Chap 4
- Frequency, Probability and Percentage Graphs
- Normal Distribution
- z scores
Frequency, Probability and
Percentage Graphs
• Draw a frequency graph for the following
distribution
2, 6, 6, 11, 11, 11, 18, 18, 18, 29, 29, 29, 37, 37, 43
X
2
6
11
18
29
37
43
f
1
2
3
4
3
2
1
• Draw a probability graph for the following
distribution
2, 6, 6, 11, 11, 11, 18, 18, 18, 18, 29, 29, 29, 37, 37, 43
X
2
6
11
18
29
37
43
f
1
2
3
4
3
2
1
Probability
1/16 = 0.0625
2/16 = 0.125
3/16 = 0.1875
4/16 = 0.25
3/16 = 0.1875
2/16 = 0.125
1/16 = 0.0625
• Draw a percentage graph for the following
distribution
2, 6, 6, 11, 11, 11, 18, 18, 18, 29, 29, 29, 37, 37, 43
X
2
6
11
18
29
37
43
f
1
2
3
4
3
2
1
Probability
1/16 = 0.0625
2/16 = 0.125
3/16 = 0.1875
4/16 = 0.25
3/16 = 0.1875
2/16 = 0.125
1/16 = 0.0625
Percentage
0.0625 x 100= 6.25
0.125 x 100 = 12.5
Based on your probability graph
•
•
•
•
What is the probability of finding scores
a) more than 18 in this distribution?
b) less than 6 in this distribution?
c) more than 37 in this distribution?
Normal Distribution
• Is the distribution of scores that has the following
characteristics:
• 1)Is unimodal, meaning has one hump
• 2) symmetic – the right half of the curve is a mirror image
of the left half
• 3) has the same value of mean, median and mode
• 4) is asymptotic – the tails of the distribution never touch
the abscissa or the x-axis
• 5) has 68% of the scores of the distribution lie between µ
± σ.
• 6) has the following mathematical model
Normal Distribution
Frequency /
Probability /
Percentage
13.59%
34.13%
34.13%
13.59%
Scores
1σ
1σ
1σ
1σ
Mean
Median
Mod
50% of the scores are found on each side of the mean
68.26% of the scores are found between 1 standard deviation from the mean
z scores
• A z score is a measure of how many standard
deviations a raw score is from the mean of the
distribution
• Suppose a normal distribution has a mean of 20
and a standard deviation (SD) of 4, the score 24
is one SD above the mean. As such the score 24
would be a 1z score above the mean (z = +1)
• What z score will be assigned to a score of 16?
(Ans: z = -1)
A z score is also called a standard unit or standard score
When all the raw scores of a normal distribution have been transformed to z
scores, the resulting distribution is called standard normal distribution.
The standard normal distribution has a mean of 0 and a SD of 1.
Standard Normal Distribution
Frequency /
Probability /
Percentage
13.59%
2z
34.13%
34.13%
1z
1z
SD = 1
13.59%
SD = 1
SD = 1
Mean = 0
Median
Mod
SD = 1
2z
z scores
How to convert a raw score to a z
score?
Population
Sample
Z=X-µ
σ
X
Z = X - ___
s
Where
X = the raw score to be transformed
X = the mean of the sample
s = the sample SD
µ = the mean of the population
σ = the population SD
Example
• For a population distribution with µ = 4.80
and σ = 2.14, what is the z score of a raw
score of 6?
Z=X-µ
σ
Z = 6 – 4.8 = +.56
2.14
The raw score of 6 has a standard score z = +.56
Which means that the score 6 is .56 standard deviations above (+)
the mean
Characteristics of z scores
• 1) A z-score distribution is obtained by transforming
every raw score to a z score.
• 2) A z-score distribution always has a mean of 0 and a
standard deviation of 1.
• 3) All raw score that fall below the mean have some z
value that is negative; all raw scores that fall above the
mean has positive z scores.
• 4) A raw score that is one SD above the mean has a z
score of +1 and a raw score that is one SD below the
mean has a z score of -1.
• 5) A raw score that is the same as the mean has a z
value of 0
• 6) The z –score distribution will have the same shape as
the raw score distribution
Convert the following raw scores to z scores
given mean = 8 and SD = 4
•
•
•
•
•
a) 12
b) 8
c) 4
d) 16
e) 6
Area under the curve
Area is a measure of the probability of finding a score
Standard Normal Distribution
Frequency /
Probability /
Percentage
13.59%
-2 z
34.13%
34.13%
1z
-1z
SD = 1
13.59%
SD = 1
SD = 1
2z
z scores
SD = 1
Mean = 0
Median
Mod
68.26% of the scores fall between -1z and +1z. This means the probability of a
randomly selected score to fall between - 1z and +1z is .6813
Area under the curve
Standard Normal Distribution
Frequency /
Probability /
Percentage
2.28
13.59%
-2 z
34.13%
34.13%
1z
-1z
SD = 1
13.59%
SD = 1
SD = 1
2.28
2z
z scores
SD = 1
Mean z = 0
Median
Mod
50% of the z scores exceed the Mean = 0. This means the probability of a
randomly selected score to be found above the Mean is .50
Area under the curve
z
Area between
Mean and z
Area
Beyond z
0
0
+1
.3413
.5 - .3413 = .1587
+2
.4772
.5 - .4772 = .0228
+1.5
.50
(see table for normal
distribution – get the area beyond z) = .0668
The area beyond the z score is given by the Normal Distribution Table
Open your Table book and learn how to use it how to use the Normal
Distribution Table
Exercise 1
• You are given a normal distribution with
the mean of 12 and SD of 4. Using this
distribution, calculate the percentage and
probability of scores that
• a) fall between 8 and 16
• b) exceed 18
• c) fall below 6
Draw a normal distribution graph to show these positions
What percentage of scores fall between the
z scores of +.25 and + 1.20?
Frequency /
Probability /
Percentage
+.25
Mean z = 0
Median
Mod
+1.20
z scores
Exercise 2: Draw the graph to show
the area involved
• What percentage of scores fall between the z scores of –
a) -.25 and - 1.20
• b) -.20 and +.20
• c) below the z score of -1.96
• d) above the z score of +1.96
• e) below the z score of -1.96 and above the z score of
+1.96
• f) -1.2 and +2.5
• g) below the z score of -.4 and above the z score of -.6
Exercise 3 : Draw the graph to
show the area involved
• Given the Mean of a population is 100 and the SD = 25,
what percentage of scores fall between the scores
• a) 75 and 125
• b) above 125
• c) below 75
• d) above 80
• e) above 110
• f) between 80 and 100
• g) between 60 and 130
• h) below 60 and above 130
Homework
•
You obtained the following data on students’ self concept for your
Masters Project Paper Research.
23, 42, 45, 25, 43, 12, 23, 32, 33, 30,
32, 26, 26, 33, 47, 55, 40, 28, 23, 16,
47, 33, 35, 50, 48, 8, 30, 23, 35, 29,
36, 33, 16, 33, 18, 26, 37. 33, 15, 33
a)
Describe the data using measures of central tendency: Mean,
median and mode
b) Describe the data using measures of variability: Range, Mean
deviation, Variance and Standard Deviation.
c) Assuming the distribution is normal, use the normal distribution
table and answer the following questions:
i) what percentage of scores exceed the score 35.
ii) what percentage of scores lie between 30 and 46.
iii) what percentage of scores lie between 20 and 50
iv) what is the probability of obtaining a score greater than 50?
v) what is the probability of obtaining a score greater that 53?
d) Draw graphs for each question in c) to show the area involved.
e) Present your descriptive statistical findings in a table (use the APA
format).