average number of CDs per student was 73 with a standard deviation
of 24. After the scores were standardized, the people doing the survey
discovered that DJ Chunky had a z-score rating of 2.9. How many CDs
does Chunky have?

8. North American women have a mean height of 161.5 cm and a
standard deviation of 6.3 cm.
(a) A car designer designs car seats to fit women taller than
159.0 cm. What is the z-score of a woman who is 159.0 cm tall?
z = -0.3968
(b) The manufacturer designs the seats to fit women with a maximum
z-score of 2.8. How tall is a woman with a z-score of 2.8? 179.14 = x

9. The Normal Distribution
A Normal Distribution is a frequency distribution that can be
represented by a symmetrical bell-shaped curve which shows that
most of the data are concentrated around the centre (i.e., mean) of
the distribution. The mean, median, and mode are all equal. Since
the median is the same as the mean, 50 percent of the data are lower
than the mean, and 50 percent are higher. The frequency distribution
showing light bulb life, for example, shows that the mean is 970
hours, and the hours of life for all the bulbs are spread uniformly
about the mean.

10. The Normal Distribution
The diagram above represents a normal distribution. In real life, the
data would never fit a normal distribution perfectly. There are,
however, many situations where data do approximate a normal
distribution. Some examples would include:
(Note that all the examples represent continuous data.)

11. • the heights and weights of adult males in North America
World Strong Man Competition 2007
by flickr user highstrungloner

12. • the times for athletes
to swim 5000 metres
United States Olympic Triathlon Trials
by flickr user Diamondduste

13. • the speed of cars on a busy highway
by flickr user El Fotopakismo

14. • the weights of
quarters produced at
the Winnipeg Mint
IMG_3677.JPG by
flickr user JonBen

15. The diagram shows a normal
distribution with a mean of 28
and a standard deviation of 4.
The values represent the
number of standard
deviations above and below
the mean. Replace the
numbers with raw scores.

16. Properties of a Normal Distribution
The 68-95-99 Rule
Generally speaking, approximately:
• 68% of all the data in a normal distribution lie within the 1
standard deviation of the mean,
• 95% of all the data lie within 2 standard deviations of the
mean, and
• 99.7% of all the data lie within 3 standard deviations of the
mean.

17. Properties of a Normal Distribution
The curve is symmetrical about the mean. Most of the data are
relatively close to the mean, and the number of data decrease as
you get farther from the mean.

18. More Properties of a Normal Distribution
• 99.7% of all the data lies within approximately 3 standard deviations
of the mean.
• All normal distributions are symetrical about the mean.
• Each value of mean and standard deviation determines a different
normal distributions. (see below)
• The area under the curve always equals one.
• The x-axis is an asymptote for the curve.
Frequency
Scores
Interactivate Normal Distribution

19. The data below shows the ages in years of 30 trees in an area of
natural vegetation.
37 15 34 26 25 38 19 22 21 28
42 18 27 32 19 17 29 28 24 35
35 20 23 36 21 39 16 40 18 41
Determine whether the data approximate the normal distribution.
USING the 68 -95-97 RULE

20. HOMEWORK
The following are the number of steak dinners served on 50
consecutive Sundays at a restaurant.
41 52 46 42 46 36 44 68 58 44
49 48 48 65 52 50 45 72 45 43
47 49 57 44 48 49 45 47 48 43
45 56 61 54 51 47 42 53 44 45
58 55 43 63 38 42 43 46 49 47
Draw a suitable histogram that has five bars.

21. HOMEWORK
The frequency table shows the ages of all the students in Senior 4
Math at Newberry High. Find the mean, μ. Then calculate the
percent of students older than the mean age. How does this
compare to the percent of students older than the mean age if the
distribution were a normal distribution?
Based on this answer, does it seem that the students' ages
approximate a normal distribution?
Age of Student 15 16 17 18 19 20 21 22
# of Students 1 7 42 24 7 4 2 1

22. Now let's try a problem involving Grouped Data
A machine is used to fill bags with beans. The machine is set to add
10 kilograms of beans to each bag. The table shows the weights of
277 bags that were randomly selected.
wt in kg 9.5 9.6 9.7 9.8 9.9 10.0 10.1 10.2 10.3 10.4 10.5
# of bags 1 3 13 25 41 66 52 41 25 7 3
(a) Are the weights normally distributed? How do you know?
(b) Do you think that using the machine is acceptable and fair to
the customers? Explain your reasoning.