Download Commentary on Normal Curve transformations

Commentary on Normal Curve transformations
1. The Normal Curve : You must keep in mind that the normal curve is a theoretical
distribution. It is assumed that if you could obtain all possible scores from a population,
charting those scores would result in a normal curve. See the features of a normal curve
under the “application” column of your class notes.
2. Under the normal curve are percentages where all scores may fall based upon equal
deviations from the center. Remember that under the normal curve, your mean, median &
mode all fall in the same place – at the 0 mark: 0 meaning the center of the distribution
with no deviations for the mean, median & mode. Notice that the percentages under the
normal curve are the same on either side of the center. One of the features of the normal
curve is that one side is the mirror image of the other. Unlike raw scores, the values under
the normal curve are static, or set & do not change based upon results of a particular
measurement allowing us then to make probability statements. The exact percentages &
distances under the normal curve are provided for you at the bottom of your class notes.
If you scroll down to the end of your class notes, you will see a more legible bell curve
with the standard percentages listed based upon each half standard mark from the center
or the mean. Additional percentages based upon standard deviations are also provided
under the second bell curve.
3. Z-Scores – You can see the different sections cut out under the normal curve are equidistant from the center & coordinate with values between -3 & 3. These values are zscores & represent deviations of values from the center of the normal distribution or
normal curve.
a. So far we have been working with raw scores. Raw scores are fitting in their
description: they are the true, raw data associated with whatever specific measure
they are married to. For instance, the total amount that you could receive on Quiz 1
was 99.99 (rounded up to 100). That meant that for each problem, you could receive a
score of 3.33. All of these values are raw scores.
b. There is little you can do with raw scores alone, other than developing & creating
descriptive measures. There are times though, when we want to do more with
numbers. We want to compare those with other measures or values from other areas.
c. To do this, we will begin by transforming raw scores into Z-scores, or standard
scores. The purpose of a z-score, or standard score is to identify & describe the exact
location of every score in a distribution. See the complete definition in your class
notes. Remember what an X-value is? An X-value is one particular raw score or value
in your data set.
d. Once raw scores are converted into standard scores, the values can then be
compared to other raw scores that were converted to similar standard values. When
you convert raw scores to standard scores, you are converting into the normal curve
with your mean, median & mode all in the center. But, the z-score is not the only
standard score. A z-score has a mean of 0 & standard deviation of + or – 1. There
are other standard scores such as SAT scores or IQ scores (see your class notes).
4. How do we convert raw scores to standard deviation scores? There is a formula used,
one for a population & one for a sample (notice the differences in the notations of each).
See the formulas in your notes followed by the example. There are also formulas there for
converting standard scores back into raw scores, for finding the mean or even the
standard deviation.
5. Find the portion of the class notes that list 3 normal curves with certain portions of them
highlighted in the center column. Although determining whether 50% or more of the
distribution is being covered tells you whether or not you are working with the body or
the tail (when 50% or less is covered), another way to determine this is by asking
yourself whether or not you are looking to the left of the identified z-score or the right of
the identified z-score. If the identified z-score falls within the + (positive) side of the
distribution (to the right of your ‘0’ or mid-point), the space to the left of the z-score will
indicate the “proportion in the body – greater than 50% of the distribution.” The space to
the right of the z-score will indicate the “proportion in the tail – less than 50% of the
distribution.” If the z-score falls within the – (negative) side of the distribution (to the left
of your ‘0’ or mid-point), the space to the left of the z-score will indicate the “proportion
in the tail – less than 50% of the distribution.” The space to the right of the z-score will
indicate the “proportion in the body – greater than 50% of the distribution.”
You have a third option, or the proportion between the z-score & ‘0,’ or the mid-point.
This is used sometimes if you are interested in the placement of a z-score in comparison
to the average.
6. The Unit Normal Table: This table allows you to locate the position of specific scores &
also the distance from one score to another. If you look at the bell curves at the end of
your class notes, the values & percentages are based upon either 1 or ½ a single standard
value. The unit normal table gives you decimals (which can be converted to percentages)
associated with each specific score. By using the unit normal table, based upon a normal
distribution, you can compute probability values.
Using the Unit Normal Table
1. There are a few resources available to you containing the Unit Normal Table. The
most useful is in the back of your text, Appendix B. There is also a web site available
for you under the Week 6 Blackboard material, but your UNT in the back of the text
is your best resource.
2. As the values in percentages are given in accordance with each standard deviation (or
half standard deviation) underneath the normal distribution curve, the unit normal
table provides exact values in accordance with each z-score. Remember that a zscore represents an exact deviation of each value from the mean by converting
standard scores into z-scores with a mean of 0 & SD of 1. Thus, we can then
compare “different” raw scores by converting them to the same standard scores.
3. As you view the Unit Normal Table, there are several columns. The first, lists zscores (up to 4.00). The second lists the proportion under the normal curve in the
body (where more than 50% of the area is being searched). The next is the proportion
in the tail (where less than 50% of the area is being searched). Then, you have a
column that represents the area between the mean & any point beyond the mean. All
Unit Normal Tables do not provide as much information.
