Data & Probability

Data/Statistics

Mean: Average

Median: Middle of an ordered list Exact middle for an odd # of items Average of the middle two for an even # of items

Mode: Most frequent

Range: Highest - Lowest

Terms

items of # Total

all of Sum

Helps you to see where the majority of the data lies, as each part is 25% of the data

Lowest and highest values = endpoints Median of the data = center of the box Median of the lower part and upper part =

edges of the box

Box & Whisker

low Q1 median Q3 high

lowest 25% 2nd 25% 3rd 25% highest 25%

the box contains 50% of the data

Outliers are 1.5 . IQR from the ends of the box IQR = Q3 – Q2

Extreme Outliers are 3∙IQR from the ends of the box

The high and the low are not always Outliers, not all data sets contain outliers.

Box & Whisker Plot

Relatively evenly distributed (normal) data

Skewed left (longer left tail)

Skewed right (longer right tail)

Skew is determined by the tail

Box & Whisker

Draw boxplot for the following test scores: 98, 75, 80, 74, 92, 88, 83, 60, 72, 99

Ordered list: 60, 72, 74, 75, 80, 83, 88,92, 98, 99Draw a number linePlot the end pointsFind the medianFind the median of the first halfFind the median of the second halfDraw the box around the “three” mediansConnect the box with “whiskers” to the endpoints

60 70 80 90 100

Boxplot

Displays all data

Stem Leaf 1st #(s) Last #

Stem & Leaf

Similar to a stem and leaf plot but does not necessarily retain the precise values of the data

Given: 10, 18, 21, 26, 30, 31, 38, 40 Stem and Leaf Dot Plot

1 0, 8 2 1, 6 3 0, 1, 8 1 2 3 4 4 0

Dot Plot

10 2 5 720 1 630 5 8 9 940 2 3 5 7 850 260 3 6

• the median the middle of the 17 values or 309

• the first quartile the middle of the first half or (201+206)/2=203.5

• the third quartile the middle of the second half or (407+408)/2=407.5

• the inter-quartile range the difference of the quarter points 407.5-203.5=204

• the mode the most frequent 309

• the percentile for 305 305 if the 5th item, 5/17=.294 * 100= 29.4 or the 29th percentile

• the value closest to the 60th percentile 60/100=x/17 .6 = x/17 .6*17 = x 10.2 = x the 10th item (402) is closest to the 60th Percentile

• Find the standard deviation enter all the data in L1 press STAT calc, choose one-var stat St. dev. =Ϭx

EXAMPLE:Given a stem and leaf plotFIND:

Shows how many and approximate values of the data

If the points follow a pattern, you can find the regression line

Scatter Plot

0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

Press 2nd + 7 1 2 (clears everything) Press 2nd 0 x-1 find diagnostics on press

enter Press Stat enter X’s go in L1 Y’s go in L2 Press Y=, arrow up press enter, zoom 9

To enter and plot points:

Decide what pattern the point appear to be following

Press STAT arrow over to calc Choose the correct pattern

4 for linear 5 for quadratic 0 for exponential

Press variable, arrow to y-vars, press 1, press 1, enter

Write down the value of r Press Y= write down the equation Press graph to see the fit

To find the regression line:

Predicting knowing x Set the window to be large enough for the given

value Graph Press 2nd trace (calc) Choose 1 (value) Enter the value and press enter

Estimating knowing y Set the window to be large enough for the given

value Enter the value in Y2= Press 2nd trace (calc) Choose 5 (intersect) Press enter three times

You may also substitute values into the equation

Find the equation for the following data and determine the value when x = 2 and when x = 7

Regression Equations

x y-1 -5

0 -2

1 0

3 1

4 3

5 4

6 6

Scatterplot—enter data in stat edit

Linear regression values

Graph to make sure the line fits the pattern

Use the calculations and enter a value of 2

Use the calculations and enter a value of 7

Click on the calculator to see how to find a regression line

Now try it for your self, checking along the way to see if you have the same values/screen shots as below—click each time you are ready to check your calculations.

Probability

How can we determine all the possible outcomes of a given situation?

TREE DIAGRAM—an illustrative method of counting all possible outcomes.

List all the choices for the 1st event

Then branch off and list all the choices for the second event for each 1st event, etc.

outcomes possible all

responses desired of # TheyProbabilit

A restaurant offers a salad for $3.75. You have a choice of lettuce or spinach. You may choose one topping, mushrooms, beans or cheese. You may select either ranch or Italian dressing. How many days could you eat at the restaurant before you repeat the salad?

Lettuce

spinach

mushrooms

beans

cheese

mushrooms

beans

cheese

ranch

Italianranch

Italianranch

Italianranch

Italianranch

Italianranch

Italian

While the tree diagram is beneficial in that it lists every possible outcome, the more options you have the more difficult it is to draw the diagram.

Fundamental counting Principle—is a mathematical version of the tree diagram, it gives the # of possible ways something can be accomplished but not a list of each way.

