math 110 sec 14-1 lecture: statistics-organizing and visualizing data statistics the study of the...

MATH 110 Sec 14-1 Lecture: Statistics-Organizing and Visualizing Data

STATISTICSThe study of the collection, analysis, interpretation,

presentation and organization of data.


STATISTICSThe study of the collection, analysis, interpretation,

presentation and organization of data.

DATAData is a set a values of quantitative or qualitative variables.

For purposes of what we will study here,let’s just think of data as a set of numerical information.

MATH 110 Sec 14-1 Lecture: Statistics-Organizing and Visualizing Data We will begin by looking at ways to organize data.


25 viewers evaluated the latest episode of CSI. The possible evaluations are

(E)xcellent, (A)bove average, a(V)erage, (B)elow average, (P)oorAfter the show, the 25 evaluations were as follows:

A V V B P E A E V V A E P B V V A A A E B V A B V

We will begin by looking at ways to organize data.





Often information that we need to analyze is not easy to deal with in its raw form.

We will begin by looking at ways to organize data.



One way of organizing discrete data into a more useful format is aFREQUENCY TABLE

We will begin by looking at ways to organize data.25 viewers evaluated the latest episode of CSI.

The possible evaluations are (E)xcellent, (A)bove average, a(V)erage, (B)elow average, (P)oor

After the show, the 25 evaluations were as follows:A V V B P E A E V V A E P B V V A A A E B V A B V



One way of organizing discrete data into a more useful format is aFREQUENCY TABLE

Let’s use construct a frequency table for the data above.




MATH 110 Sec 14-1 Lecture: Statistics-Organizing and Visualizing Data We will begin by looking at ways to organize data.




A frequency table organizes the data by counting the number of occurrences (the frequency) of

each possible outcome.

EVALUATION FREQUENCY



E 4






There are 4

‘Excellent’



E 4A 7






There are 7 ‘Above Average’



E 4A 7V 8





each possible outcome.There are

8 ‘Average’



E 4A 7V 8B 4





each possible outcome. There are 4 ‘Below Average’



E 4A 7V 8B 4P 2






2‘Poor’



E 4A 7V 8B 4P 2

TOTAL 25






25in all



E 4A 7V 8B 4P 2

TOTAL 25




This is the frequency table

for the data above.



E 4A 7V 8B 4P 2

TOTAL 25




If we take a frequency

table



E 4A 7V 8B 4P 2

TOTAL 25





table…and replace counts

with probabilities, we get a RELATIVE FREQUENCY table.



E 4A 7V 8B 4P 2

TOTAL 25




EVALUATION PROBABILITY

EAVBP

TOTAL






E 4A 7V 8B 4P 2

TOTAL 25




EVALUATION PROBABILITY

EAVBP

TOTAL 1






E 4A 7V 8B 4P 2

TOTAL 25




EVALUATION RELATIVE FREQUENCY

EAVBP

TOTAL 1




Often the ‘Probability’ column will be labeled ‘Relative

Frequency’.



E 4A 7V 8B 4P 2

TOTAL 25




EVALUATION RELATIVE FREQUENCY

EAVBP

TOTAL 1




MATH 110 Sec 14-1 Lecture: Statistics-Organizing and Visualizing Data Sometimes there are so many different outcomes that a frequency or

relative frequency table is not useful without grouping the data.


20 health care workers take an assessment with these scores:62 53 67 68 61 51 66 66 64 6159 58 56 64 67 68 57 65 69 60

Sometimes there are so many different outcomes that a frequency or relative frequency table is not useful without grouping the data.


Let’s construct a frequency table using equal sized intervals starting ’50-54’.






Let’s construct a frequency table using equal sized intervals starting ’50-54’. SCORES FREQUENCY




Let’s construct a frequency table using equal sized intervals starting ’50-54’. SCORES FREQUENCY50 - 54




Let’s construct a frequency table using equal sized intervals starting ’50-54’. SCORES FREQUENCY50 - 54 2




Let’s construct a frequency table using equal sized intervals starting ’50-54’. SCORES FREQUENCY50 - 54 255 - 59




Let’s construct a frequency table using equal sized intervals starting ’50-54’. SCORES FREQUENCY50 - 54 255 - 59 4




Let’s construct a frequency table using equal sized intervals starting ’50-54’. SCORES FREQUENCY50 - 54 255 - 59 460 - 64




Let’s construct a frequency table using equal sized intervals starting ’50-54’. SCORES FREQUENCY50 - 54 255 - 59 460 - 64 6




Let’s construct a frequency table using equal sized intervals starting ’50-54’. SCORES FREQUENCY50 - 54 255 - 59 460 - 64 665 - 69




Let’s construct a frequency table using equal sized intervals starting ’50-54’. SCORES FREQUENCY50 - 54 255 - 59 460 - 64 665 - 69 8




Let’s construct a frequency table using equal sized intervals starting ’50-54’. SCORES FREQUENCY50 - 54 255 - 59 460 - 64 665 - 69 8TOTAL 20




Let’s construct a frequency table using equal sized intervals starting ’50-54’. SCORES RELATIVE FREQUENCY50 - 54 255 - 59 460 - 64 665 - 69 8TOTAL 20

If you want a relative frequency table instead, just divide each frequency by 20.




