Univariate Statistics 2: the normal curve, zscores, and relative frequencies

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Univariate Statistics 2the normal curve,
z-scores, and relative frequencies

• Monday 7th November 2005

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Some Important Characteristics of the Normal
Curve

• The normal curve is a symmetrical distribution of
  scores with an equal number of scores above and
  below the midpoint of the abscissa (horizontal
  axis of the curve).
• Since the distribution of scores is symmetrical
  the mean, median, and mode are all at the same
  point on the abscissa. In other words, the mean
  the median the mode.
• If we divide the distribution up into standard
  deviation units, a known proportion of scores
  lies within each portion of the curve.
• Tables exist so that we can find the proportion
  of scores above and below any part of the curve,
  expressed in standard deviation units. Scores
  expressed in standard deviation units, as we will
  see shortly, are referred to as Z-scores.

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The Standard Normal Distribution.
Mean 0 Standard deviation 1
Total area under the curve 100
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Working out relative frequencies with a standard
normal distribution

• Because the area under a histogram is equal to
  100
• And because we are familiar with the shape of a
  standard normal distribution
• We can use the standard normal distribution to
  work out the relative frequencies of different
  events.

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Question What is the relative frequency of
observations below 1.18?
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Question What is the relative frequency of
observations below 1.18? That is, find the
relative frequency of the event Z lt 1.18. (Here
small z is 1.18.) Step 1 Sketch the curve.
Identify--on the measurement (horizontal/X)
axis--the indicated range of values.
The event z lt 1.18 is shaded in green. Events and
possibilities are one in the same.
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Question What is the relative frequency of
observations below 1.18? Step 2 The relative
frequency of the event is equal to the area under
the curve over the description of the event.
The blue area is the relative frequency of the
event z lt 1.18. This area appears to be
approximately 85-90. A good sketch will help
you verify your answer.
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Question What is the relative frequency of
observations below 1.18?

• Step 3
• Look at the Table that describes the area under
  the standard normal curve. Some tables list
  precise values. If so you can look up 1.18.
  Otherwise, since this is nearly 1.2, you can look
  that up for an approximation.
• Corresponding to a measurement value of z 1.18
  is an area of 0.8810.
• This is exactly the answer to the question!
  Notice that it agrees with the picture as well as
  the original "guess." For any value z the table
  supplies the area under the curve over the region
  to the left of z.
• Again, area relative frequency.

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A note on the layout of tables.
Tables showing z-scores are laid out differently.
However they all will tell you the same thing. A
common layout, thats different to Field is shown
below. If we go across from 1.1 to 0.08, we find
the score for 1.18. It says 0.8810. This is what
Field calls the Larger Portion. This table does
not give the Smaller portion. But since we know
that the two portions combined 1.0 (or 100),
we can work it out as 1 - 0.8810 0.1190 (which
is what Field lists as Smaller Portion
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Question What is the relative frequency of
observations below 1.18? Answer 0.8810 or
88.10. For a standard normal variable the
relative frequency of observations falling below
1.18 is 0.8810. (Also, for any normal
distribution, 0.8810 or 88.1 of the observations
fall below 1.18 times the standard deviation
above the mean.)
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Question What is the relative frequency of
observations below -0.63?
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Question What is the relative frequency of
observations below -0.63?

• Identify the range of values described by "below
  -0.63" (shaded green).
• Identify the area you need to find (shaded blue).
  
• Look-up the appropriate area in your table. (In
  some tables you must be careful to choose the
  "negative" portion of your table--look up -0.63.)
  
• That area is 0.2643. For a standard normal
  variable the relative frequency of observations
  falling below -0.63 is 0.2643. (Also, for any
  normal distribution, 0.2643 or 26.43 of the
  observations fall below 0.63 times the standard
  deviation below the mean. Below because -0.63 is
  negative.)

Answer 0.2643 or 26.43.
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howevera problem

• So far weve been working with a standard normal
  distribution.
• However this is not really that useful since most
  populations do not have a mean of 0 nor a
  standard deviation of 1.
• In the real world statistics needs to be able to
  work out the likelihood of other types of events,
  which have a wide range of values.
• For instance How likely is it to get a mark of
  at least 75 in an exam, if the mean is 60 and the
  standard deviation is 10?

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z-scores

• In order to address these types of problems we
  use z-scores.
• z-scores are calculated by subtracting the mean
  from any value and dividing it by the standard
  deviation.
• Z X - mean
• s
• z-scores will always have a mean of 0 and a
  standard deviation of 1.
• We can quickly see that this is true of the
  mean, since when the Xmean, the numerator will
  equal 0, and therefore z must 0.
• It may be a little less clear that it is true of
  the standard deviation.
• However if you think about the instance when X
  is one standard deviation bigger than the mean
  (i.e. X mean s)
• ? z (mean s) - mean s 1
• s s

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Example conversion to z-scores

• So, returning to the example of the test marks,
  where the question was the likelihood of getting
  a mark of 75 or better if mean60 and s10. We
  can calculate the z-score for getting 75 as
  follows
• Z 75-60
• 10
• z 1.5
• Once we know the z-score for a mark we can work
  out the relative frequency that people will score
  that amount or better in exactly the same way as
  we were when we were working with the standard
  normal distribution.

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Example continued.
The likelihood of a mark of 75 or over is the
same as a likelihood of a z-score of 1.5 or over.
It is shown by the red part of the distribution
If we look up 1.5 we find the area is 0.0670.
Therefore there is a 6.7 probability of getting
a mark of 75 or better.
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You can also find the score that corresponds with
a certain percentage

• For example, you want to find what mark you would
  need to be in the top 20
• Look at the tables in reverse, finding what
  z-score corresponds to a p of .20. ?
  approximately z.84
• Then work out the mark that corresponds to z.84
• X mean (.84 standard deviation)
• 60 (.8410) 68.4
• Therefore you would need a mark of 68.4 to be
  in the top 20

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• Example You want to find the scores people get
  in the middle 30.
• This would correspond with a score ranging 15
  above and below the mean.
• Some tables give values between two points, the
  one in Field does not.
• So we work out that the small area to the right
  of b is equal to (50-15) 35
• We then look at the tables in reverse. And find
  that a smaller portion corresponding to .35
  occurs at the approximate z-score .385
• Likewise the z-score 15 below the mean (at a) is
  -.385
• We can then convert these to marks
• 60 (.38510) 56.15 60 (-.38510) 63.85
• Therefore the middle 30 of marks fell between
  56.15 and 63.85