Which person is more overweight?

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Which person is more overweight?.

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Which person is more overweight?

http://en.wikipedia.org/wiki/Skewness

Normal Distribution

Bell Curve

Place the following under negatively skewed, normally distributed, or positively skewed, or random?
• 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?

• The weekly allowance of students

E) Age of people on a cruise this week

Determine if the following examples are

Normally Distributed,

Positively Skewed,

or Negatively Skewed.

• Positively Skewed
• Negatively Skewed
• Normally Distributed

[Default]

[MC Any]

[MC All]

Understanding the Mean

2009 3.17c

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”

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”

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”

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”

Where is the balance point for this

data set?

3 is the

Balance Point

X

X

X

X

X

X

Taken from Virginia Department o f Education “Mean Balance Point”

Where is the balance point for this

data set?

MEAN

Sum of the distances below the mean

1+1+1+2 = 5

Sum of the distances above the mean

2 + 3 = 5

X

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

Taken from Virginia Department o f Education “Mean Balance Point”

14

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 space between 10 and 11, we could continue this process to arrive at a decimal value for the balance point.

Move 1 Step

The Balance Point is between 10 and 11 (closer to 10).

Move 2 Steps

Move 1 Step

Move 2 Steps

Taken from Virginia Department o f Education “Mean Balance Point”

16

Place the 9 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”

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!

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?
• [Heights of people in this room]
• [Ages of people in this room]
• [The number of countries that people have been to in this room?]

[Default]

[MC Any]

[MC All]

Variability: How close the numbers are together.

• [The heights of students in this class]
• [The heights of students in this school]

[Default]

[MC Any]

[MC All]

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 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: 31

BK Whopper: 46

Taco Bell Beef Taco: 10

Subway Sub w/toppings: 44.5

Dominoes Med. Cheese Pizza: 39

KFC Fried Chicken: 19

Wendy’s Hamburger: 20

Arby’s Roast Beef Sandwich: 19

Hardee’s Roast Beef Sandwich: 10

Pizza Hut Medium Cheese Pizza: 39

What is the mean and standard deviation?

What percentage of these items have between 25-35 grams of fat?

What percentile is a Big Mac?

27.75; 13.86

NormalCDF(25,35,27.75,13.86)= 27.8%

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
• [18.23, 4.46]
• [18.23, 4.28]

[Default]

[MC Any]

[MC All]

Normal Distribution

Bell Curve

http://en.wikipedia.org/wiki/File:Standard_deviation_diagram.svg

SAT Scores

5000

600 700 800

500

200 300 400

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

15.8

B) What percentage of students score above a 600 on the SAT?

C) What percentage of students score between 300 and 500?

D) If Jane got a 700 on the SAT, what percentile would she be?

E) Mt. Tabor has 1600 students, how many students are expected

to get at least a 700?

F) What is the Zscore of 300?

G) What is the percentile of a student who got a 550 on the SAT?

H) What percentage of students got between a 650 and 750 (use the calculator)

47.7

100-2.2= 97.8

.022*1600 = 35 students

-2

NormalCDF(0,550,500,100)=69.14

NormalCDF(650,750,500,100)=6.1

IQ’s of Humans

5000

50 66.7 83.3

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?

C) If Tom’s IQ is 83.3, what percentile would he be?

D) Spring School has 1000 students, how many students are expected

to have at least a 133.3 IQ?

E) What number represents a Z-score of 1.5?

.1

15.8

.022 * 1000 = 22

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

Find the mean and standard deviation with your calculator

Is it positively skewed, negatively skewed, or normally skewed?

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?