Chapter 2

Descriptive Statistics

1Larson/Farber 4th ed.

Section 2.1

Frequency Distributions

and Their Graphs

Part 1: Frequency Distributions

2Larson/Farber 4th ed.

Frequency Distribution

Frequency Distribution: A table that shows classes or intervals of data with a count of the number of entries in each class. It is used to organize data and helps to recognize patterns.

The frequency, f, of a class is the number of data entries in the class.

Larson/Farber 4th ed.

Class Frequency

1 – 5 5

6 – 10 8

11 – 15 6

16 – 20 8

21 – 25 5

26 – 30 4

Frequency Distribution

• Each class has:– A lower class limit,

which is the least number that can belong to the class

(1, 6, 11, 16, 21, 26)– An upper class limit,

which is the greatest number that can belong to the class

(5, 10, 15, 20, 25, 30)

Class Frequency

1 – 5 5

6 – 10 8

11 – 15 6

16 – 20 8

21 – 25 5

26 – 30 4

Frequency Distribution

• The class width is the distance between lower (or upper) limits of consecutive classes.

– Example: 6 – 1 = 5

Class Frequency

1 – 5 5

6 – 10 8

11 – 15 6

16 – 20 8

21 – 25 5

26 – 30 4

Frequency Distribution

• The range is the difference between the maximum and the minimum data entries.

– Example: if the maximum data entry is 29 and the minimum data entry is 1, the range is 29 – 1 = 28

Class Frequency

1 – 5 5

6 – 10 8

11 – 15 6

16 – 20 8

21 – 25 5

26 – 30 4

Constructing a Frequency Distribution from a Data Set

1. Choose the number of classes (usually between 5 and 20)2. Find the class width.

a) Find the range of the datab) Divide the range by the number of classesc) Round up to the next convenient number

3. Find the class limits.a) Use the minimum data entry as the lower limit of the first class b) Find the remaining lower limits (add the class width to the lower

limit of the preceding class). c) Find the upper limit of the first class. Remember that classes

cannot overlap. d) Find the remaining upper class limits.

4. Tally the data5. Count the tally marks to find the total frequency for each

class

Example: Constructing a Frequency Distribution

The following sample data set lists the number of minutes 50 Internet subscribers spent on the Internet during their most recent session. Construct a frequency distribution that has seven classes.

50 40 41 17 11 7 22 44 28 21 19 23 37 51 54 42 86 41 78 56 72 56 17 7 69 30 80 56 29 33 46 31 39 20 18 29 34 59 73 77 36 39 30 62 54 67 39 31 53 44

Larson/Farber 4th ed. 8

Solution: Constructing a Frequency Distribution

1. Number of classes = 7 (given)

2. Find the class width

Larson/Farber 4th ed. 9

max min 86 711.29

#classes 7

Round up to 12

50 40 41 17 11 7 22 44 28 21 19 23 37 51 54 42 86

41 78 56 72 56 17 7 69 30 80 56 29 33 46 31 39 20

18 29 34 59 73 77 36 39 30 62 54 67 39 31 53 44

Solution: Constructing a Frequency Distribution

Larson/Farber 4th ed. 10

Lower limit

Upper limit

7Class width = 12

3. Find the class limits:

Use 7 (minimum value) as first lower limit. Add the class width of 12 to get the lower limit of the next class.

7 + 12 = 19

Find the remaining lower limits.

19

31

43

55

67

79

Solution: Constructing a Frequency Distribution

The upper limit of the first class is 18 (one less than the lower limit of the second class).

Add the class width of 12 to get the upper limit of the next class.

18 + 12 = 30

Find the remaining upper limits.

Larson/Farber 4th ed. 11

Lower limit

Upper limit

7

19

31

43

55

67

79

Class width = 1230

42

54

66

78

90

18

Solution: Constructing a Frequency Distribution

4. Make a tally mark for each data entry in the row of the appropriate class.

5. Count the tally marks to find the total frequency f for each class.

Larson/Farber 4th ed. 12

Class Tally Frequency, f

7 – 18 IIII I 6

19 – 30 IIII IIII 10

31 – 42 IIII IIII III 13

43 – 54 IIII III 8

55 – 66 IIII 5

67 – 78 IIII I 6

79 – 90 II 2Σf = 50

Example: Constructing a Frequency Distribution

The following represents census data reporting the ages of the entire population of the 77 resdients of Akhiok, Alaska. Construct a frequency distribution with 6 classes.

28 6 17 48 63 47 27 21 3 7 12 39 50 54 33 45 15 24 1 7 36 53 46 27 5 10 32 50 52 11 42 22 3 17 34 56 25 2 30 10 33 1 49 13 16 8 31 21 6 9 2 11 32 25 0 55 23 41 29 4 51 1 6 31 5 5 11 4 10 26 12 6 16 8 2 4 28

Larson/Farber 4th ed. 13

Expanding the Frequency Distribution

• There are additional features that we can add to the frequency distribution that will provide a better understanding of the data– Midpoint, relative frequency, and cumulative

frequency

Midpoint• The midpoint of a class is the sum of the

lower and upper limits of the class divided by two. The midpoint is sometimes called the class mark.

Lower class limit Upper class limitmidpoint =

2

Note: After you find one midpoint, you can find the following midpoints by adding the

class width to the previous midpoint

Relative Frequency

• The relative frequency of a class is the portion or percent of the data that falls in that class.– To find the relative frequency of a class,

divide the frequency, f, by the sample size, n.

Class FrequencyRelative Frequency =

Sample Size

f

n

Note: Relative frequency can be written as a decimal or as a percent.

Cumulative Frequency

• The cumulative frequency of a class is the sum of the frequency for that class and all previous classes. – The cumulative frequency of the last class is

equal to the sample size.

Example: Midpoints, Relative and Cumulative Frequencies

• Using the frequency distribution, find the midpoint, relative frequency, and cumulative frequency.

Class Frequency, f

7 – 18 6

19 – 30 10

31 – 42 13

43 – 54 8

55 – 66 5

67 – 78 6

79 – 90 2

Solution: Midpoints, Relative and Cumulative Frequencies

Expanded Frequency Distribution

Larson/Farber 4th ed. 19

Class Frequency, f

MidpointRelative

frequencyCumulative frequency

7 – 18 6 12.5 0.12 6

19 – 30 10 24.5 0.20 16

31 – 42 13 36.5 0.26 29

43 – 54 8 48.5 0.16 37

55 – 66 5 60.5 0.10 42

67 – 78 6 72.5 0.12 48

79 – 90 2 84.5 0.04 50Σf = 50 1

n

f

Example: Midpoints, Relative and Cumulative Frequencies

• Using the frequency distribution, find the midpoint, relative frequency, and cumulative frequency.

Class Frequency, f

0 – 10 5

11 – 21 13

22 – 32 16

33 – 43 7

44 – 54 11

55 – 65 3

Solution: Midpoints, Relative and Cumulative Frequencies

Expanded Frequency Distribution

Class Frequency, f Midpoint

Relative Frequency

Cumulative Frequency

0 – 10 27 5 0.3506 27

11 – 21 13 16 0.1688 40

22 – 32 16 27 0.2078 56

33 – 43 7 38 0.0909 63

44 – 54 11 49 0.1429 74

55 – 65 3 60 0.0390 77

Σ f = 50 Σ ≈ 1

Homework

• P41 #1, 3 – 6, 15, 16