NORMAL DISTRIBUTION

Objectives

At the end of the lesson students can:1. Identify the characteristics of normal curve;2. Compute for the value of the z-scores; and 3. Interpret the results of the values computed

in response to the problem computed

NORMAL DISTRIBUTION• The distribution of some human abilities and

characteristics such as mental ability tends to follow a certain specific shape called the normal distribution.

• When the distribution is normal, most of the observations (about 68%) tend to converge at the middle of the observation and the rest are distributed to the left and right ends of the distribution.

• The normal curve is bell-shaped.• In a normal distribution, the mean, median and mode

values are equal and coincide at one point when the graph is drawn.

The Normal Curve

X̅ = md=mo

The Standard Normal Distribution

• The normal curve is a graph of:y = 1 e (1/2 )z² σ √2π

where: z is the z-score σ the population standard deviation

Population standard deviation formula:

σ²N = ∑(x – μ)² N

Properties of the Theoretical 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. A normal distribution curve is unimodal.4. The curve is symmetric about the mean, which is

equivalent to saying that its shape is the same on both sides of a vertical line passing through the center.

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

Properties of the Theoretical Normal Distribution…. continued

6. The curve never touches the X-axis. Theoretically, 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 normal curve is equal to 1. or 100%..

8. 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 deviation is 0.95 or 95% and within 3 standard deviation is about 0.997 or 99.7%.

Area under normal distribution curve

-4 -3 -2 -1 0 1 2 3 40

10

20

30

40

50

60

Y-Values

Y-Values

68%95%

99%

Z-scores

• A distribution which is not normal can be normalized by changing all the scores in the distribution into the z-scores.

• The graph using z-scores as points is a normal curve.• The total area under a normal curve is 1.• At the vertex of the normal curve lie the mean, median

and mode values..• Since it is a bell-shaped, the right and left side of the

curve are symmetrical with respect to a vertical axis.• The area of the curve to the right of the vertical axis is

0.5 and the area under normal curve to the left is also 0.5.

Z-scores…

• The z-score of 0 lies at the vertex.• All z-scores to the right side are positive and those at

the left side are negative.• The formula of z-score:

z = x – μ σwhere: z = z-score x = score

σ = population standard deviation μ = mean

Exercise

1. Convert the following scores to z-scores where μ = 75 and σ = 3.

a) 75b) 80c) 58

Exercise 2. The following are the scores of 27 students in a biology quiz:

12 10 9 10 12 15 15 1615 20 22 23 10 12 10 1416 17 18 20 20 21 10 1223 10 10

a) Convert scores into z-scores using μ = 14.9 and σ = 4.5.b) What percent of the class obtained scores higher than 20?c) How many students obtained a score less than 20?d) How many students scored between 10 and 20?e) How many students scored between 20 and 23?

3. What is the z-score that marks the upper 33% of the area under normal curve?

Answer score Z-score Area

9 -1.31 0.4049

10 -1.09 0.3621

12 -0.64 0.2389

14 -0.20 0.0793

15 0.02 0.0080

16 0.24 0.0948

17 0.47 0.1808

18 0.69 0.2549

20 1.13 0.3708

21 1.36 0.4131

22 1.58 0.4429

23 1.80 0.4641

• Given: μ = 14.9 σ = 4.5

Questions

• How many percent of students scored between 9 and 15?

• How many students scored between 9 and 15?

b. What percent of the class obtained scores higher than 20?

• The z-score of 20 = 1.13 or z=1.13• Draw a graph of the normal curve where z=1.13 is located.• Locate in the area under normal curve (table) when z=1.13• What is the area? A = ____ (0.3708 or 37.08%)• What is the percentage of students who scored more than

20? (area of half of a normal curve is A= 0.5000, while the area of z=1.13 is A=0.3708. solve: 0.5000 – 0.3708 = 0.1292 or 12.92%.

• The desired are is lying to the right of z-1.13 will be shaded (higher than 20)

• Therefore, 12.92% of the students scored higher than 20.

z=0

Graph of the class scored higher than 20

Area covered by the percentage of the students scored higher than 20

How many students scored more than 20?

• The area of half of a normal curve is A= 0.5000, while the area of z=1.13 is A=0.3708.

• Solve: 0.5000 – 0.3708 = 0.1292• Multiply 0.1292 by the total number of students

(based from the problem, there are 27 number of students (n=27)

• 0.1292 x 27 = 3.4884 or 3• Therefore, there are 4 students who scored

higher than 20.

