Normal Distributions

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Normal Distributions

• Basics

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General Outline

• The Normal Curve
• Properties
• Limitations
• The Standard Normal Curve
• Properties
• Why its useful

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General Outline

• z-scores
• What they are
• How we calculate them
• Applying z-scores
• z-scores for non-normal distributions
• Transformed standard scores

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New Statistical Notation

• The _________________ of a number is the size of
  that number, regardless of its sign. That is, the
  absolute value of __________ and the absolute
  value of _____________

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Frequency Distribution of Attractiveness Scores
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z-Scores

• Like any raw score, a ____________________________
  _______________. A z-score also automatically
  communicates the raw scores distance from the
  mean
• A z-score describes a raw scores location in
  terms of how _______________ the mean it is when
  measured in ___________________

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The Normal Curve

• Results of studies are more accurate with more
  people.
• As your of observations reaches infinity, you
  get a frequency distribution that is called the
  _________________________________.

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Properties of the Normal Curve

• __________________
• ____________________
• The upper half is a mirror image of the lower
  half
• Values of the mean, median, and mode are the
  _______________

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Normal Curve

• Theoretical distribution of population scores
• Equation
• Y N/v2p se (X µ)2/2 s2
• Where
• Y frequency of a given value of X
• X any score in the distribution
• µ mean of the distribution
• s standard deviation of the distribution
• N total frequency of the distribution
• p a constant of 3.1416
• e a constant of 2.7183

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• OK, now that Ive scared you
• We actually rarely need to know the exact
  equation for the normal curve
• What you do need to know is that the normal curve
  is _____________________________ generated

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Leading to the Standard Normal Curve

• The shape and location of the normal curve
  depends on the mean and standard deviation of the
  population of interest
• Which means you could have a ______________differe
  nt ________________ for a _______________
  different types of observations

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• This is not very practical
• So mathematicians came up with the
  ____________________________.

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The Standard Normal Curve

• Properties we know ________________about this
  distribution!
• ____________________________________
• Area under the curve always equals 1.0, and we
  know all proportions under the curve

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The Standard Normal Curve Contd

• Each point along the x-axis corresponds with
  something called a _______________.
• We can make our scores (or observations) map onto
  this normal curves by transforming them into
  __________________.

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Example

• IQ scores
• Lets say you have an IQ score of 132
• Is this good or bad?
• It is hard to say unless we have a
  _____________________ to compare the score
• Reference group helps us know if this score is
  __________________________

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z-scores

• A z-score indicates how many _____________________
  _____ an observation is above or below the
  _____________.
• Also called a standard score

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Characteristics of z-scores

• Distribution has same _____________ as the raw
  scores
• Scores maintain ____________positions
• All z-scores are not ______________shaped
• The mean of the z-score always equals zero
• Standard deviation always 1

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Z- score

• The formula for population is
• z (X µ) s
• z the z-score youre calculating
• X _______________
• µ mean
• s ___________________
• Often we use z-scores to compare our score to the
  _________________

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Z-scores

• Lets practice using our IQ example
• IQ 132
• µ 100, s 16
• What is our z-score?
• __________________
• This number is called our ________________________
  __

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Your turn

• Your score is a 95.
• The population mean is 90 with a standard
  deviation of 10.
• What is your z-score?

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A z-Distribution

• A _____________________ is the distribution
  produced by transforming all raw scores in the
  data into z-scores.

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z-scores

• If you know the z-score, you can find out how
  many observations fall above or below that score
  by using the ____________________, which is also
  called a z-table.
• In other words, you can get a ____________________

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z-Distribution of Attractiveness Scores
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Steps for Finding Proportions

• Draw a ________________ and shade in the target
  area.
• Plan your solution according to the standard
  normal table.
• Convert X (your _____________) into z (your
  _________________ score)
• Find the target area by referring to your
  _____________________

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What does Z-table tell us?

• Column A _______________
• Column B proportion between mean and z-score
• Column C ___________________

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Examples

• For the following z-score, lets find the
  percentage of scores that lie beyond z.
• Z 2.05
• Go to table 1 in Appendix C beginning on page 450
• Since we are looking for what lies beyond z, use
  column C
• We find ________________
• Multiply by 100 and get 2 (so 2 of scores are
  above a z of _______________

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Examples

• Lets try another one.
• Z score .55
• What percentage lies between this z-score and the
  mean?
• This time look at column B.
• ____________________

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Your turn

• For the following z-score, what percentage lies
  beyond the z?
• Between mean and z?
• 1.74

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What about negative zs?

• If the z-score is negative, the value between the
  mean and z remains the same.
• The only difference is that we are
  _______________the table.
• we are now talking about what lies below z in
  column C.
• This will give us _____________________ for the
  negative z-score

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Examples

• -.45
• Area between mean and z .1736
• Percentile __________________ percentile
• What about -2.30?
• Area between mean and z ______
• Percentile ____________________________

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Percentile

• If you want to know the percentile for a positive
  z-score, you can take the area between the mean
  and the z-score and add .50.
• Remember, the normal curve is _____________ and
  half the scores fall below the mean while ½ fall
  above the mean.

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• So, to find the corresponding percentile, add
  .50.
• Lets say our raw score 84 µ 80 and s 12
• Our z-score .33
• In our table, we find that the area between the
  mean and the z-score ___________
• Add .50 to this value and we find that our
  percentile __________________________

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Your turn

• The z-score 1.16
• How many scores fall between the z and the mean?
• How many scores fall below this z-score (or what
  is the percentile)?

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Finding a raw score from a z (working backwards)

• Remember
• z (X µ) s
• So, to solve for X, we would do this
• .52 (X 80)/ 12
• X 80 12(.52) _______________

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Between scores

• We can also find the proportion of scores that
  fall between 2 scores.
• For example 100 and 95 (µ 80 s 12)
• Z-scores 2.5 1.25, respectively
• From column B Area (z 2.5) .4938 and Area (z
  1.25) .3944
• So, _______________________________

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z-scores for Non-normal Distributions

• Sometimes a distribution is not normal (i.e., it
  is ______________)
• In this case, the distribution of z-scores will
  also be ________________.
• This distribution will still have a mean of 0 and
  a standard deviation of 1, but
• You cant use the standard normal table to find
  proportions.
• However, you can still make interpretations based
  on the z-score.

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Standard Scores

• Standard scores can be any score that is relative
  to a ________________________________.
• z-scores can be transformed into other types of
  standard scores ( z ).
• T-score, GRE score, SAT score, IQ score, etc.

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Sampling Distribution of Means

• A distribution which shows all possible sample
  means that occur when an infinite number of
  samples of the same size N are randomly selected
  from one raw score population is called the
  _____________________________.

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Central Limit Theorem

• The _________________ tells us the sampling
  distribution of means
• forms an approximately _______________________,
• has a m equal to the m of the underlying raw
  score ____________________, and
• has a standard deviation that is mathematically
  related to the standard deviation of the
  ________________________.

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Standard Error of the Mean

• The standard deviation of the sampling
  distribution of means is called the
  _________________________. The formula for the
  true standard error of the mean is

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z-Score Formula for a Sample Mean

• The formula for computing a z-score for a sample
  mean is

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Example

• Using the following data set, what is the
  z-score for a raw score of 13? What is the raw
  score for a z-score of -2?