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Chapter 4-Describing Data: Displaying and Exploring Data
Jie Zhang, Ph.D. Student
Account and Information Systems Department
College of Business Administration
The University of Texas at El Paso
[email protected]
Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP
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mailto:[email protected]
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Learning Objectives• LO1 Construct a dot plot.
• LO2 Construct and describe a stem-and-leaf display.
• LO3 Identify and compute measures of position.
• LO4 Construct and analyze a box plot.
• LO5 Compute and describe the coefficient of skewness.
• LO6 Create and interpret a scatter diagram
• LO7 Develop and explain a contingency table
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Dot Plots
• A dot plot groups the data as little as possible and the
identity of an individual observation is not lost.
• To develop a dot plot, each observation is simply displayed as
a dot along a horizontal number line indicating the possible values
of the data.
• If there are identical observations or the observations are
too close to be shown individually, the dots are “piled” on top of
each other.
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Dot Plots - Examples• The Service Departments at Tionesta Ford
Lincoln Mercury and Sheffield Motors,
Inc., two of the four Applewood Auto Group Dealerships, were
both open 24 days last month. Listed below is the number of
vehicles serviced during the 24 working at the two Dealerships.
Construct dot plots and report summary statistics to compare the
two dealerships.
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Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP
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0
5
10
15
20
25
30
35
40
45
50
0 5 10 15 20 25 30
compare Tinesta and Sheffield
Column1 Column2
Mean 31.291667 Mean 35.791667
Standard Error 0.8394252 Standard Error 0.8866546
Median 32 Median 35.5
Mode 32 Mode 36
Standard Deviation 4.1123268 Standard Deviation 4.3437028
Sample Variance 16.911232 Sample Variance 18.867754
Kurtosis -0.634428 Kurtosis -0.410937
Skewness -0.133631 Skewness 0.6333923
Range 16 Range 14
Minimum 23 Minimum 30
Maximum 39 Maximum 44
Sum 751 Sum 859
Count 24 Count 24
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• How to Make a Dotplot
• Descriptive Statistics in Excel
• How to Make Scatterplot with Groups
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https://www.youtube.com/watch?v=N7HHmTpccZIhttps://www.youtube.com/watch?v=Czgl_VZH370https://www.youtube.com/watch?v=Czgl_VZH370https://www.youtube.com/watch?v=PTRTva-assc
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Stem-and-Leaf
• In Chapter 2, frequency distribution was used to organize data
into a meaningful form.
• A major advantage to organizing the data into a frequency
distribution is that we get a quick visual picture of the shape of
the distribution.
• There are two disadvantages, however, to organizing the data
into a frequency distribution:
1)The exact identity of each value is lost
2)Difficult to tell how the values within each class are
distributed.
One technique that is used to display quantitative information
in a condensed form is the stem-and-leaf display.
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• Stem-and-leaf display is a statistical technique to present a
set of data. Each numerical value is divided into two parts. The
leading digit(s) becomes the stem and the trailing digit the leaf.
The stems are located along the vertical axis, and the leaf values
are stacked against each other along the horizontal axis.
• Advantage of the stem-and-leaf display over a frequency
distribution -the identity of each observation is not lost.
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Stem-and-leaf Plot Example• Listed in Table 4–1 is the number of
30-second radio advertising
spots purchased by each of the 45 members of the Greater Buffalo
Automobile Dealers Association last year.
• Organize the data into a stem-and-leaf display. Around what
values do the number of advertising spots tend to cluster? What is
the fewest number of spots purchased by a dealer? The largest
number purchased?
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Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP
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The usual procedure is to sort the leaf values from the smallest
to largest.
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Self-Review 4-1
2. The rate of return for 21 stocks is;
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8.3 9.6 9.5 9.1 8.8 11.2 7.7 10.1 9.9 10.8
10.2 8.0 8.4 8.1 11.6 9.6 8.8 8.0 10.4 9.8 9.2
Organize this information into a stem-and-leaf display.(a)How
many rates are less than 9.0?(b)List this rates in the 10.0 up to
11.0 category(c)What is the median?(d)What are the maximum and the
minimum rates of return?
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Measures of Position
• The standard deviation is the most widely used measure of
dispersion.
• Alternative ways of describing spread of data include
determining the location of values that divide a set of
observations into equal parts.
• These measures include :
• quartiles, (divide into four parts)
• deciles, (10 equal parts)
• and percentiles. (100 equal parts)
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To formalize the computational procedure, let Lp refer to the
location of a desired percentile. So if we wanted to find the 33rd
percentile we would use L33 and if we wanted the median, the 50th
percentile, then L50.
The number of observations is n, so if we want to locate the
median, its position is at (n + 1)/2, or we could write this as (n
+ 1)(P/100), where P is the desired percentile.
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Percentiles - Example
• Listed below are the commissions earned last month by a sample
of 15 brokers at Salomon Smith Barney’s Oakland, California,
office.