Example:

Jani can choose from gray or blue jeans, a navy, white, green or stripped shirt and running shoes, boots or loafers? How many outfits can she wear?

_______ _______ _______pants shirts shoes

2 3 3 =18

Permutations—all the possible ways a group of objects can be arranged or ordered

Example:There are four different books to be placed in order on

a shelf. A history book (H), a math book (M), a science book (S), and an English book (E). How many ways can they be arranged?

24 WAYS 4 • 3 • 2 • 1 =

24

H, M, S, E

H, M, E, S

H, S, E, M

H, S, M, E

H, E, M, S

H, E, S, M

M, E, S, H

M, E, H, S

M, S, H, E

M, S, E, H

M, H, E, S

M, H, S, E

S, M, E, H

S, M, H, E

S, H, M, E

S, H, E, M

S, E, M, H

S, E, H, M

E, M, S, H

E, M, H, S

E, H, M, S

E, H, S, M

E, S, M, H

E, S, H, M

A permutation of n objects r at a time follows the formula

)!(

!

rn

nPrn

)!25(

!525 P

Example:

!3

!5

!3

!345

20

This can be done on your calculator with the following keystrokes:

Type the number before the PPress mathOver to prb Choose number 2 nPrEnter the number after the PPress enter.

Combinations P-3

How can you determine the difference between a permutation and a combination?

Combinations—the number of groups that can be selected from a set of objects--the order in which the items in the group are selected does not matter

Example: How many three person committees can be formed from a group of 4 people—Joe, Jim, Jane, and Jill

Joe, Jim , JillJoe, Jill, JaneJoe, Jim Jane

Is Joe, Jane, JimA different committee

Jim, Jane, Jill

)!(!

!

rnr

nCrn

Formula:

)!34(!3

!434 C

)!1(!3

!4

ways4)!1(!3

!34

What is the difference between replacement and repetition?

Repetitions and Circular Permutations P-2

Combinations

This can be done on your calculator with the following keystrokes:

Type the number before the CPress mathOver to prb Choose number 3 nCrEnter the number after the CPress enter.

Replacement—using the same object again (nr)

Example:The keypad on a safe has the digits 1- 6

on it how many:a) four digit codes can be formed

_____ _____ _____ _____

b) four digit codes can be formed if no 2 digits can be the same

_____ _____ _____ _____

6 6 6 6

6 5 4 3

Repetition—occurs when you have identical items in a group

Example:Find all arrangements for the letters in the word

TOOL

____ ____ ____ ____ TOOL OLOT LOTO

TOLO OLTO LOOTTLOO OTOL LTOO

OTLOOOTLOOLT

We would expect 24 but since you can’t distinguish between the two O’s all possibilities with

the O’s switched are removed

4 3 2 1

Formula for repetitions:

where s and t represent the number of times an item is

repeated

EXAMPLE:How many ways can you arrange the letters

in BANANAS

A N

The factorial key is also found my pressing math and arrowing over to PRB

!!

!

ts

n

!2!3

!7

?2

1

3

4

Circular Permutation—arranging items in a circle when no reference is made to a fixed point

Example:How many ways can you arrange the numbers 1-4 on a spinner?

We would expect 4! Or 24 ways but we only have 6

Circular permutations are always (n-1)!

A1

2

3

4

B1

2

4

3

C1

3

2

4

D1

3

4

2

E1

4

2

3

B1

4

3

2

?2

1

3

4

D

If all outcomes are successful, the probability will be 1

If no outcomes are successful, the probability will be 0

SoProbability is 0 ≤ P ≤ 1

outcomes possible all

responses desired of # TheyProbabilit

Examples:What is the probability of getting an ace

from a deck of 52 cards?4 aces so

What is the probability of rolling a 3 on a 6 sided die?

there is 1 3 on 6 sides so

13

1

52

4

6

1

What is the probability of rolling an even number?

2,4, 6 are even so

What is the probability of getting 2 spades when 2 cards are dealt at the same time?

at the same time indicates use of a combination

—hint there are 13 spades

2

1

6

3

17

1

252

213 C

C

What is the probability of getting a total of 5 when a pair of dice is rolled?

+ 1 2 3 4 5 6

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

Draw the following chart for the sum of all rolls and count how many have a sum of 5

36

5

Compound Probability P-5

What is meant by compound probability?

OR: P(A or B) = P(A) + P(B) – P(A and B)

Example:What is the probability of getting a 2 or a 5

on the roll of a die?

Exclusive Events: events that do not have bearing on each other

3

1

6

2

6

1

6

1

What is the probability of drawing an ace or a heart?

ace + heart – ace of hearts

+ - =

Events are inclusive if they have overlap!

52

4

52

13

52

1

13

4

52

16

AND: indicates multiplication

Examples:What is the probability of tossing a three of

the roll of a die and getting a head when you toss a coin?

three and a head

* =

These events are independent—have no effect on the outcome of the other

6

1

2

1

12

1