1. Frequency Distributions
Sections 2-1 & 2-2
Frequency Distributions

2. Important Characteristics of Data
When analyzing data, there are five characteristics that are important to look at.
1. Center: A representative or average value that indicates where the middle of the data set is located
2. Variation: A measure of the amount that the values vary among themselves
3. Distribution: The nature or shape of the distribution of data (such as bell-shaped, uniform, or skewed)
4. Outliers: Sample values that lie very far away from the vast majority of other sample values
5. Time: Changing characteristics of the data over time
We will come back to each of these in more detail.

3. Analyzing Data
In order to analyze data characteristics, you must first organize it in some way. In this chapter, we will talk about some different ways to organize and present data.

4. Frequency Distributions
breaks data into categories, and tells how many times each category is represented
frequency distributions allow us to organize data and look at trends

5. Qualitative Frequency Distribution
For qualitative data, list each category and how many people/items fall into that category.
Let’s say I survey 10 people, asking their eye color. Here are the results in a frequency distribution:
Eye Color Frequency
Blue
Brown
Green
Hazel

6. Quantitative Frequency Distribution
For quantitative data, you can separate your range of possible values into classes. Count how many people/items fall into each class.
Let’s say we’ve studied the ages of 45 people in a church group:
Age Frequency
classes

7. The following slides explain the different parts of the table in a quantitative frequency distribution.

8. are the smallest numbers in each class
Lower Class Limits
are the smallest numbers in each class
Age Frequency
Lower class limits

9. are the largest numbers in each class
Upper Class Limits
are the largest numbers in each class
Age Frequency
Upper class limits

10. Class Width
is the difference between each lower class limit and the next one
Age Frequency 10
Class width = 10
Note: the class width here is NOT 9

11. number separating classes
Class Boundaries
number separating classes
(think of this as the point at which you would round one way or the other)
Age Frequency
-0.5
9.5
19.5
29.5
39.5
49.5
59.5
69.5
79.5
89.5
99.5
Class boundaries

12. Class Midpoints
are the averages of the lower and upper class limits, or the halfway points
Age Frequency
4.5
14.5
24.5
34.5
44.5
54.5
64.5
74.5
84.5
94.5
Class midpoints

13. Constructing A Frequency
Distribution Table
1. Decide on the number of classes (should be between 5 and 20, often given).
2. Calculate class width (round up to a nice number).
Starting point: Choose the smallest value as the lower limit of the first class (or round down to a nice number), and use the class width to find the rest of the lower limits.
4. Fill in the upper class limits
Count how many data items fall into each class, and fill in the frequencies.
class width 
(highest data value) – (lowest data value)
number of classes

14. Example
Data: 3, 5, 7, 9, 10, 12, 15, 23, 27, 32, 14, 25, 45, 17, 12
Construct a frequency distribution with 4 classes.
= 10.5  11, first class starts at 3
class width =
45 – 3 4
Class Frequency
3 – – – –

15. Relative Frequency Distribution
change to percentages
Age Frequency Divide Age Relative Frequency
/45 = %
/45 = %
/45 = %
/45 = %
/45 = %
/45 = %
/45 = %
/45 = %
/45 = %
/45 = %
Total Frequency = 45

16. Sections 2-3 through 2-5
Statistical Graphics

17. Visualizing Data
We will be looking at different ways to visually represent frequency distributions. Visual representations can help in analyzing data.
Objective: To use visual representations to analyze center, variation, distribution, and outliers.

18. Visualizing Qualitative Data
Bar graphs
Pie Charts

19. Bar Graphs
A bar graph represents qualitative data with a bar for each category. There are many ways to make a bar graph. A Pareto chart is a bar graph with the bars arranged in decreasing order according to frequencies.

20. Pie Chart A graph depicting qualitative data as slices of a pie
Good for comparing what portion of the whole population falls into each category.

21. Visualizing Quantitative Data
Histograms
Frequency Polygons
Stem-and-leaf Plots

22. Histograms
A histogram is the quantitative version of a bar graph. The width of the bars represent the class width and the heights are the frequencies or the relative frequencies. For histograms, the bars touch and the bottom is labeled like a number line with the class boundaries.
Age Frequency

23. Types of Histograms Uniform Bimodal Symmetrical Skewed Left
(all frequencies the same)
Bimodal
(no visible trend)
Symmetrical
Skewed Left
(most data to left of high point)
Skewed Right
(most data to right of high point)

24. Frequency Polygon
Uses line segments connected to points directly above class midpoint values
Note: labels with midpoints this time, not boundaries

25. Stem-and-Leaf Plots
Represents data by separating each value into two parts: the stem (such as the leftmost digit) and the leaf (such as the rightmost digit)
Ages
Stem Leaves
These ages are 1, 1, 2, 6, 7
These ages are 12, 14
No one in their 80’s
Provides visual representation of the frequency distribution while maintaining original data in case you need it.

26. Analyzing Graphs
Any of these graphs can be used to analyze quantitative data. For example, we can use this histogram to analyze the center, variation, distribution, and outliers for the data. (See next slide)

27. Analyzing Graphs
Center: Use the histogram to estimate the average or middle age. Here, it would be about 40.
Variation: Using the class boundaries, we can see that the ages vary from about 0 to 99.
Distribution: Roughly symmetrical, except that the youngest group does not follow the pattern.
Outliers: The bar to the far right seems to contain outliers, or people unusually older than the others.

28. Misleading Graphs
Notice both graphs depict the same information, but in drastically different ways
Notice that the first graph portrays not much of a difference between Men and Women’s earnings. The second graph portrays a significant difference. ALWAYS READ THE NUMERICAL VALUES, DON’T JUST LOOK AT THE PICTURE!