Using the 68-95-99.7 Rule Normal Quantile Plots

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Using the 68-95-99.7 RuleNormal Quantile Plots
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Learning Objectives

• By the end of this lecture, you should be able
  to
• Do various calculations involving areas under the
  density curve using the 68-95-99.7 rule
• Identify the mathematical technique used to help
  confirm (thought not guarantee!) that our
  distribution is indeed Normal.

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A few numbers worth memorizing(though not just
yet)

• Because we use the Normal distribution SO much,
  it is worth memorizing the approximate areas from
  the Normal table that correspond to a few
  different z-scores.
• I say approximate, because the values are rounded
  off.
• Look at the areas shown here but dont memorize
  them just yet.
• z -2 ? about 2.2
• z -1 ? about 16
• z 1 ? about 84
• z 2 ? about 98
• What I do want you to memorize are the 3 numbers
  shown in a famous rule on the next slide.

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The 68-95-99.7 Rule for Normal Distributions

• This is essentially a shortcut for a mental
  ballpark of the areas under the normal curve. It
  is definitely worth memorizing.
• The area between -1 and 1 standard deviations
  corresponds to about 68 of the observations.
• The area between -2 and 2 standard deviations
  corresponds to about 95 of the observations.
• The area between -3 and 3 standard deviations
  corresponds to about 99.7 of the observations.

You WILL be asked to use these numbers on quizzes
and exams. Please note that on your exams you
will not be provided with the three numbers (68,
95, 99.7).
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Examples The 68-95-99.7 Shortcut Rule for
Normal Distributions
The z0 line (black line) is very helpful in
doing many of these calculations.

• Now lets play around with these numbers by
  answering some questions. All numbers refer to
  z-scores (i.e. standard deviations)
• What percentage of observations lie between -1
  and 1?
• Answer As we just discussed, the number of
  observations between -1 and 1 standard
  deviations is 68.
• What percentage lie between 0 and 1?
• Answer Recall that z0 represents 50. So, if -1
  to 1 is 68, then 0 to 1 is half of that, which
  is 34.
• This is an important one. Make sure you
  understand how to do it!!
• There are a few ways to think of it Look at the
  area between z0 (the black line) and z1. Note
  that is is half of the area between -1 and 1.
• If you need to visualize it (and you should!!),
  then shade in the area between z0 and z1.
• What percentage of observations lie below 1?
• Answer To do this, look at your z0 line. Make
  sure you recognize that the area to the left of
  z0 represents 50 of observations. Now, how many
  observations are between 0 and 1? Recall from
  the previous question that this is 34.Therefore,
  from 0 to 1 34, and below 0 is 50, so the area
  to the left of 1 represents 84 of observations.

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Examples The 68-95-99.7 Shortcut Rule for
Normal Distributions

• More examples
• What percentage of observations lies between -2
  and 1?
• Answer Use your midline! I would solve this by
  adding the area between -2 and 0 (half of 95) to
  the area between 0 and 1 (half of 68) ? 47.5
  34 81.5
• What percentage of observations lies between 0
  and 3?
• Answer Half of the area between -3 and 3 (99.7)
  which is 49.85.
• What percentage of observations lies below -2?
• Answer While this too can be answered in a few
  different ways, I would like you to make sure you
  can do it this way
• Look at the area between -2 and 2. Our
  shortcut tells us that this contains 95 of
  observations.
• This means that the area above 2 and below -2
  together compromise 5 of observations. So the
  area above 2 2.5 of observations, and the
  area below -2 also comprises 2.5 of
  observations.
• Answer 2.5
• What percentage of observations lies above 3?
• Answer Use the same technique as was just
  discussed
• Between -3 and 3 makes up 99.7.
• Therefore below -3 and above 3 makes up 0.3.
• Therefore below -3 is 0.15 and above 3 0.15

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Examples The 68-95-99.7 Shortcut Rule for
Normal Distributions

• One more!
• What percentage of observations lies below 2
  standard deviations?
• Answer Repeat the process from before to
  determine the area on either side of 2 and -2.
  That value was 2.5. If 2.5 of values lie above
  2, then 97.5 of observations lie below it.
• Answer 97.5

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The 68-95-99.7 Shortcut Rule for Normal
Distributions

• What percentage of women are between 62 and 67
  inches tall?
• Answer Corresponds to -1 to 1 SDs, that is,
  about 68
• What is the range of heights between which about
  95 of women fall?
• Answer About -2 to 2 SDs, so, about 59.5 to
  69.5 inches tall.
• What is the range of heights between which nearly
  all (over 99) of women fall?
• Answer A quick answer would simply to pick the
  -3 to 3 SD range (57-72).

