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Standardizing Data and Normal Model(C6 BVD)
C6: Z-scores and Normal Model
* Imagine a list of data, such as (1,3,5,7,9).
* If you add/subtract something to all the data, what
happens to center (mean)? Spread (Sx)?
* If you multiply/divide all the data by something,
what happens to center? Spread?
* If you subtracted the mean from all the data, what
would the mean of the transformed list be?
* If you divided all data in that list by Sx, what would
the new standard deviation be?
* When you transform the data by subtracting the mean
and dividing by Sx, the new list of data has a mean of 0
and a standard deviation of 1. You can do this to any
data, no matter the shape of the distribution, units,
etc.
* If we then use the standard deviation as a “yard stick”
to see how extraordinary a particular value is, we can
compare values from any data sets, no matter how
different the original distributions were. We can
compare 100m dash times with discus tosses, etc.
* Z = (value – mean)/Sx
* A z-score tells you how many standard deviations
above/below the mean a result is. The farther away it
is from the mean, the more extraordinary or unusual it
is.
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* Sometimes the overall pattern of a large number of
observations is so regular we can describe it by a
smooth curve, called a density curve.
* The area under a density curve is always 1.
* The area under the curve between any two intervals is
the proportion of all observations that fall in that
interval.
* Median – divides curve into equal areas.
* Mean – the balance (see-saw) point.
* Median = Mean if the curve is symmetric.
If it isn’t,
mean is pulled in the direction of skew (the long tail).
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* Normal curves are a very useful class of density
curves. They are symmetric, unimodal, bellshaped. They are described by N(mean, standard
deviation) –these are parameters, not statistics
* The points of inflection are one standard deviation
to either side of the mean.
* There are an infinite number of normal curves.
Your z-table is for the STANDARD NORMAL CURVE
which has been transformed to a mean of zero and
standard deviation of 1 (i.e. standardized to use
with z-scores).
* 68-95-99.7 rule
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* The distribution of heights for U.S. women can be
modeled by N(64.5,2.5)
* What % have heights over 67?
* Between 62 and 72 inches?
* What if z-score is somewhere between the standard
deviations? – Use z-table or calculator -Distributions menu – normalcdf(lower bound, upper
bound)
* Less than 5 feet? Z = -1.8
* Remember: area in table is LEFT-side area.
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* Example: Blood cholesterol level in mg/dl of
teens boys can be described by N(170,30).
What is the first quartile of the distribution?
* 1st quartile – 25th percentile.
* Find .2500 or closest in z-table – read z.
* Calculator – use invnorm(.25) – must write
percentile as decimal.
* Use z-score equation z = (x-170)/30 to solve for
x.
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* Not every density curve that looks normal really is
normal. Never say something is “normal” if is really is
only approximately normal or just unimodal/symmetric.
* How to check:
* 1. Plot data in a dotplot, stemplot or histogram.
Is data
unimodal, symmetric, bell-shaped?
* 2.
Does the 68-95-99.7 rule work? – Find mean and
standard deviation. Are about 68% of data points within 1
Sx of mean? (etc.)
* 3. Can use Normal Probability Plot on TI-calculator – look
for straight diagonal line.
* 4.
If data are not approximately normal, you can still
find z-scores, but you cannot use 68-95-99.7 rule or ztable to find probabilities/areas/proportions under the
density curve.
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