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Chapter 3 Displaying and SummarizingQuantitative Data
Dot Plots
DefinitionA dot plot is the representation of a set of data over a number line.The number of dots over a number represents the relative quantity ofthe value.
ExampleThe following ate the test scores for a particular high school studentin their math class over the course of an academic year.
64 73 85 74 8371 56 83 76 8583 87 92 84 9592 95 92 91
Dot Plots
DefinitionA dot plot is the representation of a set of data over a number line.The number of dots over a number represents the relative quantity ofthe value.
ExampleThe following ate the test scores for a particular high school studentin their math class over the course of an academic year.
64 73 85 74 8371 56 83 76 8583 87 92 84 9592 95 92 91
Solution
Grades of a High School Student
Grades
Freq
uenc
y
1
2
3
50 60 70 80 90 100
• • • •• ••
•
•
•
•
• •• •
•
••
•
Pros and Cons
What’s GoodGives a good idea of distribution
Preserves all of the data points
What’s Not GoodTedious to plot
Can be hard to read
Not practical for large data sets
Pros and Cons
What’s GoodGives a good idea of distribution
Preserves all of the data points
What’s Not GoodTedious to plot
Can be hard to read
Not practical for large data sets
Pros and Cons
What’s GoodGives a good idea of distribution
Preserves all of the data points
What’s Not GoodTedious to plot
Can be hard to read
Not practical for large data sets
Pros and Cons
What’s GoodGives a good idea of distribution
Preserves all of the data points
What’s Not GoodTedious to plot
Can be hard to read
Not practical for large data sets
Pros and Cons
What’s GoodGives a good idea of distribution
Preserves all of the data points
What’s Not GoodTedious to plot
Can be hard to read
Not practical for large data sets
Distributions
DefinitionA distribution is a representation of data vs. frequency. It shows allpossible values and how often they occur.
Now we want to concern ourselves with the analysis of the graphs.We can analyze these in a much more constructive way that we couldwith the graphs of categorical variables. Here we are analyzing thedistribution represented by the graph.
Distributions
DefinitionA distribution is a representation of data vs. frequency. It shows allpossible values and how often they occur.
Now we want to concern ourselves with the analysis of the graphs.We can analyze these in a much more constructive way that we couldwith the graphs of categorical variables. Here we are analyzing thedistribution represented by the graph.
Distribution Analysis
1 Center: Which class contains the central element(s)
2 Shape: Number of peaks, skewness3 Spread: Range=max-min
In our example, we can see a couple of things:
Range: Highest value - lowest valueHere, the range would be 95− 56 = 39.Center: The central value(s) is the center. It could be a value or aclass, depending on the type of graph.Here, the center is the 10th value, since there are 19 data points inthe set. The value we seek is 84.Shape: How many peaks are there? Is it roughly in the middle orto one side?Here we have one peak, so we would say the distribution isunimodal. That peak is to the right, so the tail stretches out to theleft. We would say this graph is left skewed.
Distribution Analysis
1 Center: Which class contains the central element(s)2 Shape: Number of peaks, skewness
3 Spread: Range=max-min
In our example, we can see a couple of things:
Range: Highest value - lowest valueHere, the range would be 95− 56 = 39.Center: The central value(s) is the center. It could be a value or aclass, depending on the type of graph.Here, the center is the 10th value, since there are 19 data points inthe set. The value we seek is 84.Shape: How many peaks are there? Is it roughly in the middle orto one side?Here we have one peak, so we would say the distribution isunimodal. That peak is to the right, so the tail stretches out to theleft. We would say this graph is left skewed.
Distribution Analysis
1 Center: Which class contains the central element(s)2 Shape: Number of peaks, skewness3 Spread: Range=max-min
In our example, we can see a couple of things:
Range: Highest value - lowest valueHere, the range would be 95− 56 = 39.Center: The central value(s) is the center. It could be a value or aclass, depending on the type of graph.Here, the center is the 10th value, since there are 19 data points inthe set. The value we seek is 84.Shape: How many peaks are there? Is it roughly in the middle orto one side?Here we have one peak, so we would say the distribution isunimodal. That peak is to the right, so the tail stretches out to theleft. We would say this graph is left skewed.
Distribution Analysis
1 Center: Which class contains the central element(s)2 Shape: Number of peaks, skewness3 Spread: Range=max-min
In our example, we can see a couple of things:
Range: Highest value - lowest valueHere, the range would be 95− 56 = 39.
Center: The central value(s) is the center. It could be a value or aclass, depending on the type of graph.Here, the center is the 10th value, since there are 19 data points inthe set. The value we seek is 84.Shape: How many peaks are there? Is it roughly in the middle orto one side?Here we have one peak, so we would say the distribution isunimodal. That peak is to the right, so the tail stretches out to theleft. We would say this graph is left skewed.
Distribution Analysis
1 Center: Which class contains the central element(s)2 Shape: Number of peaks, skewness3 Spread: Range=max-min
In our example, we can see a couple of things:
Range: Highest value - lowest valueHere, the range would be 95− 56 = 39.Center: The central value(s) is the center. It could be a value or aclass, depending on the type of graph.Here, the center is the 10th value, since there are 19 data points inthe set. The value we seek is 84.
Shape: How many peaks are there? Is it roughly in the middle orto one side?Here we have one peak, so we would say the distribution isunimodal. That peak is to the right, so the tail stretches out to theleft. We would say this graph is left skewed.
