STANDARD DEVIATION

1
STANDARD DEVIATION THE NORMAL MODEL

• What is a normal distribution?
• The normal distribution is pattern for the
  distribution of a set of data which follows a
  bell shaped curve.
• This distribution is sometimes called the
  Gaussian distribution in honor of Carl Friedrich
  Gauss, a famous mathematician.
• The bell shaped curve has several properties
• The curve concentrated in the center and
  decreases on either side. This means that the
  data has less of a tendency to produce unusually
  extreme values, compared to some other
  distributions.
• The bell shaped curve is symmetric. This tells
  you that he probability of deviations from the
  mean are comparable in either direction.

2
STANDARD DEVIATION THE NORMAL MODEL

• When you want to describe probability for a
  continuous variable, you do so by describing a
  certain area.
• A large area implies a large probability and a
  small area implies a small probability. Some
  people don't like this, because it forces them to
  remember a bit of geometry (or in more complex
  situations, calculus). But the relationship
  between probability and area is also useful,
  because it provides a visual interpretation for
  probability.
• Here's an example of a bell shaped curve. This
  represents a normal distribution with a mean of
  50 and a standard deviation of 10.

3
DESCRIBING DISTRIBUTION NUMERICALLY
4
STANDARD DEVIATION THE NORMAL MODEL
5
Formula

• Standardizing normal variables
• Formula

6
68-95-99.7 Rule

• 68 of the observations are within 1 standard
  deviation unit
• 95 of the observations are within 2 standard
  deviation unit
• 99.7 of the observations are within 3 standard
  deviation unit
• http//davidmlane.com/hyperstat/normal_distributio
  n.html

7
Example

• Some IQ tests are standardized to a Normal model
  with a mean of 100 and a standard deviation of
  16.
• a) Describe the 68-95-99.7 rule for this problem
• b) About what percent of people should have IQ
  scores above 116?
• c) About what percent of people should have IQ
  scores between 68 and 84?
• d) About what percent of people should have IQ
  scores above 132?
• e) About what percent of people should have IQ
  scores above 120?
• f) About what percent of people should have IQ
  scores below 90?
• g) About what percent of people should have IQ
  scores between 95 and 130?
• h) A person is a genius if his/her IQ belong to
  the top 10 of the all IQ scores. What minimum IQ
  score qualifies you to be a genius?

8
Answers to the Example

• Some IQ tests are standardized to a Normal model
  with a mean of 100 and a standard deviation of
  16.
• b) 16
• c) 13.5
• d) 2.5
• e) About what percent of people should have IQ
  scores above 120?
• Z (120 -100)/16 1.25
• Find P(Z gt 1.25) from standard normal chart or
  your TI calculator.
• Answer 1-.8944 .1056
• f) About what percent of people should have IQ
  scores below 90?
• Z (90 -100)/16 -0.625
• Find P(Z lt -0.625) from standard normal chart or
  your TI calculator.
• Answer .26
• g) About what percent of people should have IQ
  scores between 95 and 130?
• Z (95 -100)/16 -0.3125
• Z (130 -100)/16 1.875

9
Example

• In 2006 combined verbal and math SAT scores
  followed a normal distribution with mean 1020 and
  standard deviation 240.
• Suppose you know that Peter scored in the top 3
  of SAT scores. What was Peters approximate SAT
  score?
• Answer 1471.2

10
Using the TI-83 to Find a Normal Percentage
Always draw a picture!

• The TI-83 provides a function named normalcdf
• Press 2nd, DISTR (found above VARS)
• Scroll to normalcdf ( and press ENTER, or press
  2.
• If z has a standard normal distribution
• Percent(a lt z lt b) normalcdf ( a , b )
• Example to find P( -1.2 lt z lt .8 ), press 2nd,
  DISTR, 2, then -1.2 , .8 )
• Note that the comma between -1.2 and .8 must be
  entered
• Read .6731
• To find Percent( z lt a ), enter normalcdf ( -5 ,
  a )
• Example normalcdf( -5 , 1.96 ) gives .9750
• To find Percent( z gt a ), enter normalcdf ( a , 5
  )
• Example normalcdf( -1.645 , 5 ) gives .9500

?
-1.2
.8
?
1.96
?
-1.645
11
Using the TI-83/84 for Normal Percentages Without
Computing z-Scores

• We can let the TI find its own z-scores
• Find Percent(90 lt x lt 105) if x follows the
  normal model with mean 100 and standard deviation
  15
• Percent(90 lt x lt 105) normalcdf( 90 , 105 , 100
  , 15)
  .378
• Notice that this is a time-saver for this type of
  problem, but that you may still need to be able
  to compute z-scores for other types of problems!

12
Suppose Were Given a normal Percentage and Need
A z-score?

• IQ scores are distributed normally with a mean of
  100 and a standard deviation of 15. What score
  do you need to capture the bottom 2?
• That is, we must find a so that Percent(x lt a)
  2 when x has a normal distribution with a mean
  of 100 and a standard deviation of 15.
• With the TI 83/84 a invNorm( .02, 100
  , 15) 69.2