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Section16.2 – Normal Distributions
Normal Curves and Normal Distributions
A set of data whose distribution has a bell-shapedcurve issaid to have a NORMALDISTRIBUTION.
A perfect bell-shaped curve is called a NORMALCURVE.
The area under the curve is equal to 1.0 probability or 100%.
SHAPE: Symmetry: A normal curve is symmetrical along a vertical line splitting the bell in half.
CENTER: Mean equalsMedian: μ = M
In a perfectly normal distribution, the line of symmetry
is the median Mand the mean μof the data.
- 50% of the data are greater than or equal to the mean
- 50% are less than or equal to the mean.
Standard Deviation andPoint of Inflection. Apoint of inflectionPon a curve is the point where the concavity of the curve changes, and is one standard deviation σaway from the center (mean or median) of the data. P = μ ± σ
Example #1:Find the mean, median and standard deviation of each normal distribution shown. The point of inflection of the curve is P is the point of inflection of the curve.
a.μ = ______b.μ = ______c. μ = ______
M = ______M = ______M = ______
σ = ______σ = ______σ = ______
Example #2: Consider the following normal distributions where P and P′ are the points of inflection of the curve. Find the mean and standard deviation. (round to the nearest tenth).
2a.μ = ______2b.μ = ______2c. μ = ______
σ = ______σ = ______σ = ______
Section 16.3 – Standardizing Normal Data
Standardized Value, Z-Value, or Z-Score:
To standardize the comparison of data in we refer to data values based on the number of standard deviations they are above or below the mean. This standardized value is often called the z-value or z-score.
EXAMPLE: Given mean is 8 and standard deviation is 3
11 has z-score = 1.0 because 11 is 3 above 8 or one standard deviation above the mean
5 has z-score = -1.0 because 5 is 3 below 8 or one standard deviation below the mean
z-value Calculation:
The z-value calculation is the difference of the data value (X) and mean (μ) all over the standard deviation (σ).
Example #1: Consider a normally distributed data set with a mean μ = 45 ft. and a standard deviation σ = 10 ft. Find the z-value for the given data values.
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1a) X = 55 ft.
1b) X = 20 ft.
1c) X = 50 ft.
1d) X = 21.58 ft.
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Example #2: Consider a normally distributed data set withμ = 18.5 and σ = 1.2.
Find the z-values.
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2a) X = 22.1
2b) X = 16.34
3c) X = 19.3
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Solving for data point given the standardized value: z-value to X.
Example #3: Consider a normally distributed data set with mean μ = 235.7 meters and standard deviation σ = 41.58 meters. Find the data value X that corresponds to the standardized value
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3a) z = –3.45?
3b) z = 2.6?
3c) z = 0.3?
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Example #4: Consider a normally distributed data set with μ = 12.5 and σ = 0.35. Find data value X that corresponds to the standardized value?
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4a) z = 1.5?
4b) z = -2.5?
4c) z = 3?
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Finding Standard Deviation and Mean based on z-scores
Example #5: In a normal distribution, the data value of 45 has a z-score of 2 and the date value of 27 has a z-score of -1. What is the mean and standard deviation?
Example #6: In a normal distribution, the data value of 4.8 has a z-score of -3 and the date value of 17.2 has a z-score of 1. What is the mean and standard deviation?
Example #7: In a normal distribution, the data value of 173 has a z-score of -2 and the date value of 238 has a z-score of 3. What is the mean and standard deviation?
Section 16.4 68-95-99.7 Rule
In a set of data with a normal distribution, approximately
68% of the data fall within one standard deviation of the mean(between z = –1 and z = 1)
95% of the data fall within two standard deviations of the mean (between z = –2 and z = 2
99.7% of the data fall within three standard deviations of the mean(between z = –3 and z = 3)
1. Consider the normal distribution shown in the graph.
1a. Find the mean and the standard deviation of the distribution.
μ = ______σ = ______
1b. What percent of the data is greater than X = 65? ______
1c. Suppose the size of the population is N = 114. Approximately how many data values are less than 53?
2. Consider the normal distribution shown in the graph.
2a. Find the mean and the standard deviation of the distribution.
μ = ______σ = ______
2b. What percent of the data is greater than X = 52? ______
2c. Suppose the size of the population is N = 127. Approximately how many data values are there between 52 and the mean?
3. Consider the normal distribution shown in the graph.
3a. Find the mean of the distribution.
μ = ______
3b. What percent of the data is greater than X = 79.75? ______
3c. Suppose the size of the population is N = 89. Approximately how many data values are there between 72.25 and 79.75?
If normal curve is not provided, use number line for the data since z-value and percentages never change.
Data Values
Z-values
5. A normal distribution has a standard deviation σ = 6.1 cm and 84% of the data fall above 47.1 cm.
Find the mean.
μ = ______
6. A normal distribution has a standard deviation σ = 9.6 cm and 84% of the data fall below 49.7 cm.
Find the mean.
μ = ______
7. A normal distribution has mean μ = 11.8 and standard deviation σ = 3.4. Approximately what percent of the data fall between 1.6 and 22.0?
8. A normal distribution has mean μ = 12.5 and standard deviation σ = 3.1. Approximately what percent of the data fall between 6.3 and 18.7?
9. In a normal distribution with mean μ and standard deviation σ, what percent of the data
a. fall below the value μ + σ? b. between μ – 3σ and μ – 2σ? c. above μ – 3σ?