Distribution Analysis
1 Center: Which class contains the central element(s)2 Shape: Number of peaks, skewness3 Spread: Range=max-min
In our example, we can see a couple of things:
Range: Highest value - lowest valueHere, the range would be 95− 56 = 39.Center: The central value(s) is the center. It could be a value or aclass, depending on the type of graph.Here, the center is the 10th value, since there are 19 data points inthe set. The value we seek is 84.Shape: How many peaks are there? Is it roughly in the middle orto one side?Here we have one peak, so we would say the distribution isunimodal. That peak is to the right, so the tail stretches out to theleft. We would say this graph is left skewed.
Stem-and-Leaf Plots
Similarities to Dot Plots
Gives idea of distribution
Preserves data
Not practical for large data sets
Differences from Dot Plots
Used for quantitative variables
Easier to read actual data elements
Can be used for comparisons of two data sets
Stem-and-Leaf Plots
Similarities to Dot Plots
Gives idea of distribution
Preserves data
Not practical for large data sets
Differences from Dot Plots
Used for quantitative variables
Easier to read actual data elements
Can be used for comparisons of two data sets
Stem-and-Leaf Plots
Similarities to Dot Plots
Gives idea of distribution
Preserves data
Not practical for large data sets
Differences from Dot Plots
Used for quantitative variables
Easier to read actual data elements
Can be used for comparisons of two data sets
Stem-and-Leaf Plots
Similarities to Dot Plots
Gives idea of distribution
Preserves data
Not practical for large data sets
Differences from Dot Plots
Used for quantitative variables
Easier to read actual data elements
Can be used for comparisons of two data sets
Stem-and-Leaf Plots
Similarities to Dot Plots
Gives idea of distribution
Preserves data
Not practical for large data sets
Differences from Dot Plots
Used for quantitative variables
Easier to read actual data elements
Can be used for comparisons of two data sets
Stem-and-Leaf Plot Example
ExampleUsing the same data set as we did for the dot plot, construct astem-and-leaf plot.
First thing we need to do is order the data elements.
56 64 71 73 7476 83 83 83 8485 85 87 91 9292 92 95 95
Stem-and-Leaf Plot Example
Grades for a High School Student98765
These would be the stems for our plot.
Note: Repetition is extremely important.
Stem-and-Leaf Plot Example
Grades for a High School Student98765
These would be the stems for our plot.
Note: Repetition is extremely important.
Stem-and-Leaf Plot Example
Grades for a High School Student9 1 2 2 2 5 58 3 3 3 4 5 5 77 1 3 4 66 45 6
Here we get the exact same answer for the range and the center,although we only give the class in which the center lies, so we wouldsay that the center is in the 80’s. We get that the shape is againunimodal and skewed left. It may look different, but since itrepresents the same distribution, we expect similar answers.
Notice that the values on the right are essentially in columns - this iswhat allows us to quickly see which classes have more elements.
Stem-and-Leaf Plot Example
Grades for a High School Student9 1 2 2 2 5 58 3 3 3 4 5 5 77 1 3 4 66 45 6
Here we get the exact same answer for the range and the center,although we only give the class in which the center lies, so we wouldsay that the center is in the 80’s. We get that the shape is againunimodal and skewed left. It may look different, but since itrepresents the same distribution, we expect similar answers.
Notice that the values on the right are essentially in columns - this iswhat allows us to quickly see which classes have more elements.
More Stem-and-Leaf Plots
What if we had a 3 digit number? Suppose the student got a 100 onthe next exam?
Grades for a High School Student10 09 1 2 2 2 5 58 3 3 3 4 5 5 77 1 3 4 66 45 6
Stem-and-Leaf Plots for Comparisons
ExampleSuppose we wanted to compare the careers of Babe Ruth and MarkMcGwire in terms of their yearly home run totals to determine whichplayer was the more consistent long ball hitter. Make a back-to-backstem-and-leaf plot to make the is determination.
Ruth: 54, 59, 35, 41, 46, 25, 47, 60, 54, 46, 49, 46, 41, 34, 22McGwire: 49, 32, 33, 39, 22, 42, 9, 9, 39, 52, 58, 70, 65, 32, 29
Stem-and-Leaf Plots for Comparisons
Ruth v. McGwire76543210
We set up the graph with one set of data increasing out to the rightand the other increasing out to the left. This way we have aside-by-side comparison of the data sets.
Stem-and-Leaf Plots for Comparisons
Ruth v. McGwire7 0
0 6 59 4 4 5 2 8
9 7 6 6 6 1 1 4 2 95 4 3 2 2 3 9 95 2 2 2 9
10 9 9
Who is more consistent and why?
Histograms
Used for quantitative variables
Tracks frequency and shows distribution
Does not preserve individual values
Good for a large number of values
Bars must be vertical and must touch
Histograms
Used for quantitative variables
Tracks frequency and shows distribution
Does not preserve individual values
Good for a large number of values
Bars must be vertical and must touch
Histograms
Used for quantitative variables
Tracks frequency and shows distribution
Does not preserve individual values
Good for a large number of values
Bars must be vertical and must touch
Histograms
Used for quantitative variables
Tracks frequency and shows distribution
Does not preserve individual values
Good for a large number of values
Bars must be vertical and must touch
Histograms
Used for quantitative variables
Tracks frequency and shows distribution
Does not preserve individual values
Good for a large number of values
Bars must be vertical and must touch
Histograms
ExampleFor our test scores example, construct a histogram and analyze thedistribution.
It is easier if the values are in order as we will be grouping them intoclasses.
