Download z-scores powerpoint

Which person is more overweight?
http://www.google.com/imgres?um=1&hl=en&biw=1280&bih=518&tbm=isch&tbnid=dNfWVpb
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The mean is above the median
The mean is below the median
http://en.wikipedia.org/wiki/Skewness
34 %
.5%
2%
Normal
Distribution
Bell Curve
34 %
2%
13.5%
13.5%
Mean
.5%
Determine if the following examples are
Normally Distributed,
Years of Teaching Experienc e at
Forsyth High School
Positively Skewed,
or Negatively Skewed.
6
5
4
3
Frequency
1
Number of Shoes
Owned per Person
0
0-5
1
6-10
6
11-15
10
16-20
11
21-25
9
>26
8
2
0-4
5-9
10-14
15-19
20-25
26-30
30 +
IQ's of Randomly Selected
People
<50
51-60
61-70
71-80
81-90
91-100
101-110
111-120
121-130
131-140
141-150
>150
20
15
10
5
0
Place the following under negatively
skewed, normally distributed, or positively
skewed, or random?
A) The amount of potato chips in a bag
B) The sum of the digits of random 4-digit numbers?
C) The number of D1’s that students in this class have
gotten?
D) The weekly allowance of students
E) Age of people on a cruise this week
Which of the following best describes
the ages of people on Earth?
A. Positively Skewed
B. Negatively Skewed
C. Normally Distributed
Understanding the Mean
2009 3.17c
7
Taken from Virginia Department o f Education “Mean Balance Point”
Where is the balance point for this
data set?
X
8
X
X
X
X
X
Taken from Virginia Department o f Education “Mean Balance Point”
Where is the balance point for this
data set?
X
X
X
X
9
X
X
Taken from Virginia Department o f Education “Mean Balance Point”
Where is the balance point for this
data set?
X
X
X
10
X
X
X
Taken from Virginia Department o f Education “Mean Balance Point”
Where is the balance point for this
data set?
X
X
X
X
X
X
Taken from Virginia Department o f Education “Mean Balance Point”
11
Where is the balance point for this
data set?
3 is the
Balance Point
X
12
X
X
X
X
X
Taken from Virginia Department o f Education “Mean Balance Point”
Where is the balance point for this
data set?
Sum of the distances below
the mean
1+1+1+2 = 5
X
13
MEAN
Sum of the distances above the mean
2+3=5
X
X
X
X
X
Taken from Virginia Department o f Education “Mean Balance Point”
Where is the balance point for this
data set?
Move 2 Steps
Move 2 Steps
Move 2 Steps
Move 2 Steps
4 is the Balance Point
14
Taken from Virginia Department o f Education “Mean Balance Point”
We can confirm this by calculating:
2 + 2 + 2 + 3 + 3 + 4 + 5 + 7 + 8 = 36
36 ÷ 9 = 4
The Mean is the Balance Point
15
Where is the balance point for this
data set?
If we could “zoom in” on the
Move 1 Step
The Balance Point is between
10 and 11 (closer to 10).
space between 10 and 11, we
could continue this process to
arrive at a decimal value for the
balance point.
Move 2 Steps
Move 1 Step
Move 2 Steps
16
Taken from Virginia Department o f Education “Mean Balance Point”
• Place the 9 sticky notes as a group so that exactly
3 are ‘16’, and one is ’12’. Place the remaining
five numbers so that the balance point is 16.
Then find the sum of deviations from the center.
• Place the 9 sticky notes as a group so that exactly 1
is ‘11’, two are ‘17’ and two are ’16’. None of the
remaining ones have ’16’ Place the remaining four
numbers so that the balance point is 16. Then find
the sum of the deviations from the center.
Taken from Virginia Department o f Education “Mean Balance Point”
Variability:
How far the data is spread out
Standard deviation:
Sx for a sample size (we’ll use)
for the population.
Way to measure the variability.
Closer to zero is better!
x
Zscore:
How many standard deviations something
is from the mean. The higher the absolute
value of it, the more unique it is.
Score= Z*(Std. dev) + mean
Which of the following will have the
highest variability?
A. [Heights of people in this room]
B. [Ages of people in this room]
C. [The number of countries that people have
been to in this room?]
Variability: How close the numbers are together.
Which would have a lower standard
deviation? (Be prepared to explain):
A. [The heights of students in this class]
B. [The heights of students in this school]
Sum of Distances from Center:
-2,-2,-2,-1,-1,0,1,3,4 = 0
Sum of Squares of distances:
4,4,4,1,1,0,1,9,16=40
from center:
Average (with one less member) of the
squares of the distance from the center: VARIANCE 40/8=5
Square root of the VARIANCE:
2.23
so the STANDARD DEVIATION (Sx) is 2.23
Now find the STANDARD DEVIATION of your Poster
21
Calculator Method
• To Find the Standard Deviation and Mean
•
•
•
•
•
•
1) Put the numbers into STAT EDIT
2) Do STAT CALC 1-VAR STATS.