$2,038 $1,758 $1,721 $1,637 $2,097 $2,047 $2,205 $1,787 $2,287
$1,940 $2,311 $2,054 $2,406 $1,471 $1,460
• Locate the median, the first quartile, and the third quartile
for the commissions earned.
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Step 1: Organize the data from lowest to largest value$1,460
$1,471 $1,637 $1,721
$1,758 $1,787 $1,940 $2,038
$2,047 $2,054 $2,097 $2,205
$2,287 $2,311 $2,406
Step 2: Compute the first and third quartiles. Locate L25 and
L75 using
205,2$
721,1$
lyrespective positions,
12th and4th at the located are quartiles thirdandfirst
theTherefore,
12100
75)115( 4
100
25)115(
75
25
7525
L
L
LL
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• In the previous example the location formula yielded a whole
number. What if there were 6 observations in the sample with the
following ordered observations: 43, 61, 75, 91, 101, and 104 , that
is n=6, and we wanted to locate the first quartile?
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75.1100
25)16(25 L
Locate the first value in the ordered array and then move .75 of
the distance between the first and second values and report that as
the first quartile. Like the median, the quartile does not need to
be one of the actual values in the data set.The 1st and 2nd values
are 43 and 61. Moving 0.75 of the distance between these numbers,
the 25th percentile is 56.5, obtained as 43 + 0.75*(61- 43)
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Example: percentiles with Excel
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Box Plot• A box plot is a graphical display, based on quartiles,
that helps us
picture a set of data.
• To construct a box plot, we need only five statistics:
1. the minimum value,
2. Q1(the first quartile),
3. the median,
4. Q3 (the third quartile), and
5. the maximum value.
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Boxplot - Example
• Alexander’s Pizza offers free delivery of its pizza within 15
miles. Alex, the owner, wants some information on the time it takes
for delivery. For a sample of 20 deliveries, he determined the
following information:
1. Minimum value = 13 minutes
2. Q1 = 15 minutes
3. Median = 18 minutes
4. Q3 = 22 minutes
5. Maximum value = 30 minutes
• Develop a box plot for the delivery times.
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• Step1: Create an appropriate scale along the horizontal
axis.
• Step 2: Draw a box that starts at Q1 (15 minutes) and ends at
Q3 (22
minutes). Inside the box we place a vertical line to represent
the median (18 minutes).
• Step 3: Extend horizontal lines from the box out to the
minimum value (13 minutes) and the maximum value (30 minutes).
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Box plot in R• The data was extracted from the 1974 Motor Trend
US magazine, and
comprises fuel consumption and 10 aspects of automobile design
and performance for 32 automobiles (1973–74 models).
Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP
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Skewness
• In Chapter 3, measures of central location (the mean, median,
and mode) for a set of observations and measures of data dispersion
(e.g. range and the standard deviation) were introduced
• Another characteristic of a set of data is the shape.
• There are four shapes commonly observed:
1. symmetric,
2. positively skewed,
3. negatively skewed,
4. bimodal.
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Skewness - Formulas for Computing
• The coefficient of skewness can range from -3 up to 3.
A value near -3, indicates considerable negative skewness.
A value such as 1.63 indicates moderate positive skewness.
A value of 0, which will occur when the mean and median are
equal, indicates the distribution is symmetrical and that there is
no skewness present.
• Professor Karl Pearson(1857-1936)
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Commonly Observed Shapes
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Skewness – An Example
• Following are the earnings per share for a sample of 15
software companies for the year 2010. The earnings per share are
arranged from smallest to largest.
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• Compute the mean, median, and standard deviation. Find the
coefficient of skewness using Pearson’s estimate.
• What is your conclusion regarding the shape of the
distribution?
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Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP
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0171225
18395433
225115
9544016954090
1
95415
2674
222
..$
).$.($)(
Skewness the Compute :4 Step
3.18 is largest to smallest from arranged data, of set the in
value middle The
Median the Find :3 Step
.$)).$.($...).$.($
Deviation Standard the Compute :2 Step
.$.$
Mean the Compute :1 Step
s
MedianXsk
n
XXs
n
XX
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Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP
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What’s does the value of skewness mean? Can you get any idea
from the graph below?
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Self-Review 4-4 P123
• A sample of five data entry clerks employed in the Horry
County Tax Office revised the following number of tax records last
hour:
73, 98, 60, 92, and 84
(a) Find the mean, median, and the standard deviation
(b) Compute the coefficient of skewness using Person’s
method
(c) calculate the coefficient of skewness using the software
method,
then compare its value with Person’s method
(d) what is your conclusion regarding the skewness of the
data?
Jie Zhang, QMB 2301 Fundamentals of Business Statistics, UTEP
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Describing the relationship between two variables-Scatter
Diagram
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Questions:Can you get some intuitive idea from the left tree
graphs? Or more precisely, what the relationship between the two
variable?