56 64 71 73 7476 83 83 83 8485 85 87 91 9292 92 95 95
Histograms
We first want to create a frequency table. This is a collection ofnon-overlapping classes and the frequency of observation in each ofthose classes. We need to determine the following in this order:
Number of classesThe rule of thumb with the number of classes is to use the square rootof the number of observations in the data set.
√19 ≈ 4.36
So, we can use 4 or 5 classes. I tend to go up to the next integer to besure I have enough classes. So we will use 5 for our graph.
Histograms
We first want to create a frequency table. This is a collection ofnon-overlapping classes and the frequency of observation in each ofthose classes. We need to determine the following in this order:
Number of classesThe rule of thumb with the number of classes is to use the square rootof the number of observations in the data set.
√19 ≈ 4.36
So, we can use 4 or 5 classes. I tend to go up to the next integer to besure I have enough classes. So we will use 5 for our graph.
Histograms
Size of each classWe want them to be the same width so that the taller classes will beknown to have the most elements. If not then we have to find the areaof each rectangle to determine relative size.To find the size, we divide the ‘range’ by the number of classes.
size =95− 56 + 1
5=
385
= 7.6
We could use 7.6 for the class width or we can go to the next largestinteger. Where we may have extra if we round up, it is better than nothaving enough of a range in the classes to cover all of the data. Forthe sake of simplicity, we will use 8.
Histograms
Endpoints of each classWe start the smallest class with a left endpoint of 56, since that wasour minimum. Then, to find the next left endpoint, add 8 to 56.Continue in this manner until we have 5 classes.
Grade Range Frequency56-64-72-80-88-
Histograms
Then, we subtract 1 from each left endpoint to find the right endpointof the previous class.
Grade Range Frequency56-6364-7172-7980-8788-95
Finally, we count how many elements go in each class.
Grade Range Frequency56-63 164-71 272-79 380-87 788-95 6
Histograms
Then, we subtract 1 from each left endpoint to find the right endpointof the previous class.
Grade Range Frequency56-6364-7172-7980-8788-95
Finally, we count how many elements go in each class.
Grade Range Frequency56-63 164-71 272-79 380-87 788-95 6
Histograms
Grades of a High School Student
Grades
Freq
uenc
y
2
4
6
8
56 64 72 80 88 96
We see the same range and shape. Here, we’d have no choice but togive the class only for the center as we would lose the ability to seeindividual values.
Using The Calculator
We can make some graphs on the TI-series graphing calculator. Oneof the options we have is to make a histogram.
The advantages to using technology are that we don’t have to makefrequency tables or figure out how many classes we need, etc.
We do have to keep in mind, however, that the number of classes maybe different than when we make the graph by hand. We are usingmore approximations when we work by hand than when we usetechnology. But, this is an acceptable difference as long as the methodwe use is valid.
Using The Calculator
We can make some graphs on the TI-series graphing calculator. Oneof the options we have is to make a histogram.
The advantages to using technology are that we don’t have to makefrequency tables or figure out how many classes we need, etc.
We do have to keep in mind, however, that the number of classes maybe different than when we make the graph by hand. We are usingmore approximations when we work by hand than when we usetechnology. But, this is an acceptable difference as long as the methodwe use is valid.
Using The Calculator
We can make some graphs on the TI-series graphing calculator. Oneof the options we have is to make a histogram.
The advantages to using technology are that we don’t have to makefrequency tables or figure out how many classes we need, etc.
We do have to keep in mind, however, that the number of classes maybe different than when we make the graph by hand. We are usingmore approximations when we work by hand than when we usetechnology. But, this is an acceptable difference as long as the methodwe use is valid.
How To Make Histograms On The TI
1 In the STAT menu, select EDIT
2 Input all of the data in the same column
3 Press 2nd and then MODE to quit to a blank screen
4 Press 2nd and Y= to get into the STATPLOT menu5 Make sure all of the plots are off (if need be, use option 4)6 Press ENTER on one of the plots7 Turn the plot ON with the ENTER key8 Select the histogram, which is the third graph in the top row9 Make sure the XList is correct and then press ZOOM and then
9 , which is the option for ZOOMSTAT
If you want to get data from the graph, like endpoints for classes orfor the frequency for a class, you can press TRACE and then use thearrows to scroll around.
How To Make Histograms On The TI
1 In the STAT menu, select EDIT2 Input all of the data in the same column
3 Press 2nd and then MODE to quit to a blank screen
4 Press 2nd and Y= to get into the STATPLOT menu5 Make sure all of the plots are off (if need be, use option 4)6 Press ENTER on one of the plots7 Turn the plot ON with the ENTER key8 Select the histogram, which is the third graph in the top row9 Make sure the XList is correct and then press ZOOM and then
9 , which is the option for ZOOMSTAT
If you want to get data from the graph, like endpoints for classes orfor the frequency for a class, you can press TRACE and then use thearrows to scroll around.
How To Make Histograms On The TI
1 In the STAT menu, select EDIT2 Input all of the data in the same column
3 Press 2nd and then MODE to quit to a blank screen
4 Press 2nd and Y= to get into the STATPLOT menu5 Make sure all of the plots are off (if need be, use option 4)6 Press ENTER on one of the plots7 Turn the plot ON with the ENTER key8 Select the histogram, which is the third graph in the top row9 Make sure the XList is correct and then press ZOOM and then
9 , which is the option for ZOOMSTAT
If you want to get data from the graph, like endpoints for classes orfor the frequency for a class, you can press TRACE and then use thearrows to scroll around.
