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Econ 5
Introduction to Statistics
Asatar Bair, Ph.D. Department of Economics City College of San Francisco abair@ccsf.edu
Lectures on Chapter 2
DESCRIPTIVE STATISTICS:
Summarizing Qualitative Data
! Frequency Distribution
! Relative Frequency
! Percent Frequency Distribution
! Bar Graph
! Pie Chart
Frequency Distribution
! A frequency distribution is a tabular summary of
a set of data showing the frequency (or number) of
items in each of several non-overlapping classes.
! The objective is to provide insights about the data
that cannot be quickly obtained by looking only at
the original data.
Example: Marada Inn
Guests staying at Marada Inn were asked to rate the quality of their accommodations. The ratings provided by a sample of 20 quests are shown below. Below Average Average Above Average Above Average Above Average Above Average Above Average Below Average Below Average Average Poor Poor Above Average Excellent Above Average Average Above Average Average Above Average Average
Example: Marada Inn
Frequency Distribution Quality rating Frequency Poor 2 Below average 3 Average 5 Above average 9 Excellent 1 total 20
Relative Frequency and
Percent Frequency Distributions
! The relative frequency of a class is the fraction or
proportion of the total number of data items
belonging to the class.
! A relative frequency distribution is a tabular
summary of a set of data showing the relative
frequency for each class.
Relative Frequency and
Percent Frequency Distributions
! The percent frequency of a class is the relative
frequency multiplied by 100.
! A percent frequency distribution is a tabular
summary of a set of data showing the percent
frequency for each class.
Example: Marada Inn
Frequency Distribution Quality rating Relative Frequency Percent Frequency Poor 0. 10 10 Below average 0. 15 15 Average 0. 25 25 Above average 0. 45 45 Excellent 0. 05 5 total 1. 00 100
Wide short bar graphs emphasize similarity 0 3 6 9 Poor Below average Average Above average Excellent Quality ratings
Pie Chart
The pie chart is a commonly used graphical device for presenting relative or percentage frequency distributions for qualitative data. First draw a circle; then use the relative or percentage frequencies to subdivide the circle into sectors that correspond to the relative frequency for each class. Since there are 360 degrees in a circle, a class with a relative frequency of 0.25 would consume 0.25( 360 ) = 90 degrees of the circle.
Example: Marada Inn
Excellent 5% Above average 45% Average 25% Below average 15% Poor 10% Use of color in presenting pie charts Excellent 5% Above average 45% Average 25% Below average 15% Poor 10%
To highlight the positive features of this data
Use of color in presenting pie charts Above average 45% Excellent 5% Average 25% Below average 15% Poor 10%
To highlight the negative features of this data
Use of flashy 3D pie charts Poor Below average Average Above average Excellent
To highlight a certain slice of the pie
Exploded wedges also draw attention Below average Poor Average Excellent Above average
Summarizing Quantitative Data
- (^) Frequency Distribution
- (^) Relative Frequency and Percent
Frequency Distributions
- (^) Dot Plot
- (^) Histogram
- (^) Cumulative Distribution
Relative and Percent Frequency Distribution
Cost ($)
Relative
Frequency
Percent
Frequency
Total 1. 00 100
Dot Plot
- One of the simplest graphical summaries of
data is a dot plot.
- A horizontal axis shows the range of data
values.
- Then each data value is represented by a dot
placed above the axis.
Dot Plot
50 60 70 80 90 100 110
Cost ($) Most of the data is in this range.
Histogram
- Another common graphical presentation of quantitative data is a histogram.
- The variable of interest is placed on the horizontal axis and the frequency, relative frequency, or percent frequency is placed on the vertical axis.
- A rectangle is drawn above each class interval with its height corresponding to the interval’s frequency, relative frequency, or percent frequency.
- Unlike a bar graph, a histogram has no natural separation bet ween rectangles of adjacent classes.
Histogram
0 5 10 15 20 50- 59 60-69 70- 79 80- 89 90- 99 100- 109 Cost ($)
Relative Frequency Histogram
0
- 07
- 14
- 21
- 28
50- 59 60-69 70- 79 80- 89 90- 99 100- 109
- 10
- 14 0. 14
- 26
- 04 Cost ($)
Cumulative Distribution
- The cumulative frequency distribution shows the number of items with values less than or equal to the upper limit of each class.
- The cumulative relative frequency distribution shows the proportion of items with values less than or equal to the upper limit of each class.
- The^ cumulative percent frequency distribution^ shows the percentage of items with values less than or equal to the upper limit of each class.
Cumulative Frequency
Cost ($)
Cumulative
Frequency
Cumulative
Relative Frequency
Hudson Auto Repair
Stem and Leaf Display for Cost of Parts
Crosstabulations and Scatter Diagrams
Thus far we have focused on methods that are used to summarize the data for one variable at a time. Next we explore methods of understanding the relationship bet ween t wo variables.
Crosstabulation: The number of Finger Lakes
homes sold for each style and price for the past two years is shown below. Price Home Style Colonial Ranch Split A-Frame Total less than $100,000 18 6 19 12 55 $100,000+ (^12 14 16 3 ) Total 30 20 35 15 100 Problem with crosstabulation
Crosstabulation data are often combined to
form an aggregate crosstabulation;
this presents a possible danger;
relationships that appear in the aggregate
may be contradicted by the unaggregated
data;
this is called Simpson’s Paradox.
Crosstabulation: Simpson’s Paradox
Verdict Judge Total Luckett Kendall Upheld 129 (86%) 110 (88%) 239 Reversed 21 (14%) 15 (12%) 36 Total 150 125 275
It looks like Kendall’s doing a better job
Crosstabulation: Simpson’s Paradox
Verdict Judge Luckett Total Common Pleas Municipal Court Upheld 29 (91%) 100 (85%) 129 Reversed 3 (9%) 18 (15%) 21 Total 32 118 150 But Luckett actually has a better record in both courts. Verdict Judge Kendall Total Common Pleas Municipal Court Upheld 90 (90%) 20 (80%) 110 Reversed 10 (10%) 5 (20%) 15 Total 100 25 125 Example: Panthers Football Team Scatter Diagram The Panthers football team is interested in investigating the relationship, if any, between interceptions made and points scored. Interceptions Points scored 1 14 3 24 2 18 1 17 3 27
Panthers Football Team
0 5 10 15 20 25 30 1 2 3
Scatter Diagram
Bin range
what you do here is to define the class widths
you want to use;
Excel can do this automatically, but it has
very bad judgement and the results will be
worthless;
look at the data and decide what the upper
bounds for each class should be
Bin range For this example, I’m using the “Norris” data on the CD; if you want your classes to look like this, you enter just the upper boundary in each cell: 49, 59, 69, 79, 89, 99, 109, 119 then go to the bin range field and highlight these cells
Frequency distribution hit “OK”, and Excel gives you this I like to rename the “Bin” fields “40- 49 ”, “50- 59 ”, etc. I also rename the “More” field “Total” and add up the column above so the whole thing looks like this Histogram Now go to “Insert” and select “Chart” or hit the button Select the “Clustered column” enter the title for the x and y axes, and for the whole chart, then hit OK; to get rid of the gap bet ween the bars, double-click on one of the bars on the finished chart, then go to “Options” and enter zero under “Gap width”.
be sure to label the axes and give it a title;
another masterpiece of statistics!
Frequency Histogram: Norris Electronics 0 10 20 30 40 50 60 70 40 to 49 50 to 59 60 to 69 70 to 79 80 to 89 90 to 99 100 to 109 110 to 116 Hours until Burnout Frequency
