Download Chapter 12 Statistics and Probability

Chapter 12 Statistics and Probability
12 – 1
Measures of Central Tendency (3 M’s)
Objective- 1. Know that the 3 measures of central tendency are mean, median, and mode
and how to find each.
2. Know which measure of central tendency better represents a given set of
data.
Example:
A company’s employees stated that the average weekly salary was $200. The owners
argued that wages averaged $300. The personnel department insisted the wages
were $250 a week. Each group is correct. Why.
Wages:
200 200 200 200 200 200 200 200 200
250 250 250 250 250 250
350 350
500 500
1000
Each group is using a different Measure of Central Tendency. Think of the 3 M’s.
Mean Median Mode
Mean - average
Median – middle
Mode – most often
The employees are using Mode. Most are making $200 a week and they are looking for a
raise.
The owners are using Mean.
The personnel Department is using Median. Half the employees make less than $250 per
week, the other half makes more than $250 per week.
Because of the one salary of $1000, which is so much more than the other wages, the
mean is skewed higher.
Using the median is better in this case.
Mode is preferred when the data is not numerical.
Median tells that half the items are above and half below the average. Use median when
extreme measures distort the mean.
Mean use when no unusually large or small numbers distort the results.
Range lets you know how far apart the numbers in the data are spaced. Find the range by
subtracting the smallest number from the largest number in the data.
12-2 Line Plots and Frequency Tables
Objective- know how to organize data to more easily find mean, median and mode. The
methods of organizing the data are: line plots, frequency tables, and stem & leaf plots.
To make a line plot you have to make a number line. Determine the range to know if you
should count by 1’s, 2’s, 5’s or 10’s on the number. Then make the numbers from the
data above the number line using x’s.
Using the above list of employee wages displayed on a line plot the $1000 salary is
shown as an ‘outlier.’ Here we counted by 50’s on the number line, except for the
outlier.
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
200
250
300
350
400
450
500
1000
A line plot shows data on a number line. You place an ‘x’ for each response above the
category.
A frequency table us a listing of data that pairs each data item with the number of
times it occurs. You can use tally marks to determine the frequency of each number if
there is a large amount of data. Then add the tally marks to find the frequency.
n
f
200
9
250
6
350 500 1000
2
2
1
You can use the data from the line plots and frequency tables to find mean, median, and
mode and range.
12-3 Stem and Leaf Plots
Objective- know how to organize large amounts of data in a the stem and leaf plot to
make it easier to find mean, median, and mode.
The digit on the far right, usually the ones (units) place, is called the ‘leaf.’ The
remaining digits are called the ‘stem.’
ExampleFood
Fat Content (g)
Deluxe burger
33
Small burger
10
cheeseburger
17
Fish sandwich
26
Chicken fingers
24
Pizza
15
Apple turnover
24
Stem
1
2
3
Leaf
0 5 7
4 4 6
3
To make the stem and leaf plot
1. Choose a stem (tens digit or both
hundreds and tens digits)
2. Write the leaf values (ones digit)
in order from least to greatest
The zero under leaf represents 10, the 5 represents 15, etc.
To find the mean add all the numbers and divide by the
number of ‘leaves’
To find the median for 7 ‘leaves’ find the middle number
The mean is 21.3, the median is 24 and the mode is 24.
For 3 digit numbers the stem is two digits (hundreds and tens place) and the one digits are
the ‘leaves’
Example 2:
345, 421, 362, 459, 450, 346, 428, 425, 349, 459, 420, 366
Stem
34
36
42
45
Leaf
5 6 9
2 6
0 1 5 8
09 9
Mean:
402.5
Median: 420.5
Mode: 459
Back to Back Stem and Leaf Plot
Objective- know how to display two sets of data on one stem and leaf plot and then find
the median and mode for each set of data.
Example:
Set A: 25, 23, 33, 36, 42, 44
Set B:
19, 16, 23, 34, 26
A
Leaf
5
4
3
2
Stem
1
2
3
B
Leaf
6 9
3 6
4
Set A: median 34.5
mode: none
Set B: median 23
Mode none
Remember when making a back-to-back stem and leaf plot that the lowest ‘leaf’ number
is always closest to the stem.
Histograms
Objective: know how to make s frequency table using evenly spaced intervals and use
the table to make a special kind of bar graph called a histogram.
A histogram is a bar graph that has no space between the intervals.
Example:
Jasmine compared the prices of ten brands of sneakers at the outlet stores.
Make a frequency table from the data. Decide what interval to use.
$32.88, $42.58, $27.52, $21.40, $34.72, $30.15, $49.60, $30.92, $46.47, $29.48
Decide on the interval to use. Shown are 2 example with intervals of 10 and 5.
Price
20-29
30-39
40-49
Frequency
3
4
3
4
Price
21-25
26-30
31-35
36-40
41-45
46-50
Frequency
1
3
3
0
1
2
3
2
3
1
2
20-29 30-39 40-49
1
21-25 26-30 31-35 36-40 41-45 46-50
12-4 Box and Whisker Plots
Objective: create a box and whisker diagram to help analyze data. It is especially useful
in showing the distribution of data in quartiles, so you can analyze the data by looking at
25 % of the data in each of 4 sections.
Team Scores for the Nets
85
79
99
110
91
101
60
75
92
86
1. First arrange the data in order from least to greatest. Then find the
median
55 80 82 85 86
91 92 99 101 110
Since there are 10 numbers to get the median, find the average of the 2 middle numbers.
The median is 88.5
2. Separate the data into 4 groups. Find the median of lower half which
is called the lower quartile and the median of the upper half which is
called the upper quartile.
The lower quartile is 82 and the upper quartile is 99.
3. Draw a number line and set intervals based on the range. Mark the
number line with the median and quartile values. ( 82, 88.5 and 99)
4. Draw a box that extends from the lower quartile (first quartile) to
the upper quartile (third quartile). Mark the median with a vertical
line through the box. Then draw “whiskers” from the box to the
lowest and highest scores.
55
65
75
85
95
105
115
This shows the middle score is 88.5 the highest score of 110 and lowest score of
55. You can also say the lowest 25% of the scores are spread out between 55 and 82. The
next 25% of the scores are close together. The top 25% of the scores are between 99 and
110. 75% of the team’s scores are above 82.
12-5 Counting Principle
Objective: know that the counting principle states that the number of outcomes for an
event with two or more stages equals the product of the number of outcomes at each
stage.
Ex. High school schedule choices electives for the first half of the year include art,
technology and sewing. For the second half of the year the selections are cooking,
drama, woodshop and creative writing. How many different ways can you make
your schedule?
You can solve this problem by making a list, a tree diagram, or using the counting
principle.
cooking
Tree diagram
woodshop
Art
drama
Creative writing
Tech
sew
There are 12 branches for the twelve selections that are possible.
To use the counting principle find the product of the outcomes of each stage. There are 3
choices for the first half of the year and 4 choices for the second half.
3  4 = 12
Factorial
Objective: know how to find the number of possible arrangement for a number of items
by finding the factorial of the number of items.
A factorial is the product of all whole numbers from ‘n’ to 1. The symbol is n!
6! = 5 x 4 x 3 x 2 x 1 = 720
So 6 people can be arranged in 720 different ways.
Ex.
Five medalists from a competition are posing for a newspaper photographer. How many
different ways can they line up for the picture?
__5_
__4__ __3__ __2__ __1_
After one person stands in the first position there are only 4 medalists to choose from to stand in
the second position.
The answer is 5  4  3  2  1=120
ways.
5! = 120
4!= 4  3  2  1
= 24
Ex.
A car dealer has 5 parking spaces on the showroom floor to display cars. Eight new cars
have arrived. How many different ways can the dealer display the cars.
__8_ ___7_ __6__ ___5_ __4__
8  7  6  5  4= 6720 ways
To show this using factorial you would write
8  7  6  5  4  3  2  1
8!
=
3  2  1
3!
The 3, 2 and 1 in the numerator cancel the 3, 2 and 1 in the denominator.
Permutations and Combinations
A permutation is an arrangement or listing in which order is important.
How many ways can the starting line up for a relay be arranged if there are 4 selected
runners out of six students.
6!
2!
The symbol for permutations in the calculator is p(n,r)
p=permutation n=number of things r=number at one time
6  5  4  3 = 360
p(6,4) 6 runners, 4 picked at a time
A combination is an arrangement or listing where order is not important.
Examples include picking 3 ice cream flavors, choosing books, etc.
Selection of 5 ice cream flavors Choice of 3 scoops for a cone
Vanilla, chocolate, strawberry, mocha, pistachio
List:
VCS
(VSC)
VSP
(VPS)
VSM
(VMS)
VPM
(VMP)
VCM
(VMC)
VCP
(VPC)
This is a listing of all the possibilities if vanilla is chosen as the first flavor. Since order
does not matter the duplicates are shown using parentheses.
Using factorial 5  4  3 = 60 There are 60 possibilities if ordered mattered, but to
eliminate the duplicates divide by 3  2  1
5  4  3 60