How To Make Histograms On The TI
1 In the STAT menu, select EDIT2 Input all of the data in the same column
3 Press 2nd and then MODE to quit to a blank screen
4 Press 2nd and Y= to get into the STATPLOT menu
5 Make sure all of the plots are off (if need be, use option 4)6 Press ENTER on one of the plots7 Turn the plot ON with the ENTER key8 Select the histogram, which is the third graph in the top row9 Make sure the XList is correct and then press ZOOM and then
9 , which is the option for ZOOMSTAT
If you want to get data from the graph, like endpoints for classes orfor the frequency for a class, you can press TRACE and then use thearrows to scroll around.
How To Make Histograms On The TI
1 In the STAT menu, select EDIT2 Input all of the data in the same column
3 Press 2nd and then MODE to quit to a blank screen
4 Press 2nd and Y= to get into the STATPLOT menu5 Make sure all of the plots are off (if need be, use option 4)
6 Press ENTER on one of the plots7 Turn the plot ON with the ENTER key8 Select the histogram, which is the third graph in the top row9 Make sure the XList is correct and then press ZOOM and then
9 , which is the option for ZOOMSTAT
If you want to get data from the graph, like endpoints for classes orfor the frequency for a class, you can press TRACE and then use thearrows to scroll around.
How To Make Histograms On The TI
1 In the STAT menu, select EDIT2 Input all of the data in the same column
3 Press 2nd and then MODE to quit to a blank screen
4 Press 2nd and Y= to get into the STATPLOT menu5 Make sure all of the plots are off (if need be, use option 4)6 Press ENTER on one of the plots
7 Turn the plot ON with the ENTER key8 Select the histogram, which is the third graph in the top row9 Make sure the XList is correct and then press ZOOM and then
9 , which is the option for ZOOMSTAT
If you want to get data from the graph, like endpoints for classes orfor the frequency for a class, you can press TRACE and then use thearrows to scroll around.
How To Make Histograms On The TI
1 In the STAT menu, select EDIT2 Input all of the data in the same column
3 Press 2nd and then MODE to quit to a blank screen
4 Press 2nd and Y= to get into the STATPLOT menu5 Make sure all of the plots are off (if need be, use option 4)6 Press ENTER on one of the plots7 Turn the plot ON with the ENTER key
8 Select the histogram, which is the third graph in the top row9 Make sure the XList is correct and then press ZOOM and then
9 , which is the option for ZOOMSTAT
If you want to get data from the graph, like endpoints for classes orfor the frequency for a class, you can press TRACE and then use thearrows to scroll around.
How To Make Histograms On The TI
1 In the STAT menu, select EDIT2 Input all of the data in the same column
3 Press 2nd and then MODE to quit to a blank screen
4 Press 2nd and Y= to get into the STATPLOT menu5 Make sure all of the plots are off (if need be, use option 4)6 Press ENTER on one of the plots7 Turn the plot ON with the ENTER key8 Select the histogram, which is the third graph in the top row
9 Make sure the XList is correct and then press ZOOM and then9 , which is the option for ZOOMSTAT
If you want to get data from the graph, like endpoints for classes orfor the frequency for a class, you can press TRACE and then use thearrows to scroll around.
How To Make Histograms On The TI
1 In the STAT menu, select EDIT2 Input all of the data in the same column
3 Press 2nd and then MODE to quit to a blank screen
4 Press 2nd and Y= to get into the STATPLOT menu5 Make sure all of the plots are off (if need be, use option 4)6 Press ENTER on one of the plots7 Turn the plot ON with the ENTER key8 Select the histogram, which is the third graph in the top row9 Make sure the XList is correct and then press ZOOM and then
9 , which is the option for ZOOMSTAT
If you want to get data from the graph, like endpoints for classes orfor the frequency for a class, you can press TRACE and then use thearrows to scroll around.
How To Make Histograms On The TI
1 In the STAT menu, select EDIT2 Input all of the data in the same column
3 Press 2nd and then MODE to quit to a blank screen
4 Press 2nd and Y= to get into the STATPLOT menu5 Make sure all of the plots are off (if need be, use option 4)6 Press ENTER on one of the plots7 Turn the plot ON with the ENTER key8 Select the histogram, which is the third graph in the top row9 Make sure the XList is correct and then press ZOOM and then
9 , which is the option for ZOOMSTAT
If you want to get data from the graph, like endpoints for classes orfor the frequency for a class, you can press TRACE and then use thearrows to scroll around.
Histograms
ExampleThe EPA lists most sports cars in its “two-seater” category. The tablebelow gives the city mileage in miles per gallon. Make and analyze ahistogram for the the city mileage.
Model Mileage Model MileageAcura NSX 17 Insight 57Audi Quattro 20 S2000 20Audi Roadster 22 Lamborghini 9BMW M Coupe 17 Mazda 22BMW Z3 Coupe 19 SL500 16BMW Z3 Roadster 20 SL600 13BMW Z8 13 SLK230 23Corvette 18 SLK 320 20Prowler 18 911 15Ferrari 360 11 Boxster 19Thunderbird 17 MR2 25
Histograms
There are 22 cars, so we would use 4 <√
22 < 5 classes, so here Iwill choose 5. The size of each class would be
57− 9 + 15
=495
= 9.8
So we will use 10.
Mileage Frequency9 -18 1119-28 1029-38 039-48 049-58 1
Histograms
There are 22 cars, so we would use 4 <√
22 < 5 classes, so here Iwill choose 5. The size of each class would be
57− 9 + 15
=495
= 9.8
So we will use 10.
