Standard Deviation

Algebra I/Integrated I

Place the following under negatively skewed, normally distributed, or positively skewed, or

random?A) The amount of chips in a bagB) 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 studentsE) Age of people on a cruise this weekF) The shoe sizes of females in this class

0-4 5-9 10-14 15-19 20-25 26-30 30 +0

1

2

3

4

5

6

Years of Teaching Experienc e at Forsyth High School

<5051-60

61-7071-80

81-90

91-100

101-110

111-120

121-130

131-140

141-150>150

05

101520

IQ's of Randomly Selected People

Number of Shoes Owned per Person

Frequency

0-5 1

6-10 6

11-15 10

16-20 11

21-25 9

>26 8

Determine if the following examples are Normally Distributed, Positively Skewed, or Negatively Skewed.

Understanding the Mean

4

2009 3.17c

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

Where is the balance point for this data set?

X

XXX X X

5Taken from Virginia Department o f Education “Mean Balance Point”

Where is the balance point for this data set?

XXXX X

X

6Taken from Virginia Department o f Education “Mean Balance Point”

Where is the balance point for this data set?

X

XXX XX

7Taken from Virginia Department o f Education “Mean Balance Point”

Where is the balance point for this data set?

XXX

XXX

8Taken from Virginia Department o f Education “Mean Balance Point”

Where is the balance point for this data set?

XXX

X

X

X

3 is theBalance Point

9Taken from Virginia Department o f Education “Mean Balance Point”

Where is the balance point for this data set?

X

XXX X X

10

MEANSum of the distances above the mean2 + 3 = 5

Sum of the distances below the mean1+1+1+2 = 5

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

4 is the Balance Point

Move 2 Steps

Move 2 Steps Move 2 Steps

Move 2 Steps

Where is the balance point for this data set?

11Taken from Virginia Department o f Education “Mean Balance Point”

The Mean is the Balance Point

We can confirm this by calculating:

2 + 2 + 2 + 3 + 3 + 4 + 5 + 7 + 8 = 36

36 ÷ 9 = 4

12

The Balance Point is between 10 and 11 (closer to 10). Move 2 Steps

Move 2 StepsMove 1 Step

Move 1 Step

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.

13Taken from Virginia Department o f Education “Mean Balance Point”

Teacher Demonstration:Place the 9 sticky notes so that the mean is 15 and that there are no sticky notes on 15 and only 2 are on 16.

Calculate the mean, median, range, and standard deviation.

Each group is to work together but a different person is to be the leader of each situation. Situation 1: Place nine sticky notes so that the mean (balancing point) is 15; and the median is the smallest as possible. Record the data.

Situation 2: Place nine sticky notes so that the median is 15 and that the mean is the largest possible number. Record the data.

Situation 3: Place the 9 sticky notes as a group so that exactly 3 are ‘16’, and one is ’14’. Place the remaining four numbers so that the mean (balancing point) is 16. Also keep the data as close as possible. Record the data.

Situation 4: Same as situation 3 but keep the data as spread out as possible. Record the data.

Using the calculator, list the mean, median, and standard deviation ( ) for each situation.

Did #3 or #4 have the higher standard deviation?

x

Calculator Method• 1) Put the numbers into STAT EDIT• 2) Do STAT CALC 1-VAR STATS.• The is the “mean”• The is the standard deviation• The Sx 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)

xx

Grams of FatBig Mac: 31BK Whopper: 46Taco Bell Beef Taco: 10Subway Sub w/toppings: 44.5Dominoes Med. Cheese Pizza: 39KFC Fried Chicken: 19Wendy’s Hamburger: 20Arby’s Roast Beef Sandwich: 19Hardee’s Roast Beef Sandwich: 10Pizza Hut Medium Cheese Pizza: 39

Taken from Core Plus Mathematics

18

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=5Square root of the VARIANCE: 2.23 so the STANDARD DEVIATION (Sx) is 2.23

Now find the STANDARD DEVIATION of your Poster

*Standard deviation: (Sx or )

*Way to measure the variability. Closer to zero is better!

*A lower standard deviation means less variability.

*The more people with same data means lower standard deviation.

x

[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.

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]

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]

Memory GameDog, 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 calculatorB) Is it positively skewed, negatively skewed, or normally skewed?C) Find your percentile: NORMAL CDF (O, your value, mean, standard deviation)

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?

Summarize the Mathematics

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

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

Normal DistributionBell Curve

http://www.shodor.org/interactivate/activities/NormalDistribution/

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?

5000

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

SAT Scores

500 600 700 800200 300 400

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?

15.847.7

100-2.2 97.8

.022*1600 = 35 students

http://bcs.whfreeman.com/bps3e/Click on Statistical AppletsClick on Normal Distribution

Calc: NORMALCDF (mean, standard deviation, lower value, upper value)

5000

The IQ’s are Normally Distributed with a mean of 100 and a standard deviation of 16.667.

IQ’s of Humans

100 116.7 133.3 150 50 66.7 83.3

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?

2.2.1

15.8

.022 * 1000 = 22100+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?

15.80.3

Memory GameDog, 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 calculatorB) Is it positively skewed, negatively skewed, or normally skewed?C) Find your percentile: NORMAL CDF (O, your value, mean, standard deviation)

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?

Summarize the Mathematics

• 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

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

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