Chapter 16 – NormalCurve Practice Problems
1. Find the z-value for a given data point X in a normal distribution of data with mean μ = 100 and standard deviation σ = 10
1a.X = 140z = ______1b.X = 112z = ______
1c.X = 95z = ______1d.X = 83z = ______
2. Find the data point X that corresponds to the given z-value in a normal distribution of data with
μ = 180.3 ft. and standard deviation σ = 30.9 ft.
2a. z = –4 X = ______2b.z = 0.8 X = ______
2c. z = 4.5 X = ______2d.z = 0 X = ______
3. In a normal distribution, the data value X1 = 30 has the standardized value z1 = –2 and the data value X2 = 130 has the standardized value z2 = 3. Find the mean and standard deviation.
4. In a normal distribution, the data value X1 = 30 has the standardized value z1 = –2 and the data value X2 = 75 has the standardized value z2 = 1. Find the mean and standard deviation.
5. In a normal distribution, the data value X1 = 20 has the standardized value z1 = –3 and the data value X2 = 115 has the standardized value z2 = 2. Find the mean and standard deviation.
6. Consider the normal distribution shown in the graph.
6a. Find the mean of the distribution.
μ = ______
6b. What percent of the data is less than X = 95.50? ______
7)The number of problems missed on a quiz follows a normal distribution with a mean of 15 and a standard deviation of 4.
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7a.What percent of students missed between 11 – 19 problems?
7b.What percent of students missed more than 23 problems?
7c.What percent of students less than 23 problems?
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8) The final exam scores normal distribution with a mean of 80 and a standard deviation of 6.
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8a. What percent of students scored between 80 - 92?
8b. What percent of students scored between 68 -86?
8c. What percent of students scored higher than 74?
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9)The class mean of height is 60 inches. We know that 68% of the students are between 56 and 64 inches.
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9a. What is the standard deviation?
9b. 50% of the students are shorter than …?
9c. 16% of students are taller than …?
9d. 84% of students are taller than …?
Honors Discrete Practice Multiple Choice
CHAPTER 16: NORMAL CURVE
Pick the MOST ACCURATE Answer Choice
For #1 – 11:250 students in a math class take the final exam. The scores on the exam have an approximately normal distribution with center μ = 75 and standard deviation σ = 10.
1)The number of students scoring 75 or more is approximately
- 75B. 83C. 125D. 158
2)The average score on the exam was approximately
A. 10B. 75C. 85D. 95
3)Approximately 95% of the class scored between
A. 55 and 95B. 0 and 95C. 45 and 100D. 65 and 85
4)Assuming there were no outliers, the lowest score on the exam was around
A. 0B. 10C. 45D. 75
5)Approximately what percent of the students scored between 65 and 75?
A. 10%B. 34%C. 68%D. 95%
6)Peter’s score on the exam places him in the 16th percentile of the class. Peter’s score on the exam is approximately?
A. 16B. 45 C. 55D. 65
7)Carol scored 85 points on the exam. In approximately what percentile does this score place her?
A. 34thB. 68thC. 84thD. 95th
8)A z-value of 1.8 corresponds to a test score of
A. 18B. 57C. 76.8D. 93
9)A score of 95 corresponds to a standardized value (z-value) of
A. – 2 B. 2C. 8.5D. 20
10)A score of 60 corresponds to a standardized value (z-value) of
A. – 5 B. – 1.5 C. 1.5D. 5
11)Approximately what percentage of test scores had standardized values between -3 and 3?
A. 50%B. 68%C. 95%D. 99%
For #12 – 14: 95% of the data for a normal curve is between data values of 24 and 84.
12)Find the mean, μ, of the normal curve
A. 12B. 39C. 48D. 54
13)Find the standard deviation, σ, of the normal curve
A. 10B. 15C. 30D. 60
14)What z-value represents the data value 84?
A. 2.0B. 3.0C. 15D. 30
For #15 – 17: 16% of the data for a normal curve is above the data value 56 and another 16% of the data for the normal curve is below the data value of 44.
15)Find the mean, μ, of the normal curve
A. 38B. 48C. 50D. 62
16) Find the standard deviation, σ, of the normal curve
A. 3B. 6C. 12D. 24
17)What z-value represents the data value 44?
A. – 1.0 B. 1.0C. – 2.0D. 2.0
For # 18 - 21: As part of a study, 800 college football players are randomly chosen and their weights taken. The distribution of weights is approximately normal. The average weight is 235 pounds and the standard deviation is 25 pounds.
18)Approximately how many players weighed 210 pounds or less?
A. 64B. 128C. 200D. 544
19)Assuming there were no outliers, the range of player weights was approximately
A. 100 lbsB. 150 lbsC. 200 lbsD. 400 lbs
20)Approximately what percentage of players weighed over 285 pounds?
A. 95%B. 97.5%C. 2.5%D. 5%
21)Approximately how many players weighed between 210 and 260 pounds?
A. 760B. 128C. 256D. 544
Honors Discrete Practice Multiple Choice
CHAPTER 16: NORMAL CURVE – Solutions
1)C
2)B
3)A
4)C
5)B
6)D
7)C
8)D
9)B
10)B
11)D
12)D
13)B
14)A
15)C
16)B
17)A
18)B
19)B
20)C
21)D