Mileage Frequency9 -18 1119-28 1029-38 039-48 049-58 1
Histograms
MPG for Sports Cars
MPG
Freq
uenc
y
3
6
9
12
9 19 29 39 49 59
Center: Boundary between the first two classesRange: 58− 9 = 49Shape: Unimodal, skewed right
Histograms
MPG for Sports Cars
MPG
Freq
uenc
y
3
6
9
12
9 19 29 39 49 59
Center:
Boundary between the first two classesRange: 58− 9 = 49Shape: Unimodal, skewed right
Histograms
MPG for Sports Cars
MPG
Freq
uenc
y
3
6
9
12
9 19 29 39 49 59
Center: Boundary between the first two classesRange:
58− 9 = 49Shape: Unimodal, skewed right
Histograms
MPG for Sports Cars
MPG
Freq
uenc
y
3
6
9
12
9 19 29 39 49 59
Center: Boundary between the first two classesRange: 58− 9 = 49Shape:
Unimodal, skewed right
Histograms
MPG for Sports Cars
MPG
Freq
uenc
y
3
6
9
12
9 19 29 39 49 59
Center: Boundary between the first two classesRange: 58− 9 = 49Shape: Unimodal, skewed right
Central Tendency
We will use three methods of measuring central tendency:1 mean2 median3 mode
Example
ExampleFind the mean, median and mode for the following data set.
11 010987 46 05 0 0 0 543 0 0 0 0 1 2 52 510 4
Solution
Mean xThis the arithmetic center.
x =1n
n∑k=1
xk
This is just a fancy way of saying to add the 16 values togetherand divide by 16. When we do we get
x = 43.5
Solution
Mean xThis the arithmetic center.
x =1n
n∑k=1
xk
This is just a fancy way of saying to add the 16 values togetherand divide by 16. When we do we get
x = 43.5
Solution
Median MThis is the geometric center. To find, we line up all of the valuesin order and find the middle one. If there is an odd number ofobservations, then the median is the one in the middle. If there isan even number of observations, the median is the mean of thetwo ‘middle’ values. Here, we have
M =32 + 35
2= 33.5
ModeThis is the value(s) that occur most often, unless all values occurthe same number of times, in which case there is no mode. Here,
mode = 30
Solution
Median MThis is the geometric center. To find, we line up all of the valuesin order and find the middle one. If there is an odd number ofobservations, then the median is the one in the middle. If there isan even number of observations, the median is the mean of thetwo ‘middle’ values. Here, we have
M =32 + 35
2= 33.5
ModeThis is the value(s) that occur most often, unless all values occurthe same number of times, in which case there is no mode. Here,
mode = 30
Solution
Median MThis is the geometric center. To find, we line up all of the valuesin order and find the middle one. If there is an odd number ofobservations, then the median is the one in the middle. If there isan even number of observations, the median is the mean of thetwo ‘middle’ values. Here, we have
M =32 + 35
2= 33.5
ModeThis is the value(s) that occur most often, unless all values occurthe same number of times, in which case there is no mode. Here,
mode = 30
The Relationship Between Mean and Median
This picture indicates a serious drawback to using means: outliers.The median is what we call resistant; an extreme value does not affectthe median. The mean, however, is not resistant.
The Relationship Between Mean and Median
This picture indicates a serious drawback to using means: outliers.The median is what we call resistant; an extreme value does not affectthe median. The mean, however, is not resistant.
When We Use Mean v. Median
1 If distribution is symmetric, then mean = median, and we use themean
2 If there are outliers or strong skewness, we use the median
When We Use Mean v. Median
1 If distribution is symmetric, then mean = median, and we use themean
2 If there are outliers or strong skewness, we use the median
Using the Mean
ExampleSuppose you got an 84, 72 and 78 on your first 3 exams and wanted toknow what grade you needed to get on the fourth exam to have at leastan 80 average?
We want an average of 80 for the 4 grades. So, we need to solve for xin
84 + 72 + 78 + x4
=234 + x
4= 80
So, we get
234 + x4
= 80⇒ 234 + x = 320⇒ x = 86
Using the Mean
ExampleSuppose you got an 84, 72 and 78 on your first 3 exams and wanted toknow what grade you needed to get on the fourth exam to have at leastan 80 average?
We want an average of 80 for the 4 grades. So, we need to solve for xin
84 + 72 + 78 + x4
=234 + x
4= 80
So, we get
234 + x4
= 80⇒ 234 + x = 320⇒ x = 86
Another Mean Example
Example
Suppose you had a 75 average through 4 tests and got an 85 on the 5th
test. What is your average now?
If we have a 75 average through 4 exams, then we have accumulated75× 4 = 300 points. So, if we wanted to know the average with this5th grade, we’d have
x =300 + 85
5=
3855
= 77
Another Mean Example
Example
Suppose you had a 75 average through 4 tests and got an 85 on the 5th
test. What is your average now?
If we have a 75 average through 4 exams, then we have accumulated75× 4 = 300 points. So, if we wanted to know the average with this5th grade, we’d have
x =300 + 85
5=
3855
= 77
Yet Another Mean Example
ExampleSuppose you had a group of 11 people and the average age was 27. Ifone of those people left, the average age of the remaining 10 was 29.What is the age of the person who left?
Total age of the 11 people: 11× 27 = 297.
Total age of the 10 people : 10× 29 = 290
Difference is 297− 290 = 7
Yet Another Mean Example
ExampleSuppose you had a group of 11 people and the average age was 27. Ifone of those people left, the average age of the remaining 10 was 29.What is the age of the person who left?
