Normal Distribution

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Normal Distribution

• HED 489 Biostatistics

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• Odds are that you've come across the normal
  distribution below. You may know it by it's more
  common namethe "bell-shaped curve." In case
  you've never seen it before, that's the normal
  distribution on the right. We spend time
  understanding the normal distribution for two
  reasons
• it forms the basis of probability, and
  probability forms the basis of all statistical
  tests
• whether the distribution is normal is one of
  theperhaps the primary determinant of which
  family of statistical tests you will apply.

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Formation of The Normal Curve

• Assume for the moment that the data in the slide
  to the right represent ages of a bunch of people.
  As you can see, young people are on the left and
  older people on the right. Most of the people are
  between 9 and 11, right? That's the big bunch in
  the middle.

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• Now, let's say that we draw a line connecting the
  top midpoint of each bar. Here's what that would
  look like

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• See the nice straight lines you get? That's
  because, for purposes of this explanation, I've
  created the distribution such that those straight
  lines would result! Now, how about if we smooth
  the lines so they become a nice curve. That's
  what this next slide shows

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• See how we have the outline of a bell-shaped
  curve. This would actually happen with these
  data. The last thing that happens when we have a
  normal distribution is that the outline becomes
  filled in with data. That's what this next slide
  shows.

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• So, if we strip away the unessential information,
  we're left with the first slide you saw in this
  lesson. Remember what it looked like?

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Normal Curve Properties

• All normal distributions share some standard
  properties
• the mean bisects the distribution exactly, such
  that the two halves of the distribution form a
  mirror image of each other.
• the standard deviation of a normal distribution
  will always be 1.0
• the "tails" of the distribution never contact the
  x-axis.

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Moving Towards Probability

• As you can see on the slide to the right, we can
  plot the location of various values of the
  standard deviation by adding or subtracting the
  standard deviation (1.0) to or from the mean (0).
  See where positive and negative standard
  deviations fall on this distribution? Normally,
  we don't go beyond 3.0 standard deviations,
  commonly abbreviated "s.d." We'll talk about why
  in just a bit.
• Remember, if we were measuring age, our s.d.
  would be in increments of years. If we were
  measuring water consumption, our increment might
  be ounces.

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• If you collect the ages of, say, 10,000 people,
  and build a frequency distribution, it will
  contain all 10,000 people, right? That goes
  without saying, doesn't it? Well, the same thing
  can be said of the normal distribution. That is,
  all of the data, or 100 of it for a particular
  variablelike agewill be contained within the
  distribution. For any normal distribution, no
  matter what the data are, 99.7 of the data will
  be contained in the space from -3 to 3 s.d. Do
  you see that in the slide on the right? See all
  the blue area between -3 and 3? That's where
  most of the data are. See how little blue there
  is to the left of -3 and to the right of 3?
  There are very few cases there. Combined, in
  fact, only .3 of the data are in those little
  "tails."

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• So, if you had those 10,000 ages on slips of
  paper, and you selected one at random, it would
  fall between 3.0 s.d. 99.7 of the time. That is
  to say there is a probability of .997 of
  selecting an age at random that falls between
  3.0 s.d. Probability, folks, is where we're
  going with this. Because the largest area under
  the curve falls between 1.0 s.d., it stands to
  reason that most of the cases will be contained
  in this area, too. As you can see from the slide,
  about 68.2 of the cases (34.1 2) fall between
  1.0 s.d.
• An additional 14 of the cases are contained
  between 1.0 and 2.0 s.d., and 14 more fall
  between -1.0 and -2.0 s.d. See that? In total,
  95.4 of the cases fall between 2.0 s.d.
• Because there's not much area under the curve
  between 2.0 and 3.0 s.d., only 2.2 of the cases
  will fall between 2.0 and 3.0 s.d., and between
  -2.0 and -3.0 s.d. Remember we said earlier that
  99.7 of the cases fall between 3.0 s.d.? Well,
  this is how we get to 99.7.
• If you didn't follow that, go back and study it
  again until you understand it.

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Taking Another Step

• This slide summarizes what you just learned. That
  is, about 68 of the cases fall between 1.0
  s.d., about 95 between 2.0 s.d., and about 99
  fall between 3.0 s.d. Now, pay close attention
  it's going to get tricky If about 95 of the
  cases fall between 2.0 s.d., then about 5 will
  be greater than -2.0 or 2.0 s.d. Do you see
  that? If all, or 100 of the cases fall somewhere
  along the curve, and you account for 95 of them
  between 2.0 s.d., then 5 are left. About 2.5
  of the cases fall to the left of 2.0 and another
  2.5 fall to the right of 2.0 s.d.
• Still with me? Ok, contemplate this if you pick
  a value at random, there is a probability of .05
  that it will be greater than -2.0 or 2.0 s.d.
  Why? Because there's a probability of .95 that it
  will be between 2.0 s.d.

