The Normal Curve

Curve by JasonUnbound

A z-score may also be described by the following formula:

z = the z-score (standardized score)x = a number in the distributionμ = the population meanσ = standard deviation for the population

z-scores

Standardizing Two Sets of ScoresTwo consumer groups, one in Vancouver and one in Halifax, recently tested five brands of breakfast cereal for taste appeal. Each consumer group used a different rating system.

Vancouver HalifaxCereal Brand Rating Cereal Brand Rating A 1 P 25 B 10 Q 35 C 15 R 45 D 21 S 50 E 28 T 70

Use z-scores to determine which cereal has the higher taste appeal rating.

HOMEWORK

T has the higher taste appeal rating.

The table shows the lengths in millimetres of 52 arrowheads.

16 16 17 17 18 18 18 18 19 20 20 21 2121 22 22 22 23 23 23 24 24 25 25 25 2626 26 26 27 27 27 27 27 28 28 28 28 2930 30 30 30 30 30 31 33 33 34 35 39 40

(a) Calculate the mean length and the standard deviation.

(d) What percent of the arrowheads are within one standard deviation of the mean length?

(c) How many arrowheads are within one standard deviation of the mean?

(b) Determine the lengths of arrowheads one standard deviation below and one standard deviation above the mean.

HOMEWORK

The table shows the lengths in millimetres of 52 arrowheads.16 16 17 17 18 18 18 18 19 20 20 21 2121 22 22 22 23 23 23 24 24 25 25 25 2626 26 26 27 27 27 27 27 28 28 28 28 2930 30 30 30 30 30 31 33 33 34 35 39 40

(a) Calculate the mean length and the standard deviation.

(d) What percent of the arrowheads are within one standard deviation of the mean length?

(c) How many arrowheads are within one standard deviation of the mean?

(b) Determine the lengths of arrowheads one standard deviation below and one standard deviation above the mean.

HOMEWORK

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

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?

HOMEWORK

North American women have a mean height of 161.5 cm and a standard deviation of 6.3 cm.

(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?

(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

179.14 = x

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.

The Normal Distribution

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.)

World Strong Man Competition 2007 by flickr user highstrungloner

• the heights and weights of adult males in North America

United States Olympic Triathlon Trials by flickr user Diamondduste

• the times for athletes to swim 5000 metres

大阪湾岸線 by flickr user El Fotopakismo

• the speed of cars on a busy highway

IMG_3677.JPG by flickr user JonBen

• the weights of quarters produced at the Winnipeg Mint

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.

The 68-95-99 Rule

• 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.

Generally speaking, approximately:

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.

Properties of a Normal Distribution

More Properties of a Normal Distribution

Interactivate 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

The data below shows the ages in years of 30 trees in an area of natural vegetation.

Determine whether the data approximate the normal distribution.

37 15 34 26 25 38 19 22 21 2842 18 27 32 19 17 29 28 24 3535 20 23 36 21 39 16 40 18 41

USING the 68 -95-97 RULE

HOMEWORKThe following are the number of steak dinners served on 50 consecutive Sundays at a restaurant.

Draw a suitable histogram that has five bars.

41 52 46 42 46 36 44 68 58 4449 48 48 65 52 50 45 72 45 4347 49 57 44 48 49 45 47 48 4345 56 61 54 51 47 42 53 44 4558 55 43 63 38 42 43 46 49 47

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?

Age of Student 15 16 17 18 19 20 21 22# of Students 1 7 42 24 7 4 2 1

HOMEWORK

Based on this answer, does it seem that the students' ages approximate a normal distribution?

Now let's try a problem involving Grouped DataA 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.