Total age of the 11 people: 11× 27 = 297.
Total age of the 10 people : 10× 29 = 290
Difference is 297− 290 = 7
Means From Frequency Tables
ExampleFind the mean of the following values.
Age Frequency21 522 823 424 125 2
We first count the total number of observations, which is 20. Then ...
x =21 ∗ 5 + 22 ∗ 8 + 23 ∗ 4 + 24 ∗ 1 + 25 ∗ 2
20=
44720
= 22.35
Means From Frequency Tables
ExampleFind the mean of the following values.
Age Frequency21 522 823 424 125 2
We first count the total number of observations, which is 20. Then ...
x =21 ∗ 5 + 22 ∗ 8 + 23 ∗ 4 + 24 ∗ 1 + 25 ∗ 2
20=
44720
= 22.35
Box Plots and the 5-Number Summary
When dealing with the median, we measure variation with the5-number summary. These 5 numbers indicate the maximum andminimum, the median and the quartiles.
In order to find the five number summary, we first line the dataelements in order. Then we find the minimum and maximum, andthen the median.
minimum smallest value of the setmaximum largest value of the set
median central(s) value of the setfirst quartile Q1 median of all values smaller than the medianthird quartile Q3 median of all values larger than the median
Box Plots and the 5-Number Summary
When dealing with the median, we measure variation with the5-number summary. These 5 numbers indicate the maximum andminimum, the median and the quartiles.
In order to find the five number summary, we first line the dataelements in order. Then we find the minimum and maximum, andthen the median.
minimum smallest value of the set
maximum largest value of the setmedian central(s) value of the set
first quartile Q1 median of all values smaller than the medianthird quartile Q3 median of all values larger than the median
Box Plots and the 5-Number Summary
When dealing with the median, we measure variation with the5-number summary. These 5 numbers indicate the maximum andminimum, the median and the quartiles.
In order to find the five number summary, we first line the dataelements in order. Then we find the minimum and maximum, andthen the median.
minimum smallest value of the setmaximum largest value of the set
median central(s) value of the setfirst quartile Q1 median of all values smaller than the medianthird quartile Q3 median of all values larger than the median
Box Plots and the 5-Number Summary
When dealing with the median, we measure variation with the5-number summary. These 5 numbers indicate the maximum andminimum, the median and the quartiles.
In order to find the five number summary, we first line the dataelements in order. Then we find the minimum and maximum, andthen the median.
minimum smallest value of the setmaximum largest value of the set
median central(s) value of the set
first quartile Q1 median of all values smaller than the medianthird quartile Q3 median of all values larger than the median
Box Plots and the 5-Number Summary
When dealing with the median, we measure variation with the5-number summary. These 5 numbers indicate the maximum andminimum, the median and the quartiles.
In order to find the five number summary, we first line the dataelements in order. Then we find the minimum and maximum, andthen the median.
minimum smallest value of the setmaximum largest value of the set
median central(s) value of the setfirst quartile Q1 median of all values smaller than the median
third quartile Q3 median of all values larger than the median
Box Plots and the 5-Number Summary
When dealing with the median, we measure variation with the5-number summary. These 5 numbers indicate the maximum andminimum, the median and the quartiles.
In order to find the five number summary, we first line the dataelements in order. Then we find the minimum and maximum, andthen the median.
minimum smallest value of the setmaximum largest value of the set
median central(s) value of the setfirst quartile Q1 median of all values smaller than the medianthird quartile Q3 median of all values larger than the median
5-Number Summary Example
ExampleFind the 5-number summary for the data from the first example.
11 010987 46 05 0 0 0 543 0 0 0 0 1 2 52 510 4
5-Number Summary Example
Since the values are already in order, we only need to calculate thevalues.
minimum 4Q1 30
Median 33.5Q3 52.5
maximum 110
Teddy Ballgame
ExampleTed Williams yearly RBI totals:145, 113, 120, 137, 123, 114, 127, 159, 97, 126, 3, 34, 89, 83, 82, 87,85, 43, 72Find the 5-number summary for this set of data,
What do we do first?
We put the values in order:3, 34, 43, 72, 82, 83, 85, 87, 89, 97, 113, 114, 120, 123, 126, 127,137, 145, 159.Then ...
Teddy Ballgame
ExampleTed Williams yearly RBI totals:145, 113, 120, 137, 123, 114, 127, 159, 97, 126, 3, 34, 89, 83, 82, 87,85, 43, 72Find the 5-number summary for this set of data,
What do we do first?
We put the values in order:3, 34, 43, 72, 82, 83, 85, 87, 89, 97, 113, 114, 120, 123, 126, 127,137, 145, 159.Then ...
Teddy Ballgame
ExampleTed Williams yearly RBI totals:145, 113, 120, 137, 123, 114, 127, 159, 97, 126, 3, 34, 89, 83, 82, 87,85, 43, 72Find the 5-number summary for this set of data,
What do we do first?
We put the values in order:3, 34, 43, 72, 82, 83, 85, 87, 89, 97, 113, 114, 120, 123, 126, 127,137, 145, 159.
Then ...
Teddy Ballgame
ExampleTed Williams yearly RBI totals:145, 113, 120, 137, 123, 114, 127, 159, 97, 126, 3, 34, 89, 83, 82, 87,85, 43, 72Find the 5-number summary for this set of data,
What do we do first?
We put the values in order:3, 34, 43, 72, 82, 83, 85, 87, 89, 97, 113, 114, 120, 123, 126, 127,137, 145, 159.Then ...