Solve for the answer and interpreta.1) How many percent of students scored less

than 10?a.2) How many students scored less than 10?b.1) How many percent of students scored less

than 20? b.2) How many students obtained a score less than

20?c.1) How many percent of students scored

between 10 and 20?c.2) How many students scored between 10 and

20?

EXERCISES 1. Convert the following scores to z-scores if the mean

is 50 and the standard deviation is 10.a. 76b. 40c. 50d. 10

2. In a certain test in English, the mean score of a group of 200 students was 56 and the standard deviation was 12. in a test in chemistry, the mean score was 68 and the standard deviation was 8. if a student scores 65 in English and 75 in chemistry, in which test did she perform better?

QUIZ1. A statistics professor reported that of the 30

students who took the quiz, the mean score is 72 and the standard deviation is 6.

a. What is the percentage of students who obtained a score less than 75?

b. How many students got a score higher than 80?c. How many percent of students got a score

between 75 and 90?

2. The weights of 20 female students aged 15 to 20 years have a mean of 120 pounds and a standard deviation of 6 pounds.A1. What percent of the students have weights below

110 pounds?A2.How many students have weight below 110?B1.What percent of the students have weights above

130 pounds?B2. How many students have weights above 130

pounds?C1. How many percent of students have weights

between 100 and 125 pounds?C2. How many students have weights between 100 and

125 pounds?

SKEWNESS

Objectives

At the end of the lesson students can:1. Identify the characteristics of skewness;2. Compute for the value of the skewness: and 3. Interpret the results of the values computed

in response to the problem computed

Skewness (Sk) • It refers to the symmetry or asymmetry of a

frequency distribution and its measure can be obtained by using the formula:

Sk = 3 (x̅- Md) s

• If the values of the mean and the median are equal, the distribution is normal and the graph is bell-shaped curve.

Types of Skewness…

1. If the observations are concentrated at the left side of the vertical axis and has fewer observations at the right, it is called positively skewed distribution. the mean is higher than the median Example: marrying age of women

2. If the observations are concentrated at the right side , you have negatively skewed distribution.The mean is lower than the medianExample: age vs education

Example

1. Calculate the degree of skewness of a distribution if the mean is 45, the median is 40, and the standard deviation is 5. Graph and describe the skewness.

Exercise • Find the degree of skewness of the scores of

students in a Statistics Exam. Graph and describe the result Scores f

91 – 95 6

86 – 90 8

81 – 85 15

76 – 80 20

71 – 75 18

66 – 70 13

61 - 65 5

QUIZ

1. Calculate the degree of skewness in a distribution if the mean is 45, the median is 40, and the standard deviation is 5.

2. Referring to the table on the right, find the degree of skewness for the of the scores of students in Math 150. Describe the graph formed.

SCORES f

60-64 4

65-69 9

70-74 10

75--79 14

80-84 12

85-89 8

90-94 3

KURTOSIS

Objectives

At the end of the lesson students can:1. Identify the characteristics of kurtosis;2. Compute for the value of the kurtosis of a

given data; and 3. Interpret the results of the values computed

in response to the problem computed

Kurtosis (Ku)

• The degree of peakedness or flatness of the curve.

• This also known as the percentile coefficient of kurtosis.

• Formula: Ku = QD/PRwhere: QD = quartile deviation

PR = Percentile Range

Kurtosis (Ku)…

• QD = (Q₃ - Q₁) / 2• PR = P₉₀ - P₁₀• When the value of Ku is:

a. Equal to 0.263, the curve is a normal curve or mesokurtic.

b. Greater than 0.263, the curve is leptokurtic or thin.

c. Less than 0.263, the curve is platykurtic or flat.

Exercise • Find the degree of kurtosis of the scores of

students in a Statistics Exam. Graph and describe the result Scores f

91 – 95 6

86 – 90 8

81 – 85 15

76 – 80 20

71 – 75 18

66 – 70 13

61 - 65 5

seatwork

1. Calculate the percentile coefficient of Kurtosis for the scores of students in Math 150. Graph and interpret.

SCORES f

60-64 4

65-69 9

70-74 10

75--79 14

80-84 12

85-89 8

90-94 3

Quiz…2. Determine the degree of skewness and

kurtosis of the given data and interpret the result.

Anxiety level of Students During the Exam

Level Frequency 15 – 19 2510 – 14 34

5 – 9 400 - 4 32