Describing, Exploring, and Comparing Data

• Summarizing a set of data with a…– Tables– Pictures– A small set of numbers that describes the data’s

• Center• Variation• Distribution• Outliers

• Comparing summaries– Several different groups– Changes over time within the same group

Frequency Table

• A list of classes of values and the count of the actual values that occur in each class.

• Favorite subjects of three Algebra 1 classes– Qualitative data

Math 4

English 11

Social Studies 9

Languages 2

Physical education 15

Science 4

Music and arts 8

Total 53

Let’s Build a Frequency Table: Page 572

Category Frequency

• M&M Colors

Frequency Table

• Average temperature for August 2009 and 2010– Qualitative data

Temperature 2009 2010

85 - 89 0 3

80 - 84 9 8

75 - 79 13 11

70 - 74 7 7

65 - 69 2 2

60 – 64 0 0

55 – 59 0 0

31 31

Characteristics of a Good Frequency Table

• Class are mutually exclusive (no overlaps)• Together the classes include all possible (or practical)

values.– Sum of all frequencies equals the number of original values– Includes classes whose frequency is 0

• Convenient or sensible class limits• The frequency counts include the whole sample (or

population)• Consistent class widths

– “Greater than” or “Less than” classes sometimes necessary

• Between 5 and 20 classes

Definitions

• Lower class boundary (LCB):– The smallest value in each class

• Upper class boundary (UCB):– The largest value in each class

• Class boundaries:– The midpoint (average) of the LCB and UCB of adjacent

classes

• Class midpoints:– The midpoint within a class. The average of a classes LCB

and UCB

• Class width:– the difference between the LCB of adjacent classes

Building a quantitative frequency table

1. Estimate the number of classes

2. Calculate class width:

– Round if convenient 3. Pick the LCB for the first class.

– Minimum value– Round if convenient

4. Continually add the class width to the LCB of one class to get the LCB of the next.

5. Calculate the UCB6. Tally the actual values in each class

max min

number of classes

Let’s Build a Frequency Table: Page 573

Category Frequency

Others

• Relative Frequency Table– Each class lists its percent of total rather than the actual

count

• Cumulative Frequency Table– Each class lists the total frequency counts from the first

class to the current class

– Start with the “lowest” class

• Relative Cumulative frequency Table– Each class lists the total percent of total from the first class

to the current class

Let’s Build a Frequency Table

Category Frequency

Cumulative Relative Relative/

Cumulative

90 – 95 0

85 - 89 3

80 - 84 8

75 - 79 11

70 - 74 7

65 - 69 2

Your turn: Let’s Build a Frequency Table

Category Frequency

Cumulative Relative Relative/

Cumulative

Using Frequency Tables

• Validating data– Does the frequency differ from what we would expect?

• Exploring data– Spotting trends

– Identifying “hot spots”

• Comparing data– Changes over time

– Comparing samples

Two-way Frequency Table

• Multiple categories for qualitative data

• Day of the week is the row data

• Gender is the column data

Favorite day Female Male

Sunday 2 0

Monday 2 2

Tuesday 1 1

Wednesday 1 1

Thursday 0 0

Friday 17 23

Saturday 17 18

Two-way Frequency Table

• The sum of the rows and columns are called the marginal frequencies

• Marginal frequencies are often expressed as percents

• The marginal frequencies are usually graphed in a bar chart

Favorite day Female Male

Sunday 2 0

Monday 2 2

Tuesday 1 1

Wednesday 1 1

Thursday 0 0

Friday 17 23

Saturday 17 18

Conditional Relative Frequency

• Calculating relative frequencies within a category

• What percent of females like Monday?

• What percent of Friday fans are male?

Favorite day Female Male

Sunday 2 0

Monday 2 2

Tuesday 1 1

Wednesday 1 1

Thursday 0 0

Friday 17 23

Saturday 17 18

Homework

• Using your class data poll, create frequency and relative frequency tables for:

1. Favorite color

2. Favorite month• Using your class data poll, create frequency, relative

frequency, and cumulative frequency tables for:

1. Math and CR SAT scores

2. Lucky numbers less than 100

Gender

0%

10%

20%

30%

40%

50%

60%

Female Male

By Favorite Day

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

Sunday Monday Tuesday Wednesday Thursday Friday Saturday

• Do male and females like different days

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Sunday Monday Tuesday Wednesday Thursday Friday Saturday

Female

Male

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Sunday Monday Tuesday Wednesday Thursday Friday Saturday

Male

Female