6-2 Standard Units Areas under Normal Distributions

1
6-2 Standard Units Areas under Normal
Distributions
2
What is a z score?

• A common statistical way of standardizing data
  on one scale so a comparison can take place is
  using a z-score. The z-score is like a common
  yard stick for all types of data. Each z-score
  corresponds to a point in a normal distribution
  and as such is sometimes called a normal deviate
  since a z-score will describe how much a point
  deviates from a mean or specification point.
• http//www.measuringusability.com/z.htm
• The z score for an item, indicates how far and
  in what direction, that item deviates from its
  distribution's mean, expressed in units of its
  distribution's standard deviation.
• http//www.sysurvey.com/tips/statistics/zscore.htm
  

3
Z score

• Is a way to standardize scores based on the mean.
  
• Tables are used to describe the areas under
  normal curves, associated with particular z
  scores.
• For instance, if Physics period 1 has an average
  test score of 74 and Physics period 7 had an
  average of 78
• How does someone who had an 83 in period 1
  compare with someone who had an average of 87 in
  period 7?
• Do they deserve the same grade? Both were
  substantially above average, although the grades
  were obviously quite different.
• Would the standard deviations of the two classes
  make a difference to you??

4
Z Score (cont)

• The conversion formula of a score to a z score is
  based on the mean and standard deviation.
• So what is the z score for someone whose score
  the average?

5
Z score (cont)

• What about a z score for an x above the mean?
• What about a z score for an x below the mean?

6
Raw Score

• If you know the z score, it can be translated
  into the appropriate raw score by
• x zs µ

7
Standard Normal Distribution

• Standardizing a z score makes the center µ 0.
• Standardizing a z score makes the spread s 1.
• The z distribution will be normal if the x
  distribution is normal.
• Areas still correspond as they did in 6-1.
• 68 of the area under the curve is 1 standard
  deviation from the mean.
• 95 of the area under the curve is 2 standard
  deviations from the mean
• 99.7 will be 3 standard deviations from the
  mean.

This is sometimes called the 68-95-99.7 rule or
The Empirical Rule
8

• The college you want to apply to says that while
  there is no minimum SAT score required, the
  middle 50 of their students have combined SAT
  scores between 1530 and 1850. What would be a
  minimum acceptable ACT score that could be
  correlated to that same range of SAT scores?
• For college bound seniors, the average combine
  SAT is 1500 and standard deviation is 250. The
  ACT average is 20.8 with a standard deviation of
  4.8

9
Why convert?

• If you can convert the raw score to a z score,
  you can use tables in the back to determine the
  area under the curve. Why?
• The areas under the curve is equal to the
  probability that the measurement falls in this
  interval.
• The textbook has a left tailed table. That is,
  the z score will correspond to a given cumulative
  area to the left of z.
• Therefore, what would be the area to the right of
  z?
• 1 area left of z.
• (It could also be the opposite of the area to the
  left of z)

10
What does this area stuff mean?

• Essentially if you have a z score, you can make
  an assumption about the of scores (i.e. the
  probability) that fall at that number or below.

11
Some notes

• A z score of 6 or 7 would be highly interesting.
  Why?
• Calculator
• normalcdf (zleft, zright) calculates the score
  between two z scores.
• If you need to do left or right of a z score,
  some books suggest using 3.49 for the end, some
  suggest 99. Use what you want.

12
ADD BVD PROBLEM

• Pg. 113