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

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

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.

Relatively evenly distributed (normal) data

Skewed left (longer left tail)

Skewed right (longer right tail)

Skew is determined by the tail

Draw boxplot for the following test scores: 98, 75, 80, 74, 92, 88, 83, 60, 72, 99Try before checking the answer belowOrdered 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

Displays all data

Stem Leaf 1st #(s) Last #

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 Create both a stem and leaf and a dotplot then

check your answer below

Stem and Leaf Dot Plot

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

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 the following stem and leaf plotFIND each of the requested items then check your answers to the right

Shows how many and approximate values of the data

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

Use the following data for the next several slides: (1, 5), (2, 11), (3, 16), (4, 20)

Press 2nd + 7 1 2 (clears everything)◦ Press 2nd + 5 1 2 for a regular TI-83

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 to plot 1press enter, zoom 9

Decide what pattern the points 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 to 3 decimal places Press graph to see the fit

Predicting knowing x try using x = 3.5 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 try using y =18 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

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 a video on how to find a regression line if you did not get the correct values

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.

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

Try the following then check below 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: try before checking your answer

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 4 3 =24

Permutations—all the possible ways a group of objects can be arranged or ordered (the way things are listed matters)

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

!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.

Now Try 8P3

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

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

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

Using the same basic steps on the calculator but choosing nCr find

8C3

Is there a difference in value for 8C3 and 8P3

For the problem above:

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

Type the number before the CPress mathOver to prb Choose number 3 nCr

Enter the number after the CPress enter.

Using the same basic steps on the calculator but choosing nCr

find 8C3

Is there a difference in value for 8C3 and 8P3

What is the difference between replacement and repetition?

Replacement—being allowed to use the same object again (nr)

Example: try each before checkingThe 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 LOTOTOLO 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 we divide by the number of individual repetitions—that is 24/2 = 12 which is what we have

4 3 2 1

Formula for repetitions:

where s and t represent the number of times different items are repeated

EXAMPLE: try then checkHow many ways can you arrange the letters

in BANANAS

A’s N’s

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

!!

!

ts

n

420!2!3

!7

?2

1

3

4

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

How many ways can 6 charms be arranged on a bracelet that does not have a clasp.

(6-1)! = 5! = 120 ways

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

Try the following examples then check below: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 one 3 on 6 sides so

13

1

52

4

6

1

Try each then check: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 the 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

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 meant by compound probability? The words or & and are in the problem

Try then checkWhat 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: try then checkWhat 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