Z-score, CDF of normal distributions

x→Z-score→CDF ***

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The empirical rule: within 1, 2, 3 SDs of the mean.

Using z=1,2,3
"Calculate Between CDF by Z"
az=-1 bz=1  slice area/probability= ________
az=-2 bz=2  slice area/probability= ________
az=-3 bz=3  slice area/probability= ________

The empirical rule "slices":

az=0  bz=1  slice area/probability= ________
az=-1 bz=0  slice area/probability= ________

az=1  bz=2  slice area/probability= ________
az=-2 bz=-1  slice area/probability= ________

az=2  bz=3  slice area/probability= ________
   Right tail from 2SD to 3SD      ~1/47 of the population
az=-3 bz=-2  slice area/probability= ________
   Left tail from -3SD to -2SD

Left Half of the 34.1% "slice" from mean to +1 SD:
az=0 bz=0.5  slice area/probability= ________
Right Half of the 34.1% "slice" from mean to +1 SD:
az=0.5 bz=1  slice area/probability= ________
                         Notice that these two halves do not have the same area.

az=-0.675 bz=0.675  slice area/probability= ________   ==IQR
                 The middle half of the data/population is within [-.675,.675] of the mean.

-4 is practically -∞     All the way from the left.
az=-4 bz=-3   area/probability= ________
                             ~1/769 of the population.
az=-4 bz=-2   area/probability= ________
az=-4 bz=-1   area/probability= ________
az=-4 bz=0    area/probability= ________
az=-4 bz=1    area/probability= ________
az=-4 bz=2    area/probability= ________
az=-4 bz=3    area/probability= ________
az=-4 bz=4    area/probability= ________



Set mean=100  SD=10
"Calculate Between CDF by X"
ax=90  bx=110  slice area/probability= ________
ax=80  bx=120  slice area/probability= ________
ax=70  bx=130  slice area/probability= ________

ax=100 bx=110  slice area/probability= ________
ax=90  bx=100  slice area/probability= ________

ax=100 bx=120  slice area/probability= ________
ax=80  bx=100  slice area/probability= ________

Left Half of the 34.1% "slice" from mean to +1 SD:
ax=100 bx=105  slice area/probability= ________

Right Half of the 34.1% "slice" from mean to +1 SD:
ax=105 bx=110  slice area/probability= ________
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Clear

"Calculate CDF"  is the cumulative area/probability from -∞ to a Z-score.
    % of data less than this Z-score
z score=-2   "Calculate CDF"   pink area/probability= ________
z score=-1   "Calculate CDF"   pink area/probability= ________
z score= 0   "Calculate CDF"   pink area/probability= ________
z score= 1   "Calculate CDF"   pink area/probability= ________
z score= 2   "Calculate CDF"   pink area/probability= ________
z score= 3   "Calculate CDF"   pink area/probability= ________
  Notice that these are almost the same as the above "Calculate Between CDF by Z" from -4 to this Z.


Set mean=100  SD=10
x datum → z-score → CDF(z):

x=100 → Z score=_______   
 Calculate CDF.  Left: P(≤z) = CDF(z) = ________   Right:P(≥z)=1-CDF(z)=_______
                               pink area < x(Z)        yellow area > x(Z)
x=90 → Z score=_______   
 Calculate CDF.  Left: P(≤z) = CDF(z) = ________   Right:P(≥z)=1-CDF(z)=_______

x=110 → Z score=_______   
 Calculate CDF.  Left: P(≤z) = CDF(z) = ________   Right:P(≥z)=1-CDF(z)=_______

x=80 → Z score=_______   
 Calculate CDF.  Left: P(≤z) = CDF(z) = ________   Right:P(≥z)=1-CDF(z)=_______

x=120 → Z score=_______   
 Calculate CDF.  Left: P(≤z) = CDF(z) = ________   Right:P(≥z)=1-CDF(z)=_______

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Clear

Going "backwards" from CDF to Z score. Given an area or probability or percentage of
the data, what is the corresponding Z score?

Left: P(≤z) = CDF(z) = .5
"InvNorm CDF (Quantile)"      z score:________ 

Left: P(≤z) = CDF(z) = .8413
"InvNorm CDF (Quantile)"      z score:________ 

Left: P(≤z) = CDF(z) = .9772
"InvNorm CDF (Quantile)"      z score:________ 


Left: P(≤z) = CDF(z) = .25
InvNorm CDF (Quantile)        z score:________     == Q1 

Left: P(≤z) = CDF(z) = .75
InvNorm CDF (Quantile)        z score:________     == Q3



Now do for a normal data set
Clear.   Set mean=100  SD=10
It determines the X of a particular CDF.

Left: P(≤z) = CDF(z) = .5
"InvNorm CDF (Quantile)"      z score:________  x:_______

Left: P(≤z) = CDF(z) = .8413
"InvNorm CDF (Quantile)"      z score:________  x:_______ 

Left: P(≤z) = CDF(z) = .9772
"InvNorm CDF (Quantile)"      z score:________  x:_______ 

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"Special areas/percentages" "Special z scores"

Symmetrically about the mean

90% of the area/data == left and right tails each 5%
Left: P(≤z) = CDF(z) = .95
"InvNorm CDF (Quantile)"      z score:________  Yellow right tail is 5%
Left: P(≤z) = CDF(z) = .05
"InvNorm CDF (Quantile)"      z score:________  Pink left tail is 5%
 (NB. these z scores are the ± of each other)
"Calculate Between CDF by Z"  using these 2 z scores.
So 90% of the area/probability/data is between [______,________]


95% of the area/data == left and right tails each 2.5%
Left: P(≤z) = CDF(z) = .975
"InvNorm CDF (Quantile)"      z score:________  Yellow right tail is 2.5%
Left: P(≤z) = CDF(z) = .025
"InvNorm CDF (Quantile)"      z score:________  Pink left tail is 2.5%
 (NB. these z scores are the ± of each other)
"Calculate Between CDF by Z"  using these 2 z scores.
So 95% of the area/probability/data is between [______,________]


99% of the area/data == left and right tails each 0.5% (half a percent, tiny)
Left: P(≤z) = CDF(z) = .995
"InvNorm CDF (Quantile)"      z score:________  Yellow right tail is 0.5%
Left: P(≤z) = CDF(z) = .005
"InvNorm CDF (Quantile)"      z score:________  Pink left tail is 0.5%
 (NB. these z scores are the ± of each other)
"Calculate Between CDF by Z"  using these 2 z scores.
So 99% of the area/probability/data is between [______,________]

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Typical kind of questions:

A population with mean μ=89.5 and SD σ=12.8

A datum x=76.2 is greater than ______% of the population.
               It is less than ______% of the population.

A datum x=96.8 is greater than ______% of the population.
               It is less than ______% of the population.

Between 76.2 and 96.8 is _______% of the population.


What datum value is at the 70%ile of the population:_______
  i.e. is larger than 70% and less than 30%
What datum value is at the 30%ile of the population:_______
What datum value is Q1 of the population:_______
What datum value is Q3 of the population:_______

What datum value is 1.2σ above the mean:_______
What datum value is 1.2σ below the mean:_______