Chap 4

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Chap 4

• - Frequency, Probability and Percentage Graphs
• - Normal Distribution
• - z scores

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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 f
• 1
• 2
• 11 3
• 4
• 29 3
• 37 2
• 43 1

3

• 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 f Probability
• 1 1/16 0.0625
• 2 2/16 0.125
• 11 3 3/16 0.1875
• 4 4/16 0.25
• 29 3 3/16 0.1875
• 37 2 2/16 0.125
• 43 1 1/16 0.0625

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• Draw a percentage graph for the following
  distribution

2, 6, 6, 11, 11, 11, 18, 18, 18, 29, 29, 29, 37,
37, 43

• X f Probability
  Percentage
• 1 1/16 0.0625
  0.0625 x 100 6.25
• 2 2/16 0.125
  0.125 x 100 12.5
• 11 3 3/16 0.1875
• 4 4/16 0.25
• 29 3 3/16 0.1875
• 37 2 2/16 0.125
• 43 1 1/16 0.0625

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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?

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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 s.
• 6) has the following mathematical model

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Normal Distribution
Frequency / Probability / Percentage
34.13
34.13
13.59
13.59
Scores
1s
1s
1s
1s
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
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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)

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A z score is also called a standard unit or
standard scoreWhen 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
34.13
34.13
13.59
13.59
1z
z scores
2 z
2 z
1z
SD 1
SD 1
SD 1
SD 1
Mean 0 Median Mod
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How to convert a raw score to a z score?
Population Z X - µ s
Sample Z X - ___ s
Where X the raw score to be
transformed the mean of the
sample s the sample SD µ
the mean of the population s the
population SD
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Example

• For a population distribution with µ 4.80 and s
  2.14, what is the z score of a raw score of 6?

Z X - µ s
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
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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

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Convert the following raw scores to z scores
given mean 8 and SD 4

• a) 12
• b) 8
• c) 4
• d) 16
• e) 6

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Area under the curve
Area is a measure of the probability of finding a
score
Standard Normal Distribution
Frequency / Probability / Percentage
34.13
34.13
13.59
13.59
1z
z scores
2 z
-2 z
-1z
SD 1
SD 1
SD 1
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
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Area under the curve
Standard Normal Distribution
Frequency / Probability / Percentage
34.13
34.13
13.59
13.59
2.28
2.28
1z
z scores
2 z
-2 z
-1z
SD 1
SD 1
SD 1
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
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Area under the curve

• z Area between
  Area
• Mean and z
  Beyond z
• 0 0
  .50
• 1 .3413
  .5 - .3413 .1587
• 2 .4772
  .5 - .4772 .0228
• 1.5 (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
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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
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What percentage of scores fall between the z
scores of .25 and 1.20?
Frequency / Probability / Percentage
z scores
.25
1.20
Mean z 0 Median Mod
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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

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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

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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
• Describe the data using measures of central
  tendency Mean, median and mode
• Describe the data using measures of variability
  Range, Mean deviation, Variance and Standard
  Deviation.
• 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).

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