4. Although it is possible to obtain values above a standard deviation of 4.00 (which is
where your table stops in the back of the text), but the probability of obtaining a score
beyond that point is so rare, that it is not included in the UNT. If you go back & look
at the percentages of scores or values that fall under the normal curve (You can find
these values in the curves listed at the bottom of your Class Notes & they are also
repeated in your Week 6 Announcement under the section that helps provide more
detailed instructions on your assignment.) You see that at the SD point of + or – 3,
only 0.1% of values occur beyond these points. Therefore, obtaining a value beyond a
SD of 4 is even less likely. A 2 standard deviation point difference is considered
SIGNIFICANT.
5. No negative z-scores are represented in the Unit Normal Table. It is assumed that
each z-score listed & corresponding proportions are the same if the z-score were a
negative number. This is one of the reasons why you would want to draw your
distribution out so you can double check & make sure that if are working with
negative values (you would be working with the left side of the bell curve). As
negatives are not listed under the Unit Normal Table, it is pertinent that when
converting raw scores to z-scores or standard scores, that you are very careful to
maintain directionality (+ symbol if you are working with positive values, or the right
side of the curve. – symbol if you are working with negative values, or the left side of
the curve). If you do not maintain & indicate directionality appropriately, then your
outcome will be in error. Again, drawing out your distribution can be valuable in
helping you to assimilate & maintain the organization of the material.
6. The values are listed as decimals, but can then be converted to percentages. For
example, if you are searching for the proportion in the body given a z-score of 0.24,
the value that corresponds with that z-score is .5948. Or, 59% of the normal curve is
covered in the body give a z-score of 0.24.
Re: Your assignment this week
The first set of questions is asking you to determine the distance under the normal curve between
two z-scores. Remember that the area under the normal curve ranges from 0% (none of the area
covered) to 100% (all of the area covered). When you look at the two normal curves presented to
you at the end of your Class Notes:
you see that two sets of scores are listed. First, are the z-scores listed on the horizontal line. Each
z-score represents a standard deviation (or as in the first curve that includes ½ SD’s) from the
mean or the center. Now in a normal curve, each side is a mirror image of the other. Those zscores with negative values fall on the left side of the mean. Those with positive values fall on
the right side of the mean. The second set of values you are looking at are % values that list the
percentage of the area covered under the normal curve between the z-scores listed. When asked
about the distance covered under the normal curve, you simply need to add up the % values
listed between the two z-score values given to you. The best way would be to mark the 2 scores
given to you & then color in the space between these two scores. For that area that is colored in,
add up the % values within that space. You will not subtract simply because you may be working
with values to the left of center, for the question is asking you to identify the area covered under
the normal curve, or the % values given to you.
The second set of questions asks you to determine the two parts of a z-score. First, the value &
second, the directionality; whether or not the score falls above the mean or below the mean.
Next, when you are asked to convert raw scores to standard scores, or z-scores, use your z-score
conversion formula:
z=x-µ
σ
When asked to convert z-scores back to raw scores, use the conversion formula for converting
back to raw scores:
X = µ + zσ
For the “Bill & Ted” question, you must convert their scores to standard scores to determine
whose score falls highest on the normal curve.
When determining the probability of obtaining a certain value, follow these steps:
(1) Draw a normal curve & mark on the curve where the value would fall (for instance, a)
states -1.00, so you would mark on the normal curve -1, or 1 SD below the mean or
the center of the curve). You don’t have to include the drawing in this assignment.
Just use it for reference.
(2) Go to the back of your text, Appendix B, the Unit Normal Table & find the z-score
1.00 under the z-score column (The Z-scores in the Unit Normal Table does not
recognize negative values since the other columns are used to determine distance
under the normal curve)
(3) The question will ask you to find the probability of obtaining a score greater than, less
than or b/t the mean & the z-score. If the question asks “greater than,” then you are
looking at the area to the right of the z-score. Color that area in under the normal
curve that you drew. Since the first problem says “greater than,” that means that you
will color in all the space from -1.00 to the right under your normal curve.
(4) You have 3 options in the Unit Normal Table to choose from. The first column says
“proportion in the body.” This means if you are looking at an area under the normal
curve of 50% or greater, this would represent the body of the curve. The next column
says “proportion in the tail.” If less than 50% of the area under the normal curve is
colored in, then you are looking at the area in the tail. Then the last column, “area b/t
the mean & the z-score” is just that. The area you would be looking at b/t the z-score
& the mean, or the very center of the distribution. For our first problem, since all of
the positive side of the normal curve is colored in including 1 SD on the negative
side, then more than 50% of the area is colored in, so we would be looking at the
proportion in the body.
(5) Write down the .xxxx value correlating w/ the z-score & the proportion you are
looking at. Then, move the decimal over 2 places the right. This will then tell you the
probability of obtaining a score that falls w/in the space you have colored in. For the
first problem, since we are looking at the area in the body, the value corresponding to
our z-score of 1.00 is .8413, then our probability would be 84.13%