Transcript Normal Distributions

Normal Distributions
Remember rolling a 6 sided dice
and tracking the results
1
2
3
4
5
6
This is a uniform distribution (with certain characteristics)
A histogram is used to display a
normal distribution
Histograms:
The horizontal axes of a histogram
contains the bins into which each
piece of data must fall
The vertical axes of a histogram
contains the frequency (number)
Bin width: The width of each interval of the
histogram.
• They should be equal.
• Try to avoid bins with a frequency of zero
• Do not “hit the post”
Not a histogram
Who likes popcorn?
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180
Find the sum of three dice in 50
rolls
Group the results 3 and 4 in one bin, 5 and 6
in another bin…and so on
Normal
Distribution
Normal Distribution
• Make a note of the characteristics
and Diagram that are on page
425.
The Normal Curve
Since a Normal Distribution is
described in terms of percentages,
we define the area under the Normal
Curve as 1 (100%)
The percentage of data that lies
between two values in a normal
distribution is equivalent to the area
that lies under the normal curve
between these two posts.
Z- Scores
Consider the following situation:
A school gives a scholarship for the
highest mark in Data Management
The student must be taking all three maths as well to receive
the award (so they may get beat in DM)
Caley, who took MDM first semester,
received 84%.
Lauren, who took MDM second
semester, received 79%
Who should get the MDM award?
It depends…
Both student’s marks must
be compared on the same
scale.
Think Canadian and American
money.
Results can be written in terms
of “standard deviations away
from the mean.”
(Z-score)
This allows for effective
comparisons
Conversion to z-score
Z=x-x
s
X: result
X: mean
s : Standard Deviation
Caley: 84%, CA: 74%, sd: 8
Z = 84 - 74
8
= 1.25
That means 84% is 1.25
standard deviations above
the mean (double check…)
Lauren: 79%, CA: 60%, sd: 9.8
Z = 79 - 60
9.8
= 1.94
That means 79% is 1.94
standard deviations above
the mean (a better relative
grade)
mean
+1
+2
+3
Example 2
Suppose you received 75% as a
final mark in a class.
You want to know what
percentage of students were
below your grade in your class.
Assume your class follows a
normal distribution.
If we assign the area under
the standard normal curve
to be 1, then the
percentage of results less
then a given data point,
will be equal to the area
under the curve to the left
of the equivalent z-score
post.
The areas under the curve are
calculated an summarized on
page 606
Convert: x = 75%, CA: 70, SD: 6
Z = 0.83
Look up 0.83 in the chart
0.7967
That means 79.67% of the grade
were below your grade of a 75%
Do example 2 and 3 on pg 426
together
pg 146 z score info
Notice:
Since the area under every normal
curve equals 1.
The percent of the data that lies
between 2 specific values, a and
b, is the area under the normal
curve between endpoints a and b
a
b
b z-score area – a z-score area
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