The x is the “mean”
The Sx is the standard deviation we will use
The n is the amount of data (good way of checking)
The ‘med’ is the median (scroll down)
To find the % of people within a certain range:
• NORMALCDF (lower value, upper value, mean,
standard deviation)
To find the percentile: The lower value is 0
• NORMALCDF (0, upper value, mean, standard
deviation)
Taken from Core Plus Mathematics
Grams of Fat
Big Mac:
BK Whopper:
Taco Bell Beef Taco:
Subway Sub w/toppings:
Dominoes Med. Cheese Pizza:
KFC Fried Chicken:
Wendy’s Hamburger:
Arby’s Roast Beef Sandwich:
Hardee’s Roast Beef Sandwich:
Pizza Hut Medium Cheese Pizza:
1) What is the mean and standard deviation?
31
46
10
44.5
39
19
20
19
10
39
27.75; 13.86
2) What percentage of these items have between 25-35 grams of fat?
NormalCDF(25,35,27.75,13.86)= 27.8%
3) What percentile is a Big Mac?
NormalCDF(0,31, 27.75, 13.86)= 57.0%
The following is the amount of black M&M’s in a bag:
12, 13,14, 15, 15, 16, 17, 20, 21,22,23,24,25
Find the mean and standard deviation
A. [18.23, 4.46]
B. [18.23, 4.28]
Normal
Distribution
Bell Curve
http://en.wikipedia.org/wiki/File:Standard_deviation_diagram.svg
The SAT’s are Normally Distributed with a mean
and a standard
deviation of 100.
A) Give a Title and fill in the bottom row
SAT Scores
34 %
.5%
200
34 %
2%
13.5%
300
13.5%
600 700
400 500 5000
2%
.5%
800
15.8
B) What percentage of students score above a 600 on the SAT?
C) What percentage of students score between 300 and 500?
47.7
D) If Jane got a 700 on the SAT, what percentile would she be? 100-2.2= 97.8
E) Mt. Tabor has 1600 students, how many students are expected
.022*1600 = 35 students
to get at least a 700?
F) What is the Zscore of 300? -2
G) What is the percentile of a student who got a 550 on the SAT?
NormalCDF(0,550,500,100)=69.14
H) What percentage of students got between a 650 and 750 (use the calculator)
NormalCDF(650,750,500,100)=6.1
IQ’s of Humans
50
66.7
83.3
5000
100
116.7
133.3
150
The IQ’s are Normally Distributed with a mean of 100 and a standard
deviation of 16.667.
2.2
A) What percentage of people have an IQ below 66.7?
B) A genius is someone with IQ of at least 150? What percentage? .1
15.8
C) If Tom’s IQ is 83.3, what percentile would he be?
D) Spring School has 1000 students, how many students are expected
.022 * 1000 = 22
to have at least a 133.3 IQ?
E) What number represents a Z-score of 1.5? 100+1.5(16.667) = 125
The following is the amount of black M&M’s in a
bag: 12, 13,14, 15, 15, 16, 17, 20, 21,22,23,24,25
What percentage is above 22.6 black M&M’s (use
the bell curve)?
15.8
0.3
Adult female dalmatians weigh an average of
50 pounds with a standard deviation of 3.3
pounds. Adult female boxers weigh an average
of 57.5 pounds with a standard deviation of 1.7
pounds. The dalmatian weighs 45 pounds and
the boxer weighs 52 pounds. Which dog is
more underweight? Explain using z scores
http://www.rossmanchance.com/applets/NormalCalcs/NormalCalculations.html
One way to measure light bulbs is to measure
the life span. A soft white bulb has a mean life
of 700 hours and a standard deviation of 35
hours. A standard light bulb has a mean life of
675 hours and a standard deviation of 50 hours.
In an experiment, both light bulbs lasted 750
hours. Which light bulb’s span was better?
http://www.rossmanchance.com/applets/NormalCalcs/NormalCalculations.html
Memory Game
Dog, cat, monkey, pig, turtle,
apple, melon, banana, orange, grape,
desk, window, gradebook, pen, graph paper,
Stove, oven, pan, sink, spatula,
Shoes, tie, bracelet, necklace, boot
A) Find the mean and standard deviation with your calculator
B) Is it positively skewed, negatively skewed, or normally skewed?
C) Find your percentile:
Debate:
• Side 1) You are trying to convince your teacher
to always curve test grades to a standard
deviation
• Side 2) You are trying to convince your teacher
to never curve test grades to a standard
deviation
Summarize the Mathematics
A) Describe in words how to find the standard deviation.
B) What happens to the standard deviation as you increase the sample
size?
C) Which measures of variation (range, interquartile range, standard
deviation) are resistant to outliers. Explain
D) If a deviation of a data point from the mean is positive, what do you
know about its value? What if the deviation is zero?
E) What do you know about the sum of all the deviations of the mean?
F) Suppose you have two sets of data with an equal sample size and
mean. The first data set has a larger deviation than the second one.
What can you conclude?
Think back to the two overweight people shown
on the first slide. How could we now determine
which one is more overweight?