Chapter 2 Frequency Distributions and Graphs A frequency

Chapter 2 Frequency Distributions and Graphs

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

The number of miles that the employees of a large department store traveled to work each day 1 18 4 9 9 2 7 16 11 18 6 3 4 12 8 7 15 5 1 8 12 15 8 9 4 13 4 6 2 14 2 17 5 10 7 6 1 18 11 3 9 14 5 4 2 5 5 2 10 6

Class Limits (in miles) 1 -3 4 -6 7 -9 10 -12 13 -15 16 -18 Frequency 10 14 10 6 5 5 Total 50 How to construct a grouped frequency Distribution?

n k 2 k Number of classes It should be between 5 and 20. Some Statisticians use “ 2 k “ rule. 1 2 2 4 3 8 4 16 5 32 6 64 7 8 9 10 128 256 512 1, 024

2 to k rule Essentially we would look to construct k classes for our frequency distribution, when the value of 2 k 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 2 k rule is a guide, not a rule. If the 2 k rule suggests you need 6 classes, also consider using 5 or 7 classes. . . but certainly not 3 or 9.

q Class interval or class width H : the highest value, L: the smallest value q Class interval can also be estimated based on # of observations

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

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

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

Example These data represent the record high temperatures for each of the 50 states. Construct a grouped frequency distribution for the data using 7 classes. n 112 110 107 116 120 100 118 112 108 113 127 114 110 120 116 115 121 117 134 118 113 105 118 122 117 120 110 105 114 118 119 118 110 114 122 111 112 109 105 106 104 112 109 110 111 114

Class limits 100 -104 Class boundaries 99. 5 -104. 5 Frequency 2 Relative frequency 0. 04 Cumulative frequency 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

Histogram A 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.

Example Construct a histogram to represent the data shown below for the record high temperature: Class boundaries Frequency 99. 5 -104. 5 2 104. 5 -109. 5 8 109. 5 -114. 5 18 114. 5 -119. 5 13 119. 5 -124. 5 7 124. 5 -129. 5 1 129. 5 -134. 5 1

18 Histogram 15 12 9 6 3 99. 5 104. 5 109. 5 114. 5 119. 5 124. 5 129. 5 o. The largest concentration is in the class 109. 5 – 114. 5.

18 Frequency Polygone 15 12 9 6 3 99. 5 104. 5 109. 5 114. 5 119. 5 124. 5 129. 5

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

Class boundaries 99. 5 -104. 5 -109. 5 -114. 5 -119. 5 -124. 5 -129. 5 -134. 5 Frequency 2 8 18 13 7 1 1 Cumulative Frequency 2 10 28 41 48 49 50

Cumulative Frequency Polygone 50 40 30 20 10 99. 5 104. 5 109. 5 114. 5 119. 5 124. 5 129. 5

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

Example The 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 Homicide Rape Robbery Assault Number 13 34 29 164

164 150 100 Assault 29 Robbery 34 Rape 13 Homicide 50 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.

Example This frequency distribution shows the number of pounds of each snack food eaten during the 1998 Super Bowl. Construct a pie graph for the data. Million Snack pounds Potato Chips 11. 2 Tortilla Chips 8. 2 Pretzels 4. 3 Popcorn 3. 8 Snack nuts 2. 5

We need to find percentages for each category and then compute the corresponding sectors so that we divide the circle proportionally. Snack Potato Chips Tortilla Chips Pretzels Popcorn Snack nuts Million pounds 11. 2 8. 2 4. 3 3. 8 2. 5 percentage 37. 33% 27. 33% 14. 33% 12. 67% 8. 33% Degree ≈134º ≈98º ≈41º ≈46º ≈30º

Snack nuts Popcorn 8% 13% Pretzels 14% Potato Chips 37% Potato Chips Tortilla Chips Pretzels Popcorn Tortilla Chips 27% Snack nuts

Stem and Leaf Plots A 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.

Example At 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 14 36 32 31 43 32 52 20 02 33 44 32 57 32 51 13 23 44 45

It is helpful to arrange the data in order but it is not required. 02, 13, 14, 20, 23, 25, 31, 32, 32, 33, 36, 43, 44, 45, 51, 52, 57 Leading digit (Stem) 0 1 2 3 4 5 Trailing digit (Leaf) 2 34 035 1222236 3445 127

EXERCISES 1 The 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 BL Y W W W G BL BL BR BL W G Y Y BR BL BR W BL BL W G W BL BR W BR BL W BL BL W W W BL W W BR Y BR BL BR G G Y BR Y G

EXERCISES 2 The ages of the signers of the Declaration of Independence of the US are shown below. 41 44 44 35 35 54 52 63 43 46 47 39 60 48 45 40 50 27 46 34 39 40 42 31 53 35 30 34 27 50 50 34 50 55 50 37 69 42 63 49 39 52 46 42 45 38 33 70 33 36 60 32 42 45 62

EXERCISES 2 (Cont. ) 1) Construct a frequency distribution using seven classes. Include relative frequency, percentage and Cumulative frequency. 2) Construct a histogram, frequency polygone, and Ogive. 3) Develop a stem-and-leaf plot for the data.

Thank You for your attention! Good Luck!