Decimals (Concepts & Computation)

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Normal Distributions &
z score
Dec. 9
Normal Distribution
•
•
•
•
Tend to cluster around the mean
Symmetrical
bell-shaped frequency polygon
Mode, median, mean all occur at the
highest point, in the center of distribution
• 3 units of SD to the left of the mean and
3 units of SD to the right of the mean
• 99.74% of polygon area lies btw –3SD and
+3SD
Skewed Distribution
• A large percentage the values is on
either left or right side
• Positively skewed
– Has a tail to the right
• Negatively skewed
– Has a tail to the left
z scores
• Used to compare values of interval or ratio
–level variables
– Rita’s score of 21 out of 24 vs. Mary’s score 85
out of 100
• Variables should be normally distributed
• z score equals raw score minus mean
divided by SD
• Can be converted to percentiles
• Known as standard scores
Areas of Normal Curve
Table
Examples of z Scores &
corresponding Areas and
percentiles
z score
.12
1.78
-2.90
1.15
-1.15
row
0.1
1.7
2.9
1.1
1.1
column
.02
.08
.00
.05
.05
area
04.78
46.25
49.81
37.49
37.49
percentile
54.78
96.25
.19
87.49
12.51
Exercise
1. On an IQ test with a mean of 100, SD
of 16, what is the percentile of IQ 104?
• Debora leads a treatment group of college
students diagnosed as chronic anxiety.
Group members have been selected for
treatment on the basis their scores on
Anxiety Scale A with a mean of 70 and SD
of 10 or Scale B with mean of 50 and SD
of 12. Gina had score of 78 on the Scale A
and Tom received 66 on the Scale B.
There is only one vacancy in the group.
Who should be in the treatment group?
Deriving Raw Scores
from Percentiles
• Lauren wishes to form a treatment
group of college students with high
anxiety levels using the results of
testing with Scale B (mean=50,
SD=12). She wants to include the top
10% of all students measured by
Scale B. What is the cutoff point?
• Subtract the percentile of mean (50) from
90 to determine the area of a normal
curve that would fall between the 90th
percentile and the mean
• Find the z score corresponding to the area
• Use the z score formula, add 50 if the z
score is positive
Exercise
One hundred students took mid-term
exam designed to measure knowledge of
statistics. Their scores on it formed a
normal distribution with a mean of 80
and a standard deviation of 6.
Instructor decided to assign A to the
top 10% of the scores. What score does
John have to get to earn?
Exercise
• 89,56,45,78,98,45,55,77,88,99,98,97,54,34,94
• 77,88,87,67,98,87,55,77,45,44,88,99,69,67,98
• Calculate the mode, median, mean, range,
variance, and standard deviation for both
sections. Which section did better overall
on the exam?