Get that Program!!
FLIP50
2.2 Standard Normal Calculations
So How do We Compare
Does It Make Sense
the Seemingly
Incomparable?
Variable
Vs.
x
POO
x
Milk
Vs.
…Using the Standard Normal
Distribution

This is a normal distribution with a µ = 0
and σ = 1
If we change our Data into
values that fit this curve,
then we can compare the
incomparable!!!
Z- Score
z
x

x = observation; µ = mean of that distribution
σ = std dev of that distribution
What’s A Z Score?
Z Score: An observation’s relative position to it’s mean
Tells how many STANDARD UNITS an observation is from
the mean
Let‘s look at dairy cow poo
production for example…
Average Dairy Cow Daily Poo Production
Frankie Farmer’s best cow
Bessy produced 147 lbs of poo
today. Where is she compared
to the rest of the cows?
147  140
z
5
1.4 Std
units
ABOVE
THE
125 130 135 140 145 150 155
This will allow us to help Frankie!!
Which one is the
better “Pooer”?
Chicken Little
Average’s 1.6lbs of
poo per day
Overall Chicken
Average: 0.9lbs
Std Dev: .3lbs
1.6  .9
z
 2.33
.3
Bessy
Average’s 152lbs of
poo per day
Overall Dairy Cow
Average: 140lbs
Std Dev: 7lbs
152  140
z
 1.71
7
Better or Worse?
A more positive Z score isn’t always
better!!
 What are some situations where that’s
true?

Finding Probabilities using table A
Change your value to a Z -Score
Draw Std Normal Curve and
mark position of z value(s)
Shade “area” looking for
(Table A gives this area…)
x
Find area in table A, do appropriate math to
find shaded region (Table A is only for areas
of less than z)
Table Practice (straight)
1)
2)
Find the following probabilities using the table:
z < -.43
.3336
z < 1.32
.9066
Find the following probabilities using the table
in a distribution with µ= 64.5 and σ = 2.5:
x < 68
.9192
x < 58
.0047
Finding Probabilities for “other”
Regions

Fancy Math
 Subtracting different regions

Visualization
z > 1.5
.0668
1
z < 1.5
.9332
Finding Probabilities for “other”
Regions

Find the area for -1.67 < z < 2.1
 Visualize, find table #’s, do the math

Visualization
-1.67 < z < 2.1
.9346
z < 2.1
z <-1.67
.9821
.0475
Back to the Farm!!!
Gertie
Produces 4.1
kg of milk
per day
Natl. Avg =
4.5kg
Std Dev. =
.25kg
4.1  4.5
z
 1.6
.25
1 - .0548 =
.9452
Going from z -> x

Natl. Avg = 67lbs.
Std Dev. = 4.3lbs
Sometimes you have a Z-Score and
need to know what the real (raw) score
is:
x  (  z )  
Willy Whitehorse has been rumored
around the barn to produce 2.7 std units
of poo. Frankie would like to know how
much poo he’ll be shoveling out tonight!!!
x  (4.3 * 2.7)  67  78.61lbs
of POO!!!
Are They Normal?

Method 1 – Histogram/StemLeaf
 Construct a histogram or stemplot
 Look for symmetry about the mean and bell
shapedeness
 Also, mark the 68,95 points on the graph and get a
“count” to see if it matches up
○ See if there are the correct percentage of values that fit
within the parameters for those percentages (1 or 2σ’s)
**Difficult to assess
for small data sets
Are They Normal?
Method 2 – Normal Probability Plot (Calculator)




Put Data into a List
1 Var Stats to compare Mean, Median
Boxplot to look for symmetry (or Histogram)
Normal Probability Plot is the last graph under “Type” in
the stat plot screen
 Zoom Stat to finish the graph
 If the points shown from a “line” pattern, the data is said
to be normal.
Are They Normal?
The50
Flip50
Program flips a coin 50
Flip
Program
times
and
 Run the
Flip records
50 Program the # of heads. It
 Fix the window on your
does
this
experiment
100
times
and
histogram (Xscale)
records
 Look at histogram
for the results.
normality
 Stat Plot (see screen
capture)
 Zoom 9 to see the Normal
Probability Plot
Homework
#’s 28-33,36,42-48;
Worksheet