Chapter 2

Descriptive Statistics

2.1 Frequency Distributions and Their Tables

A frequency distribution is a table that shows classes or intervals of data entries with a count of the number of entries in each class.

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

In the example above there are 6 classes.Each class has an upper class limit and a lower class limit.The distance between each the upper (or lower) limits of consecutive classes is the class width.The distance between maximum data entry and the minimum data entry is the range.

Your turn:What is class width for table (a)?

5

What is class width for table (b)?

15What is range for table (a)?

Table (a) Table (b)

Class Frequency Class Frequency

6-10 2 15-29 3

11-15 4 30-44 4

16-20 5 45-59 8

21-25 9 60-74 9

26-30 6 75-89 3

31-35 2 90-104 1

Not enough info. You need the raw data.Here is the raw data: 27, 28, 28, 29, 26, 27, 8, 15, 15, 33, 16, 20, 20, 20, 17, 21, 21, 21, 23, 23, 23, 24, 24, 25, 27,28, 29 ,29,29, 30, 28, 31, 26, 10, 11

Now, what is the range for table (a)? 25 = 33 - 8

Guidelines for constructing a frequency distribution table from a data set

1) Decide on the number of classes

2) Find the class width

Between 5-20 is best. Too many or too few won’t show patterns

Determine the range and divide by the number of classes…then round up to the next number that makes sense 3) Find the class limits, upper and lower

Use the minimum and add the class width…repeat No overlap!!!!!

4) Make a tally mark for each data entry OR list in order)

5) Count and put in a table

Practice data set, ACT scores : 22, 11, 17, 17, 13, 21, 25, 15, 26, 18, 24, 18, 35, 31, 20, 14,

21, 16, 24, 22, 25, 29, 16, 24, 24

On the TV series “The Walking Dead” 349 zombies met their untimely demise during the show’s first 27 episodes.

Here is the breakdown per episode (in no particular order): 13, 16, 4, 14, 8, 17, 8, 13, 11, 20, 16, 7, 6, 24, 2, 8, 12, 3, 7, 11, 9, 19, 15, 28, 15, 19, 24

Prepare a frequency distribution table from this data (number of classes should be 6).

Also……state 1) min and max 2) range 3) class width

Practice set 2

A Midpoint of a class is the sum of the lower and upper limits of the class, divided by two Using the example below - (6 + 11) / 2 = 8.5

The Relative frequency of a class is the portion or percentage of the data that falls in that class.Class frequency / Sample size = Relative Frequency

The Cumulative frequency of a class is the sum of the frequency for that class and all previous classes.The cumulative frequency for the first 4 classes in the matrix to the left is6 + 23 + 34 + 17 = 80

∑ is the upper case greek letter Sigma∑ denotes the sum

∑ f = 6+23+34+17+12+8 = 100

∑ f/n (the relative frequency) = .06 + .23 + .34 + .17 + .12 + .08 = 1

A frequency histogram is a bar graph that represents the frequency distribution of a data set. A histogram has the following properties.

1. The horizontal scale is quantitative and measures the data values2. The vertical scale measures the frequencies of the class3. Consecutive bars must touch

A Relative Frequency Histogram has the same shape as corresponding frequency histogram. The difference is that the vertical scale measures the relative frequencies not frequencies

61-80 81-100 101-120 121-140 141-160 161-1800

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Relative Frequency Histogram

Rela

tive

Fre

quency

18 33 48 63 78 9302468

1012141618

Frequency Histogram

Range

Fre

quency

Since the NFL went to a 16 game format the Cincinnati Bengals have won the following games each season: 11, 10, 9, 4, 7, 8, 11, 8, 8, 2, 6, 4, 4, 3, 7, 8, 7, 3, 3, 5, 3, 9, 8, 12, 4, 10, 7, 8, 7, 7, 12, 6, 4, 4

Practice set 2

For this data set:Decide the number of classesFind the class width (range divided by number of classes)Find the class limits (use minimum and add width

repeatedly)County tally marksFind class boundaries (usually + or – 0.5) from class limitsChoose appropriate horizontal and vertical scalesUse frequency distribution to find the height of each barComplete histogram

A frequency Polygon is a line graph that emphasizes the continuous change in frequencies.

Steps for constructing a frequency polygon1) Choose appropriate horizontal and vertical scales2) Plot points that represent the midpoints and frequency

of each class3) Connect the points and extend the sides as necessary

0 11 26 41 56 71 86 1010

2

4

6

8

10

12

14

16

18Frequency Polygon

Fre

quency

A Cumulative Frequency Graph is a line graph that dispalys the cumulative frequency of each class at it’s upper class boundary.

Steps for constructing a frequency polygon1) Choose appropriate horizontal and vertical scales2) Plot points that represent the upper class boundaries

and cumulative frequencies of each class3) Connect the points and extend the sides as necessary

0 18 33 48 63 78 930

10

20

30

40

50

60

Class boundaries are the numbers that separate classes without forming gaps between them.

Determining Class Boundaries

Class Class boundaries

7-18 The difference between the 6.5-18.5

19-30 upper limit of one class and the 18.5-30.5

31-42 lower limit of the subsequent 30.5-42.5

43-54 class….divided by two 42.5-54.5

55-66 19-18 = 1 => 1/2 54.5-66.5

67-78 66.5-78.5

79-90 78.5-90.5

79 – 0.5 = 78.5 90 + 0.5 =90.5