Solution
Minimum 3Q1 82
Median 97Q3 126
Maximum 159
Box-and-Whisker Plots
How can we visually represent this summary of the data? We use boxplots, or box-and-whisker plots.
Ted Williams’ RBI Totals
RBIs
Teddy Ballgame
20 40 60 80 100
120
140
160
Box-and-Whisker Plots
How can we visually represent this summary of the data? We use boxplots, or box-and-whisker plots.
Ted Williams’ RBI Totals
RBIs
Teddy Ballgame20 40 60 80 10
0
120
140
160
Box-and-Whisker Plots
How can we visually represent this summary of the data? We use boxplots, or box-and-whisker plots.
Ted Williams’ RBI Totals
RBIs
Teddy Ballgame20 40 60 80 10
0
120
140
160
Box-and-Whisker Plots
How can we visually represent this summary of the data? We use boxplots, or box-and-whisker plots.
Ted Williams’ RBI Totals
RBIs
Teddy Ballgame20 40 60 80 10
0
120
140
160
Box-and-Whisker Plots
How can we visually represent this summary of the data? We use boxplots, or box-and-whisker plots.
Ted Williams’ RBI Totals
RBIs
Teddy Ballgame20 40 60 80 10
0
120
140
160
Using Technology
The box plot is another that we can construct using the TI-seriesgraphing calculator. We do everything the same as when constructinga histogram until we reach the point where we choose the type ofgraph.
There are two options for box plots.
1 Second row, first graph shows outliers (we will get to those soon)2 Second row, second graph does not show outliers
We again use ZOOM and 9 to produce the graph.
Using Technology
The box plot is another that we can construct using the TI-seriesgraphing calculator. We do everything the same as when constructinga histogram until we reach the point where we choose the type ofgraph.
There are two options for box plots.
1 Second row, first graph shows outliers (we will get to those soon)
2 Second row, second graph does not show outliers
We again use ZOOM and 9 to produce the graph.
Using Technology
The box plot is another that we can construct using the TI-seriesgraphing calculator. We do everything the same as when constructinga histogram until we reach the point where we choose the type ofgraph.
There are two options for box plots.
1 Second row, first graph shows outliers (we will get to those soon)2 Second row, second graph does not show outliers
We again use ZOOM and 9 to produce the graph.
Using Technology
The box plot is another that we can construct using the TI-seriesgraphing calculator. We do everything the same as when constructinga histogram until we reach the point where we choose the type ofgraph.
There are two options for box plots.
1 Second row, first graph shows outliers (we will get to those soon)2 Second row, second graph does not show outliers
We again use ZOOM and 9 to produce the graph.
Getting Statistics
We can also find the statistics we need using the calculator relativelyeasily.
1 Input the data in the usual way
2 Press 2nd and MODE to quit to a blank screen
3 Press STAT , scroll to CALC, and select 1-Var Stats4 You will see 1-Var Stats on the screen; now select which list the
data is in by pressing 2nd and then the appropriate number 1-6,followed by the ENTER key
On this screen are some statistics we need x and Sx and if we scrolldown, we will see the 5-number summary.
Getting Statistics
We can also find the statistics we need using the calculator relativelyeasily.
1 Input the data in the usual way
2 Press 2nd and MODE to quit to a blank screen
3 Press STAT , scroll to CALC, and select 1-Var Stats4 You will see 1-Var Stats on the screen; now select which list the
data is in by pressing 2nd and then the appropriate number 1-6,followed by the ENTER key
On this screen are some statistics we need x and Sx and if we scrolldown, we will see the 5-number summary.
Getting Statistics
We can also find the statistics we need using the calculator relativelyeasily.
1 Input the data in the usual way
2 Press 2nd and MODE to quit to a blank screen
3 Press STAT , scroll to CALC, and select 1-Var Stats
4 You will see 1-Var Stats on the screen; now select which list thedata is in by pressing 2nd and then the appropriate number 1-6,followed by the ENTER key
On this screen are some statistics we need x and Sx and if we scrolldown, we will see the 5-number summary.
Getting Statistics
We can also find the statistics we need using the calculator relativelyeasily.
1 Input the data in the usual way
2 Press 2nd and MODE to quit to a blank screen
3 Press STAT , scroll to CALC, and select 1-Var Stats4 You will see 1-Var Stats on the screen; now select which list the
data is in by pressing 2nd and then the appropriate number 1-6,followed by the ENTER key
On this screen are some statistics we need x and Sx and if we scrolldown, we will see the 5-number summary.
Getting Statistics
We can also find the statistics we need using the calculator relativelyeasily.
1 Input the data in the usual way
2 Press 2nd and MODE to quit to a blank screen
3 Press STAT , scroll to CALC, and select 1-Var Stats4 You will see 1-Var Stats on the screen; now select which list the
data is in by pressing 2nd and then the appropriate number 1-6,followed by the ENTER key
On this screen are some statistics we need x and Sx and if we scrolldown, we will see the 5-number summary.
