Download 5-1 Random Variables and Probability Distributions

6-2
Standard Units
Areas under Normal
Distributions
What is a z score?
“A common statistical way of standardizing data on
one scale so a comparison can take place is using
a z-score. The z-score is like a common yard stick for
all types of data. Each z-score corresponds to a
point in a normal distribution and as such is
sometimes called a normal deviate since a z-score
will describe how much a point deviates from a
mean or specification point.”
http://www.measuringusability.com/z.htm
“The z score for an item, indicates how far and in what
direction, that item deviates from its distribution's
mean, expressed in units of its distribution's standard
deviation.”
http://www.sysurvey.com/tips/statistics/zscore.htm
Z score
Is a way to standardize scores based on the mean.
Tables are used to describe the areas under normal
curves, associated with particular z scores.
For instance, if Physics period 1 has an average test
score of 74 and Physics period 7 had an average of
78
How does someone who had an 83 in period 1
compare with someone who had an average of 87
in period 7?
Do they deserve the same grade? Both were
substantially above average, although the grades
were obviously quite different.
Would the standard deviations of the two classes
make a difference to you??
Z Score (cont)
The conversion formula of a score to a z
score is based on the mean and
standard deviation.
x μ
z
σ
So what is the z score for someone
whose score = the average?
Z score (cont)
What about a z score for an x above the
mean?
What about a z score for an x below the
mean?
Raw Score
If you know the z score, it can be
translated into the appropriate raw
score by
x = zσ + μ
Standard Normal Distribution
Standardizing a z score makes the center μ = 0.
Standardizing a z score makes the spread σ = 1.
The z distribution will be normal if the x distribution is
normal.
Areas still correspond as they did in 6-1.
68% of the area under the curve is 1 standard
deviation from the mean.
95% of the area under the curve is 2 standard
deviations from the mean
99.7% will be 3 standard deviations from the mean.
This is sometimes called the 68-95-99.7 rule or The Empirical
Rule
The college you want to apply to says that while there
is no minimum SAT score required, the middle 50% of
their students have combined SAT scores between
1530 and 1850. What would be a minimum
acceptable ACT score that could be correlated to
that same range of SAT scores?
For college bound seniors, the average combine SAT
is 1500 and standard deviation is 250. The ACT
average is 20.8 with a standard deviation of 4.8
Why convert?
If you can convert the raw score to a z score, you can
use tables in the back to determine the area under
the curve. Why?
The areas under the curve is equal to the probability that
the measurement falls in this interval.
The textbook has a left tailed table. That is, the z score
will correspond to a given cumulative area to the left
of z.
Therefore, what would be the area to the right of z?
1 – area left of z.
(It could also be the opposite of the area to the left of –z)
What does this area stuff mean?
Essentially if you have a z score, you can
make an assumption about the % of
scores (i.e. the probability) that fall at
that number or below.
Some notes
A z score of 6 or 7 would be highly
interesting. Why?
Calculator:
normalcdf (zleft, zright) calculates the
score between two z scores.
If you need to do left or right of a z
score, some books suggest using 3.49
for the end, some suggest 99. Use
what you want.
ADD BVD PROBLEM
Pg. 113