 10
3  2 1 3
There are 10 ways to choose 3 scoops of ice cream from 5 flavors.
The symbol is
C(5,3) =
c(n,r)
p (5,3)
3!
12-6 Probability
Objective: know how to calculate the probability of a given event.
Probability is the likelihood that a certain event, or set of outcomes, will occur.
P(E) = number of favorable outcomes
Total number of possible outcomes
The sample space is the set of possible outcomes.
Ex. What is the probability that a letter chosen at random from the word Mississippi is
the letter I?
The sample space is 11. The number favorable outcomes, letters that are ‘I,’ is 4.
The probability is
P(I)=
4
11
12-7 Independent and Dependent Events
Objective: know the difference between independent and dependent events and be able
to find the probability of both independent and dependent events.
Compound Events are a combination of two or more single events.
Compound events can be independent or dependent. Examples of independent events: a
1. throwing one die and then using a spinner - The first event does not effect the second.
3. choosing a marble from a bag, putting it back and then choosing again – The sample
space remains the same when you choose the second time.
Independent Events are events in which the outcome of the first has no effect on the
outcome of the other.
P(A and B) = P(A)  P(B) Find the probability of each event and multiply them.
Dependent Events are events in which the outcome of the second depends on the
outcome of the first.
P(A and B) = P(A)  P(B, given A)
Example: If there are 3 red marbles, 2 blue and 1 yellow marble in a bag, what is the
probability of choosing a blue and then another blue, (do not put the first marble back).
P(blue) =
2 1
or for the first event.
6 3
1 1 1
P(blue,blue)=  
3 5 15
P(blue)=
1
for the second event
5
One out of 15 chances of picking two blue marbles.
12-8 Making Predictions
Objective: estimate an animal population from a sample.
To estimate wildlife populations from a sample researchers capture, tag, and free animals.
Later the researchers go out to capture the same species of animal and see how many
have been tagged.
Set up a proportion and solve to get an estimate of the total population.
#tagged  animals #tagged  animals  recaptured

total  population
total  animals  recaptured
New Jersey officials want to know the approximate bear population. Park rangers
capture, tag, and set free 24 bears. Two weeks later rangers capture 38 bears. Eight of the
bears have been tagged. Estimate the number of bears in NJ.
24 8

x 38
8x = 912
x=114
Approximate number of bears is 114.