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Standard Deviation
Understanding the Mean
2009 3.17c
2
Taken from Virginia Department o f Education “Mean Balance Point”
Where is the balance point for this
data set?
X
3
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
4
X
X
Taken from Virginia Department o f Education “Mean Balance Point”
Where is the balance point for this
data set?
X
X
X
5
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”
6
Where is the balance point for this
data set?
3 is the
Balance Point
X
7
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
8
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
9
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
10
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
11
Taken from Virginia Department o f Education “Mean Balance Point”
• Place the 8 sticky notes as a group so that exactly
3 are ‘16’, and one is ’12’. Place the remaining
four 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”
[Which of the following will have the
most 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.
x
Standard deviation: (Sx or
)
Way to measure the variability. Closer
to zero is better!
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
15
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:
31
46
10
44.5
39
19
20
19
10
39
Calculator Method
•
•
•
•
•
1) Put the numbers into STAT EDIT
2) Do STAT CALC 1-VAR STATS.
The
is the “mean”
x
The Sx is the standard deviation
The
is a standard deviation we will not
use
• The n is the amount of data (good way of
checking)
• The ‘med’ is the median (scroll down)
x
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]
Normal
Distribution
Bell Curve
http://en.wikipedia.org/wiki/File:Standard_deviation_diagram.svg
http://en.wikipedia.org/wiki/Skewness
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 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
F) The shoe sizes of females in this class
Which is more likely to make a better bell curve,
measuring the heights of people in this room or
measuring the heights of people in this school?
http://www.shodor.org/interactivate/activities/NormalDistribution/
SAT Scores
200
300
400
5000
500
600
700
800
The SAT’s are Normally Distributed with a mean of 500 and a standard
deviation of 100.
A) Give a Title and fill in the bottom row
B) What percentage of students score above a 600 on the SAT? 15.8
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
E) Mt. Tabor has 1600 students, how many students are expected 97.8
to get at least a 700? .022*1600 = 35 students
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
Find the mean and standard deviation
A. [18.23, 4.46]
B. [18.23, 4.28]
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?
15.8
0.3
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?
• The more people with same data means lower
standard deviation.
• A lower standard deviation means less
variability.
• Z score is how many standard deviations you
are from the mean. The higher of the
absolute value of the z-score indicates the less
likelihood of the event happening.
Ex: Z score of 2 is more remarkable than a z
score of 1
Ex 2: A mean of 7, stdeviation of 3. A z-score
of -1.5 would be 7+(-1.5)*3 = 2.5
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?
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….
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
Think back to the two overweight people shown
on the first slide. How could we now determine
which one is more overweight?