Inflection point
mean µ 64.5 standard deviation s 2.5
N(µ, s) N(64.5, 2.5)
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The 68-95-99.7 Shortcut Rule for Normal
Distributions

• More Examples
• What percentage are taller than 67 inches?
• Answer If 68 of all women are between 62 and
  67 inches tall, this means that 32 are outside
  of that range. In other words, 16 are shorter
  than 62 inches, and 16 are taller than 67.
• What percentage are shorter than 59.5 inches?
• Answer If 95 of all women are between 59.5 and
  69.5, then 5 are outside of that range. In
  other words, 2.5 are shorter than 59.5 and 2.5
  are taller than 69.5.

Inflection point
mean µ 64.5 standard deviation s 2.5
N(µ, s) N(64.5, 2.5)
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Shortcut Rule or Z-Table?

• Students have often been confused as to which
  should be used.
• Whenever possible, use your z-table as you will
  get a much more accurate result. In particular,
  if you are given z-scores that are not anywhere
  near whole numbers (e.g. 2.332), then there is no
  shortcut to use! The shortcut can only be used
  with whole (integer) numbers between -3 and 3.
• The main purpose of learning the shortcut rule
  (in addition to the fact that they come up on all
  kinds of exams), is to encourage you develop an
  undersatnding of what you are trying to do rather
  than just jumping to calculators and z-tables.
• For this course, you will be asked to do both.

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Is the distribution truly Normal?

• Deciding whether data does indeed show a Normal
  (or, close to Normal) distribution is a very
  important question.
• All the examples weve been discussing above
  involving z-scores assume that the data is
  Normal. If the data was not Normal, all of our
  answers and calculations would be flawed.
• Recall that there are many other types of
  distributions that are not Normal. Some examples
  include skewed, bimodal, Binomial (later in the
  quarter), Poisson, etc, etc
• Each type of distribution has its own
  characteristic formulas, calculations, inference
  techniques, etc. Again, because the Normal
  distirbution is one of the most commonly
  encountered distributions, we will spend lots of
  time discussing it.
• So how to you decide if a distribution is Normal?
  
• You might be tempted to say look at a graph.
  And this is not entirely false When examining
  data, a chart is a great (if not the BEST) place
  to start!
• However, as humans, we are easily fooled. There
  are many histograms (and related density curves)
  that look Normal, but in fact, are not.
• Fortunately, we do have a statistical test that
  can help confirm (thought not guarantee) that our
  dataset does indeed appear to be Normal.

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Normal Quantile Plot

• The Normal Quantile plot is a graph that helps us
  determine if a distribution is indeed Normal
• It is a mathematical plot that we can create
  using our statistical software package of choice.
  
• Here is the method (which is provided for
  interest only)
• The data points are ranked and the percentile
  ranks are converted to z-scores with Table A. The
  z-scores are then used for the x axis against
  which the data are plotted on the y axis of the
  normal quantile plot.
• If the distribution is indeed normal the plot
  will show a fairly straight line, indicating a
  good match between the data and a normal
  distribution.
• Systematic deviations from a straight line
  indicate a non-normal distribution. Outliers
  appear as points that are far away from the
  overall pattern of the plot.

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Normal Quantile Plot shows a good fit to a
straight line the distribution of rainwater pH
values is close to normal.
Normal quantile plot is not a straight line. This
tells us that the data do not follow a Normal
distribution.
Normal quantile plots are complex to do by hand,
but they are standard features in most
statistical software.
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The normal quantile test supports normality, but
does NOT guarantee it!

• Two key points here
• If the plot is NOT straight, then your data is
  NOT normal!
• If the plot IS straight, then you have supported
  the idea that your dataset is normal. However,
  you have NOT guaranteed it!
• This concept (confirming / supportive tests) will
  come up with various other statistical concepts
  down the road. Whenever you encounter them, you
  should be sure to make use of them.

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