1. Frequency Distributions and Graphs
Descriptive Statistics
Frequency Distributions and Graphs

2. Introduction Data must be organized Present data
Frequency distribution
Present data
Statistical graphs and charts

3. Organizing Data Frequency distribution Uses classes and frequency
3 types
Categorical frequency distribution
Nominal - or ordinal- type data
Ungrouped frequency distribution
Grouped frequency distribution

4. Construct a Categorical Distribution
Make a table
Class Tally Frequency Percent
Tally the data
Count the tallies – frequency
Find the percentage
% = f/n * 100
f – frequency
n – total number

5. Example 2-1
Twenty-five army inductees were given a blood test to determine their blood type. Construct a frequency distribution for the data. AB B A O
Step 1: Make a table
Step 2: Tally the data
Step 3: Count the tallies
Step4: Find the percentage of values in each class

6. Example 2-1 Class A B O AB Tally IIIII IIII II IIII IIII IIII
Frequency 5 7 9 4
Percent
5/25 * 100 =20
7/25*100=28
9/25*100=36
4/25*100=16

7. Ungrouped Frequency Distribution
Range of data is relatively small
Range = highest value – lowest value
Single data values for each class
Steps similar to categorical distribution
Make a table
Each unique data value is a class
Tally - Frequency
Percent or Relative Frequency
Rel. freq. = # of ea. value/ total # of values

8. Example
A survey was taken on Maple Avenue. In each of 20 homes, people were asked how many cars were registered to their households. The results were recorded as follows: 1, 2, 1, 0, 3, 4, 0, 1, 1, 1, 2, 2, 3, 2, 3, 2, 1, 4, 0, 0 Construct a frequency distribution for the data.

9. Solution Step 1: Make a table # of cars tally frequency rel. freq.
Step 2: Tally – Frequency
Step 3: Percent or Rel. Freq.
# of Cars
Tally
Frequency
Rel. Freq.
IIII 4
4/20 = .20 1
IIII I 6
6/20 = .30 2 5
5/20 = .25 3
III
3/20 = .15 II
2/20 = .10

10. Grouped Frequency Distribution
Range of data is large
Classes are more than 1 unit in width
Vocabulary
Class width – difference between consecutive lower class limits or upper class limits
31 – 24 = 7
37 – 30 = 7
Class
Limits
24 – 30
31 – 37
38 – 44
Lower class limits
Upper class limits

11. Grouped Frequency Distribution
Class boundaries
Always end in a 5
1 decimal place to right of data
If data are whole numbers
- 0.5 from lower limit
+ 0.5 to upper limit
Class limits should have the same # of dec. places as the data
Class limits
Class boundaries
24 – 30
23.5 – 30.5
31 – 37
30.5 – 37.5
38 – 44
37.5 – 44.5

12. Guidelines for Classes
There should be between 5 and 20 classes
Class width should be an odd number
Midpoints will be integers instead of decimals
Classes must be mutually exclusive
No data value can fall into 2 classes
Classes must be continuous
No gaps
Even if there is no data in a class it must be included
Classes are exhaustive
All data fits into a class
Classes are equal in width
Only exception open-ended class

13. Constructing a grouped freq. distribution
Determine the number of classes
Find the highest and lowest values
Find the range
Select the # of classes (5 – 20)
Find the width
w = range/# of classes
Round up
3.2 becomes 4
3 becomes 4
Choose the lowest value as your starting point
Add the class width to this lower limit
Continue adding the width to the lower limits until you get the rest of the lower limits
Find the first upper limit
Subtract 1 from the second lower limit
Continue adding the width to find the remaining upper limits

14. Example 2-2 - done in parts
These data represent the record high temperatures for each of the 50 states. Construct a grouped frequency distribution for the data using 7 classes

15. Class limits Determine the # of classes
Find the highest and lowest values H = 134 L = 100
Find the range: H – L ; R = 134 – 100 = 34
Select # of classes (already asks for 7)
Find the class width: w = R/7; w = 34/7 = 4.9 ~ 5
Choose a starting point – lowest value = 100
Add the class width to find next lower limit
Continue adding the width until there are 7
Find the first upper limit – subtract 1 from the second lower limit 105 – 1 = 104
Add the width to the upper limits
Class limits
100 – 104
105 – 109
110 – 114
115 – 119
120 – 124
125 – 129
+ 5
+ 5
+ 5
+ 5
+ 5
+ 5
+ 5
+ 5
+ 5
+ 5
+ 5
+ 5

16. Class boundaries & Class midpoints
Class limits
Class boundaries
Class midpoints
100 – 104
99.5 – 104.5
102
105 – 109
104.5 – 109.5
107
110 – 114
109.5 – 114.5
112
115 – 119
114.5 – 119.5
117
120 – 124
119.5 – 124.5
122
125 – 129
124.5 – 129.5
127
130 – 134
129.5 – 134.5
132
Class midpoints determined by adding the 2 class limits together then dividing by 2 Or
Adding the 2 class boundaries together then dividing by 2
= 204/2 = 102

17. Constructing a grouped freq. distribution
Tally the data
Find the numerical frequencies from the tallies
Find the cumulative frequencies
Class
limits
bound
Tally
Freq
Cum. Freq
100 – 104
99.5 – 104.5 II 2
105 – 109
104.5 – 109.5
IIII III 8 10
110 – 114
109.5 – 114.5
IIII IIII IIII III 18 28
115 – 119
114.5 – 119.5
IIII IIII III 13 41
120 – 124
119.5 – 124.5
IIII II 7 48
125 – 129
124.5 – 129.5 I 1 49
130 – 134
129.5 – 134.5 50 + + + + + +