Frequency Distributions and Graphs

Chapter 2

A frequency distribution is the organization of raw data in table from, using classes and frequency.

1 2 6 7 12 13 2 6 9 518 7 3 15 15 4 17 1 14 5

4 16 4 5 8 6 5 18 5 29 11 12 1 9 2 10 11 4 109 18 8 8 4 14 7 3 2 6

The number of miles that the employees of a large department store traveled to work each day

Class Limits(in miles) Frequency

1-3 104-6 147-9 10

10-12 613-15 516-18 5

Total 50

How to construct a grouped frequency Distribution?

Number of classes It should be between 5 and 20.Some Statisticians use “ 2k “ rule.

k 1 2 3 4 5 6 7 8 9 102k 2 4 8 16 32 64 128 256 512 1,024

2 to k ruleEssentially we would look to construct k classes for our frequency distribution, when the value of 2k first exceeds the number of observations in our sample. So, if we had a sample with 39 observations, we would first consider constructing 6 classes, because 26 = 64, the first power of 2 with a value larger than the sample size of 39.

A guide, not a dictator.Strictly speaking the 2k rule is a guide, not a rule. If the 2k rule suggests you need 6 classes, also consider using 5 or 7 classes ... but certainly not 3 or 9.

Class interval or class width

H : the highest value, L: the smallest value Class interval can also be estimated

based on # of observations

H Lwk

1 3.322lnH Lw

n

Select the lower limit of the first class and set the limits of each classIt could be L or any value smaller than L. It should be an even multiple of the class interval.

There should be between 5 and 20 classes. The classes must be continuous. The classes must be exhaustive. The classes must be mutually exclusive. The classes must be equal in width.

Relative frequency

o Relative frequency of a class is the frequency of that class divided by to total number of frequency.

fRFn

ExampleThese data represent the record high temperatures for each of the 50 states. Construct a grouped frequency distribution for the data using 7 classes.

112 100 127 120 134 118 105 110 109 112110 118 117 116 118 122 114 114 105 109107 112 114 115 118 117 118 122 106 110116 108 110 121 113 120 119 111 104 111120 113 120 117 105 110 118 112 114 114

Class limitsClass

boundaries FrequencyRelative

frequencyCumulative frequency

100-104 99.5-104.5 2 0.04 2

105-109 104.5-109.5 8 0.16 10

110-114 109.5-114.5 18 0.36 28

115-119 114.5-119.5 13 0.26 41

120-124 119.5-124.5 7 0.14 48

125-129 124.5-129.5 1 0.02 49

130-134 129.5-134.5 1 0.02 50

HistogramA histogram is a graph that displays the data by using contiguous vertical bars (unless the frequency of a class is 0) of various heights to represent the frequencies of the classes.

ExampleConstruct a histogram to represent the data shown below for the record high temperature:

Class boundaries Frequency99.5-104.5 2

104.5-109.5 8109.5-114.5 18114.5-119.5 13119.5-124.5 7124.5-129.5 1129.5-134.5 1

99.5 109.5104.5 124.5119.5114.5 129.5

3

6

9

12

15

18

Histogram

oThe largest concentration is in the class 109.5 – 114.5.

99.5 109.5104.5 124.5119.5114.5 129.5

3

6

9

12

15

18

Frequency Polygone

The Ogive is a graph that represents the cumulative frequencies for the classes in a frequency distribution.

Class boundaries Frequency

CumulativeFrequency

99.5-104.5 2 2104.5-109.5 8 10109.5-114.5 18 28114.5-119.5 13 41119.5-124.5 7 48124.5-129.5 1 49129.5-134.5 1 50

99.5 109.5104.5 124.5119.5114.5 129.5

10

20

30

40

50

Cumulative Frequency Polygone

Other types of Graphs

Bar ChartBar Chart is use to represent a frequency distribution for a categorical variable, and the frequencies are displayed by the heights of vertical bars.

ExampleThe table shown here displays the number of crimes investigated by law enforcement officers in U.S. national parks during 1995. Construct a Bar chart for the data.

Type NumberHomicide 13Rape 34Robbery 29Assault 164

50

100

150H

omic

ide

Rap

e

Rob

ber

y Ass

ault

1334 29

164

Total number of crime: 234

Pie Graph A pie graph is a circle that is divided into sections or wedges according to the percentage of frequencies in each category of the distribution.

ExampleThis frequency distribution shows the number of pounds of each snack food eaten during the 1998 Super Bowl. Construct a pie graph for the data.

SnackMillion pounds

Potato Chips 11.2Tortilla Chips 8.2Pretzels 4.3Popcorn 3.8Snack nuts 2.5

We need to find percentages for each category and then compute the corresponding sectors so that we divide the circle proportionally.

SnackMillion pounds percentage Degree

Potato Chips 11.2 37.33% ≈134ºTortilla Chips 8.2 27.33% ≈98ºPretzels 4.3 14.33% ≈41ºPopcorn 3.8 12.67% ≈46ºSnack nuts 2.5 8.33% ≈30º

Potato Chips 37%

Tortilla Chips 27%

Pretzels 14%

Popcorn 13%

Snack nuts 8%

Potato Chips Tortilla Chips Pretzels Popcorn Snack nuts

Stem and Leaf PlotsA stem and leaf plot is a data plot that uses part of the data value as the stem and part of the data value as the leaf to form groups or classes.

ExampleAt an outpatient testing center, the number of cardiograms performed each day for 20 days is shown. Construct a tem and leaf plot for the data.

25 31 20 32 1314 43 02 57 2336 32 33 32 4432 52 44 51 45

It is helpful to arrange the data in order but it is not required.02, 13, 14, 20, 23, 25, 31, 32, 32, 32, 32, 33, 36, 43, 44, 44, 45, 51, 52, 57

Leading digit (Stem)

Trailing digit (Leaf)

0 21 3 42 0 3 53 1 2 2 2 2 3 64 3 4 4 55 1 2 7

EXERCISES 1The following data represent the color of men’s dress shirts purchased in the men’s department of a large department store. Construct a categorical frequency distribution, bar chart and pie chart for the data (W= white, BL= blue, BR= brown, Y= yellow, G= gray).

EXERCISES 1(Cont.)W W BR Y BL BL W W Y GW W BL BR BL BR BL BL BR YBL G W BL W W BL W BL BRY BL G BR G BR W W BR YW BL Y W W BL W BR G G

EXERCISES 2The ages of the signers of the Declaration of Independence of the US are shown below.

41 54 47 40 39 35 50 37 49 42 70 3244 52 39 50 40 30 34 69 39 45 33 4244 63 60 27 42 34 50 42 52 38 36 4535 43 48 46 31 27 55 63 46 33 60 6235 46 45 34 53 50 50  

EXERCISES 2 (Cont.)

1) Construct a frequency distribution using seven classes. Include relative frequency, percentage and Cumulative frequency.

2) Construct a histogram, frequency poly-gone, and Ogive.

3) Develop a stem-and-leaf plot for the data.

Thank You for your attention!

Good Luck!