Today’s Agenda

Attendance / Announcements Collect Projects

Return Exams

MyMathLab HW Due Sunday

Remaining Schedule

Sections 10.1, 10.2

Exam Schedule

Exam 6 (Ch 10)

Monday 4/27

Final Exam (Cumulative)

Monday 5/4

Intro to Statistics

Statistics is the science that deals with the collection and summarization of data. Methods of stat analysis allow us to make conclusions about a population based on sampling.

Statistics is more a field of Communications,

than one of Mathematics!

Intro to Statistics

1. Organize Data

2. Display Data

3. Identify the “averages” of the data

4. Identify the “spread” of the data

5. Make conclusions

Obtaining Data

• Want to represent a Population

• Collect data from a Sample

–Should be a Random Sample to be

a fair representation of the

population

Displaying Qualitative data with

Frequency Table

FREQUENCY TABLE A grouping of qualitative data into

mutually exclusive classes showing the number of

observations in each class.

Quantitative Data set

The number of data items, n = 80

Displaying quantitative data with

Frequency Distribution

A FrequencyFREQUENCY DISTRIBUTION A grouping of data into

mutually exclusive classes showing the number of

observations in each class.

Classes:

This Frequency Distribution has 7 classes

Displaying quantitative date with

Frequency Distribution

Class width: The class width is obtained by subtracting the lower limit of a class from the lower limit of the next class.

Class frequency: The number of observations in each class.

Class midpoint: A point that divides a class into two equal parts. This is the average of the upper and lower class limits.

Constructing a Frequency Table

• Step 1: Decide on the number of classes. (if not already given)

A useful recipe to determine the number of classes (k) is the

“2 to the k rule.” such that

2k > n

There were 80 vehicles sold. So n = 80. If we try k=6, which means we would use 6 classes, then 26=64, somewhat less than 80.

Hence, 6 is not enough classes. If we let k = 7, then 27 128, which is greater than 80. So the recommended number of classes is 7.

Constructing a Frequency Table

Step 2: Determine the class interval or width.

The formula is:

where k is the number of classes.

($35,925 - $15,546)/7 = $2,911

Round up to some convenient number, such as a multiple of 10 or 100. Use a class width of $3,000

kClassWidth

minmax

Tuition for a random sample of 30

private, 4-year colleges(thousands)

23 22 38 25 11 16

15 26 23 24 37 18

21 36 36 28 18 9

39 17 27 24 10 32

24 27 22 24 28 39

23 22 38 25 11 16

15 26 23 24 37 18

21 36 36 28 18 9

39 17 27 24 10 32

24 27 22 24 28 39

There are 30 Data Items, so n = 30

Where each can be called

So,

“21”, “37”, etc. are Data Values

ix254 x

Organizing Data

• Frequency Distribution Table–Organize data into Classes

• Usually between 5 - 15

–Each class must have the same Class Width

Class width* = Max data value – Min data value

Number of classes

*Round up to next highest integer

Organizing Data

Let’s make a Freq. Dist. Table with 7 classes to organize

the tuition data…Need Class Width!

28.47

939*

CW

So, each class will have a class width of 5!

Organizing Data

Note: Class width is not (9 – 5)!!!

It is the distance between the lower

limit of each class.

Make

this

column

first!

Displaying Data

1. An account ing firm selected 24 complex tax returns prepared by a certa in tax preparer. The number of

errors per return were as follows. Group the data into 5 classes, and make a frequency table and

histogram/ polygon to represent the data.

Your Class Width =

8 12 0 6 10 8 0 14

8 12 14 16 4 14 7 11

9 12 7 15 11 21 22 19

Create frequency table

Displaying Data

• Frequency Histogram (bar graph)

–Each class is its own “bar”

• No spaces between classes (bars)

–Must label each axis (classes vs. frequency)

–Use straightedge to make lines

1

3

45

67

89

2fre

quen

cy

Tuition5-

9

10-1

4

15-1

9

20-2

4

25-2

9

30-3

4

35-3

9

Displaying Data

• Frequency Polygon (line graph)

–Connects the midpoints of the top of each class.

–Then connect to ground on each side

–Use straightedge to make lines

1

3

45

67

89

2fre

quen

cy

Tuition5-

9

10-1

4

15-1

9

20-2

4

25-2

9

30-3

4

35-3

9

Characterizing Data

Displaying Data1. An account ing firm selected 24 complex tax returns prepared by a certa in tax preparer. The number of

errors per return were as follows. Group the data into 5 classes, and make a frequency table and

histogram/ polygon to represent the data.

Your Class Width =

8 12 0 6 10 8 0 14

8 12 14 16 4 14 7 11

9 12 7 15 11 21 22 19

Create frequency

histogram/polygon

Displaying Data

From worksheet:2. The data below is a random sample of 30 households regarding their monthly

electric bill. Determine the number of classes to use and make a frequency table and

histogram/polygon to represent the data.

10.2 Measures of Central Tendency

• Ways to describe “on average…”–Mean

• What is commonly thought of as “average”

–Median• The “middle” of the data

–Mode• The data value that occurs most often

We need some data…

• Number of hits during spring training for 15

Phillies players: (alphabetical order)

21 19 10 1 6

28 32 11 2 15

2 17 21 29 21

Sample Mean

n

xx

• The mean of a sample set of data

“x bar” is the sample mean.

Round to nearest hundredth. (2 decimal places)

The sum of all data values

The number of data items

• Number of hits for 15 Phillies players:

21 19 10 1 6

28 32 11 2 15

2 17 21 29 21

67.1515

211921

n

xx

Median

• The “middle” of an ordered data set

– Arrange data in order

– Find middle value

• If n is odd, simply select middle value as the

median.

• If n is even, the median value will be the

mean of the two central values (since a

“middle” does not exist)

2

1

nposition

Finding Median Values1. Median of the Phillies data.

2. Age (years) in the intensive care unit at

a local hospital.

68, 64, 3, 68, 70, 72, 72, 68

Median

• When is median a better indicator of “average” than the mean?

Mode

• The data value that appears most often– Single Mode

• One data value appears more than any other– No Mode

• No data values repeat– Multi-Mode

• There is a “tie” for the value that appears the most

Mode

• Mode of Phillies data?

• 2, 3, 3, 3, 5, 6, 6, 6, 7, 7, 10

• 18, 34, 61, 62, 85

• 9.5, 9.2, 9, 9, 9.1, 8.9

Classwork / Homework

• Worksheet

• Page 604

•1, 7, 21 – 25

• Page 614

•1 – 19 odd, 29