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Calculating the Z Score

• Calculating the z score is relatively simple. 
  Just follow the formula.

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• You will need to know the mean and the standard
  deviation.  Just punch in the score you need and
  you'll get the z score.  For example, if you
  scored 100 on the test, and the mean is 80 and
  the standard deviation is 10, you'll have the
  formula as such
• z 100 - 80 / 10 z  20 /10z 2

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The Normal Distribution So What Does the z Score
mean?

• It's time for you to open your textbook to the
  inside back cover (to Table A).  Click slide and
  it will appear on-screen (it is also shown on the
  next slide in this program). Now, what this table
  shows is the area under the curve from zero to
  any point along the curve, out to three decimal
  points, to 4.0 s.d.
• This table is also referred to a table of
  "z-scores." See the "z" in the upper left cell?
  For a normal distribution of data, the standard
  deviation is equal to z. More on that in a
  moment. First, I want to get you comfortable with
  this table.
• Remember we said that about 34 of the cases fall
  between zero and 1 s.d.? In case you forgot that,
  just look at the slide above and it will remind
  you. And, remember that we said that if we
  selected a case at random from a normal
  distribution of data that the probability is
  about .34 that the value of that case will fall
  between zero and 1.0 s.d.?

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• Remember that? Ok, now, look down the left column
  of Table A, either in the book or on the slide at
  the top. Did you get there? Now, move your finger
  one column to the right. That's the one labelled
  ".00" The value in the cell where your finger is
  pointing is .3413, right? That's where I got
  "about" 34 and "about" .34.

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• The above formula is used to calculate the z
  score.  What happens if you're interested in some
  s.d. other than 1.0, 2.0, or 3.0? Here's where
  the table really comes in handy! Say you're
  interested in the area under the curve (also
  known as "probability," right?) for a s.d. (or
  z-score, as I want you to begin thinking, as
  well) of 1.96? How do you get there. Well, run
  your finger down the first column until you get
  to 1.9, and then run across until you get to .06.
  1.9 .06 1.96. What value did you find? .4750?
  If so, you did it just right!
• Note that the Normal Curve table just shows half
  of the curve, that is, from zero to the right.
  That's because the curve is symmetrical, so if
  you want to know the area, or probability, for
  the other half of the curve, just double the
  tabled value. For example, if you wanted to know
  the probability for 1.96, you'd multiply .4750
  time 2. That would give you .9500, and you'd say
  that the probability of selecting a value at
  random that would fall between 1.96 s.d. would
  be .95. Ok? Now, I want to give you some practice
  working with the normal curve so I know that
  you've become comfortable with it.

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Calculate the percent under the curve between the
mean and a z-score (or s.d.) of -1.39.

• To do this, use the Normal Curve table. Run your
  finger down the first column until you reach 1.3.
  Run your finger along the 1.3 row until you reach
  the .09 column. The figure in that cell is the
  area under the curve from 0.00 to 1.39. It is
  also the probability of selecting a value at
  random and having it fall between 0.00 and 1.39.
  Since the curve is symmetrical, the value for the
  area 0.00 to 1.39 is the same as for 0.001.39.
  Note this is probability or area under the curve,
  not percentage. Click for table.

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Calculate the probability of selecting a value at
random greater than a z-score (or s.d.) of 2.01.

• You should be able to find the area under the
  curve, also know as probability, for a value of
  2.01. Run down the first column until you get to
  2.0. Then across the row to the .01 column.
• However, consider the curve at the lower right.
  See the figure .4772? That's the area under the
  curve for 0.00 to 2.00. What you want is the area
  beyond, or to the right of 2.01. To get that, you
  have to subtract the probability for 0.00 to 2.01
  from all of the area on the right side of the
  curve. That is, from 0.00 to infinity. What is
  this figure? Can't remember? Look at the Normal
  Curve table.
• Remember, report probability, not percentage.
  Click to get table.

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Calculate the probability of selecting a value at
random greater than a z-score of 2.00 (note ).

• This assignment requires you to consider both
  sides of the normal curve. Like the last
  assignment you need to calculate "what's left"
  under the curve from 0.00 - -2.00 and 0.00 -
  2.00. To do this, you need to determine the
  probability from 0.00 to 2.00 double that value
  to include the left side of the curve and,
  subtract that figure from the probability for the
  entire curve.
• Click to get table.

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• Calculate the z-score equivalents of these
  systolic blood pressure values 100, 120, 130,
  140, and 190, where the mean equals 126.2, and
  the standard deviation equals 18.8.  Click here
  to enter an excel file to do the work.