Let’s construct a frequency table using equal sized intervals starting ’50-54’. SCORES RELATIVE FREQUENCY50 - 54 2/2055 - 59 4/2060 - 64 6/2065 - 69 8/20TOTAL 20





Let’s construct a frequency table using equal sized intervals starting ’50-54’. SCORES RELATIVE FREQUENCY50 - 54 1/1055 - 59 1/560 - 64 3/1065 - 69 2/5TOTAL 20


MATH 110 Sec 14-1 Lecture: Statistics-Organizing and Visualizing Data If we prefer, we can turn a frequency or relative frequency table into a

BAR GRAPH.


BAR GRAPH.

EVALUATION FREQUENCYE 4A 7V 8B 4P 2

TOTAL 25

Let’s turn the frequency table we constructed earlier

into a BAR GRAPH.


BAR GRAPH.


TOTAL 25


into a BAR GRAPH.

2

4

6

8

10

Let’s put the frequencies on the

vertical axis.

FREQ

UEN

CY


BAR GRAPH.


TOTAL 25


into a BAR GRAPH.

2

4

6

8

10

…and the outcomes on the horizontal axis.

FREQ

UEN

CYE A V B P


BAR GRAPH.


TOTAL 25


into a BAR GRAPH.

2

4

6

8

10

Finally, let the height of each bar represent the

corresponding frequency.

FREQ

UEN

CYE A V B P


BAR GRAPH.


TOTAL 25


into a BAR GRAPH.

2

4

6

8

10

FREQ

UEN

CYE A V B P

MATH 110 Sec 14-1 Lecture: Statistics-Organizing and Visualizing Data If the data from a frequency or relative frequency table is continuous rather than discrete, we construct the ‘bars’ with no space between them and call the resulting graph a histogram instead of a bar graph.


We will not be focusing on the difference between discrete and continuous here. If you simply think of ‘discrete values’ as

values that are completely distinct from each other and ‘continuous values’ as values that can sort of bleed over into

each other, you will be OK for what we will be doing.




each other, you will be OK for what we will be doing. A quick example to help with this distinction:

If you are counting things, the counts are discrete.There is no doubt that 2 is different than 3.






But if you are measuring a person’s height, one person might measure and get 59.99 inches while another person might measure the same person and get

60.01 inches. (In this sense, the values sort of bleed into each other.)






But if you are measuring a person’s height, one person might measure and get 59.99 inches while another person might measure the same person and get

60.01 inches. (In this sense, the values sort of bleed into each other.)

Don’t worry if this distinction is not perfectly clear to you. It really will not have much if

any impact on what we are doing here.

MATH 110 Sec 14-1 Lecture: Statistics-Organizing and Visualizing Data Let’s construct a histogram from the frequency table below.

Pounds lost Frequency 0 to 10 14

10+ to 20 2320+ to 30 1730+ to 40 11

Total 65



10+ to 20 2320+ to 30 1730+ to 40 11

Total 65 5

10

15

20

25

FREQ

UEN

CY



10+ to 20 2320+ to 30 1730+ to 40 11

Total 65 5

10

15

20

25

FREQ

UEN

CY0 to 10 10+ to 20

Pounds Lost20+ to 30 30+ to 40

MATH 110 Sec 14-1 Lecture: Statistics-Organizing and Visualizing Data A Stem-and-Leaf plot is another visual way to display data.

MATH 110 Sec 14-1 Lecture: Statistics-Organizing and Visualizing Data A Stem-and-Leaf plot is another visual way to display data.

In constructing a stem-and-leaf display, we view each number as having two parts. The left digit is considered the stem and the right digit the

leaf. This is probably best illustrated through an example.

MATH 110 Sec 14-1 Lecture: Statistics-Organizing and Visualizing Data These are the number of home runs hit by the home run champions in the National League for the years 1975 to 1989 and for 1993 to 2007. 1975–1989: 38, 38, 52, 40, 48, 48, 31, 37, 40, 36, 37, 37, 49, 39, 47 1993–2007: 46, 43, 40, 47, 49, 70, 65, 50, 73, 49, 47, 48, 51, 58, 50

Compare these home run records using a stem-and-leaf display.



The left (BLUE) digit is considered the stem



and the right (RED) digit is considered the leaf.






1975–1989





1975–1989

Among the left (blue) digits, there are only threes, fours and fives.





1975–1989

Among the left (blue) digits, there are only threes, fours and fives.

For the leaves, write down each rightmost (red) digit in numerical order next to the

stem that it belongs to.





1975–1989





1975–1989 1993–2007





1975–1989 1993–2007

Among the left (blue) digits, there are only fours, fives, sixes and sevens.





1975–1989 1993–2007

Among the left (blue) digits, there are only fours, fives, sixes and sevens.

For the leaves, write down each rightmost (red) digit in numerical order next to the

stem that it belongs to.



1975–1989 1993–2007





1975–1989 1993–2007

We can compare these data by placing the two displays side by side.Some people call this a Back-to-back Stem-and-Leaf plot.


In summary, these are the data organization and display methods discussed

FREQUENCY TABLE

BAR GRAPH

RELATIVE FREQUENCY TABLE

HISTOGRAM

STEM-AND-LEAF PLOT/DISPLAY

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