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• Calculate the probability of selecting a blood
  pressure value at random greater than 160, where
  the mean is 126.20 and the standard deviation is
  18.80.
• To determine this value, you first need to
  calculate the equivalent z-score, as you did in
  the prior assignment. Then, determine the area
  under the curve represented by that z-score.
  Finally, calculate the area beyond, or greater
  than that value. The resulting value represents
  probability. Click here for an excel file to work
  on.

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• Calculate the probability of selecting a blood
  pressure value at random between 110 and 135
  where the mean is 126.20 and the s.d is 18.80.
  Click here for an excel file to work on.

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• We're about to make the transition from
  descriptive to inferential statistics. The heart
  of inferential statistics is "statistical
  significance." This is the probability that the
  value you ended up with when you completed your
  calculation happened is "real" or happened by
  random chance. Here's an example

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• Let's say that you're working with a group of
  people to get their blood pressure down. High
  blood pressure is unhealthy, right? One of the
  things you do is put them on a gentle exercise
  program, both to get their weight down a bit and
  to strengthen their cardiovascular system. All
  things being equal, if your program is
  successful, their blood pressure should come
  down.
• If you take the individuals' blood pressure at
  the beginning of the study, calculate the mean,
  and take it and average it again at the end, the
  mean blood pressure at the end should be lower
  than at the beginning if your program worked.
  But, how much lower does ending blood pressure
  have to be in order for us to assert with some
  confidence that our program was successful? If
  the beginning group average was 128 and the
  ending was 124, is this difference large enough
  to claim programmatic success? What if the ending
  average was 120? 110? This is the issue central
  to statistical significance is the difference
  between the two values so close to the mean of
  the normal distributionzerothat the probability
  of selecting the value at random is too high to
  accept as "real?" Or, is it so far from the mean
  that it's out in one of the tails of the
  distribution, where the chances of pulling out of
  the hat of values randomly is very small, and,
  therefore, more likely "real?"
• Here are some ways that a difference in mean
  values before and after your high blood pressure
  occur by chance alone for any particular group
  of people, their blood pressure might go down for
  reasons other than our exercise program. Maybe
  they had a high salt diet when they started and
  cut down on their sodium intake. Maybe they were
  experiencing a lot of stress and they got it
  under control. Or, maybe their blood pressure
  just went down unexpectedly.
• Conversely, maybe your program actually worked.
  If so, you should be able to select another,
  similar group, conduct the same program, and find
  similar a similar difference between starting and
  ending blood pressure values. Not exactly the
  same, but the difference should be fall in the
  same general area on the normal curve. If you
  conducted this program with 100 similar groups
  and found about the same difference, see how
  you'd be pretty confident that your program
  actually worked? Well, you probably can only run
  it once, so you need that difference value to
  fall a good long way from the mean in order to
  have confidence that your program worked. That's
  what statistical significance does for you.

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• Ok assume that the five blood pressure values
  that you worked with for the last assignment, and
  are now plotted nicely on the normal curve to the
  right actually represent the difference in mean
  blood pressure between the beginning and ending
  values for five groups of people. In other words,
  you conducted your study five times.
• Many researchers use a statistical significance
  level of .05 as their critical level. This means
  that if the value is statistically-significant,
  it will occur by chance alone, less than 5 times
  in 100. This means that if you conducted your
  study 100 times and found a mean difference about
  this large each time, you would be right in
  claiming that your program worked at least 96
  times that's not too bad for this kind of
  program.
• Another way of looking at the .05 critical value
  is that it has to fall into one of the two tails
  of the normal distribution, and not from that big
  bunch of scores in the middle. Why? Because there
  are lots of scores in the middle, and you have a
  very good chance of selecting one of them by
  chance alone. So, if the mean difference score is
  in the "bulge" of the distribution, the
  probability that it happened by chance alone is
  too great for you to assume your program worked.

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• So, on the normal curve, in order for a value to
  be statistically-significant at the 95 level or
  greater, you have to have a z-score of greater
  than 1.96. That is, the value has to come from
  the area to the left of -1.96 or from the area to
  the right of 1.96. This is the only way that you
  can reduce your odds that the value you
  calculated occurred by random chance alone less
  than 5 times in 100.
• If the value you calculate is statistically-signif
  icant at, let's say the .05 level, you write it
  like this plt.05.
• "lt" stands for "less than." If it's not
  statistically-significant, you write pgt.05. "gt"
  stands for "greater than." Because I may have
  left you wondering, the reason that we almost
  always consider both sides of the normal curve is
  because our calculated value might be greater or
  less than the mean. For the blood pressure
  example, our program may have failed so badly
  that we actually caused the average ending blood
  pressure to increase! We'll talk a bit more about
  one-tail vs two-tail tests a little later.
• I accept this may be confusing for you. I assure
  you that we'll do lots of work with probability,
  statistical significance, critical values, and
  the tables in the back of the textbook before
  we're done. I'm pretty sure it will all become
  clear before we're done.