The Geometric View
From the minimum to Q1 is the bottom 25% of the observations
From Q1 to Q3 is the middle 50% of the observations
From Q3 to the maximum of the top 25% of the observations
We can look at this in other ways too:
The top half lies above the median
The top 75% lies above Q1
The bottom 75% lies below Q3
The Geometric View
From the minimum to Q1 is the bottom 25% of the observations
From Q1 to Q3 is the middle 50% of the observations
From Q3 to the maximum of the top 25% of the observations
We can look at this in other ways too:
The top half lies above the median
The top 75% lies above Q1
The bottom 75% lies below Q3
The Geometric View
From the minimum to Q1 is the bottom 25% of the observations
From Q1 to Q3 is the middle 50% of the observations
From Q3 to the maximum of the top 25% of the observations
We can look at this in other ways too:
The top half lies above the median
The top 75% lies above Q1
The bottom 75% lies below Q3
The Geometric View
From the minimum to Q1 is the bottom 25% of the observations
From Q1 to Q3 is the middle 50% of the observations
From Q3 to the maximum of the top 25% of the observations
We can look at this in other ways too:
The top half lies above the median
The top 75% lies above Q1
The bottom 75% lies below Q3
The Geometric View
From the minimum to Q1 is the bottom 25% of the observations
From Q1 to Q3 is the middle 50% of the observations
From Q3 to the maximum of the top 25% of the observations
We can look at this in other ways too:
The top half lies above the median
The top 75% lies above Q1
The bottom 75% lies below Q3
The Geometric View
From the minimum to Q1 is the bottom 25% of the observations
From Q1 to Q3 is the middle 50% of the observations
From Q3 to the maximum of the top 25% of the observations
We can look at this in other ways too:
The top half lies above the median
The top 75% lies above Q1
The bottom 75% lies below Q3
Box-and-Whisker Plot Example
ExampleConstruct a box-and-whisker plot for the data from the first example.
minimum 4Q1 30
Median 33.5Q3 52.5
maximum 110
Solution
Some Data Set
Values
Set 125 50 75 10
0
125
Analysis of Box-and-Whisker Plots
We can also look at the distribution like we did with histograms, butin a limited way as we cannot really tell how many peaks. But we canlook at the spread and center (directly from the table) and we can lookat the skewness.
What is the range here? 106
What do we know about the distribution? Skewed right distribution.
Further, this right endpoint seems to be pretty far away, so we maythink it is an outlier. But how do we determine if it is analytically?
Analysis of Box-and-Whisker Plots
We can also look at the distribution like we did with histograms, butin a limited way as we cannot really tell how many peaks. But we canlook at the spread and center (directly from the table) and we can lookat the skewness.
What is the range here?
106
What do we know about the distribution? Skewed right distribution.
Further, this right endpoint seems to be pretty far away, so we maythink it is an outlier. But how do we determine if it is analytically?
Analysis of Box-and-Whisker Plots
We can also look at the distribution like we did with histograms, butin a limited way as we cannot really tell how many peaks. But we canlook at the spread and center (directly from the table) and we can lookat the skewness.
What is the range here? 106
What do we know about the distribution? Skewed right distribution.
Further, this right endpoint seems to be pretty far away, so we maythink it is an outlier. But how do we determine if it is analytically?
Analysis of Box-and-Whisker Plots
We can also look at the distribution like we did with histograms, butin a limited way as we cannot really tell how many peaks. But we canlook at the spread and center (directly from the table) and we can lookat the skewness.
What is the range here? 106
What do we know about the distribution?
Skewed right distribution.
Further, this right endpoint seems to be pretty far away, so we maythink it is an outlier. But how do we determine if it is analytically?
Analysis of Box-and-Whisker Plots
We can also look at the distribution like we did with histograms, butin a limited way as we cannot really tell how many peaks. But we canlook at the spread and center (directly from the table) and we can lookat the skewness.
What is the range here? 106
What do we know about the distribution? Skewed right distribution.
Further, this right endpoint seems to be pretty far away, so we maythink it is an outlier. But how do we determine if it is analytically?
Analysis of Box-and-Whisker Plots
We can also look at the distribution like we did with histograms, butin a limited way as we cannot really tell how many peaks. But we canlook at the spread and center (directly from the table) and we can lookat the skewness.
What is the range here? 106
What do we know about the distribution? Skewed right distribution.
Further, this right endpoint seems to be pretty far away, so we maythink it is an outlier. But how do we determine if it is analytically?
IQR Criterion
DefinitionThe IQR Criterion is an analytic way for us to determine if data pointsare outliers based on a 5-number summary. To determine outliers, weuse
Q1 − 1.5IQR
andQ3 + 1.5IQR
to give us endpoints of the acceptable data range, where IQR is theInterquartile Range and
IQR = Q3 − Q1
These new endpoints are sometimes referred to as fences.
IQR Criterion
DefinitionThe IQR Criterion is an analytic way for us to determine if data pointsare outliers based on a 5-number summary. To determine outliers, weuse
Q1 − 1.5IQR
andQ3 + 1.5IQR
to give us endpoints of the acceptable data range, where IQR is theInterquartile Range and
IQR = Q3 − Q1
These new endpoints are sometimes referred to as fences.
Using the IQR Criterion
So, basically what we are doing is saying that any values no furtheraway from the middle 50% than 1.5 times the range of the middle50% are acceptable. Anything outside that range is an outlier.
ExampleAre there any outliers in the previous data set?
First we find the IQR, which is
Q3 − Q1 = 52.5− 30 = 22.5
and then we consider the new endpoints (fences).
Q1 − 1.5IQR = 30− 1.5(22.5) = 30− 33.75 = −3.75
Q3 + 1.5IQR = 52.5 + 1.5(22.5) = 52.5 + 33.75 = 86.25
Since 110 is larger than this upper threshhold, we would say it is anoutlier.
Using the IQR Criterion
So, basically what we are doing is saying that any values no furtheraway from the middle 50% than 1.5 times the range of the middle50% are acceptable. Anything outside that range is an outlier.
ExampleAre there any outliers in the previous data set?
First we find the IQR, which is
Q3 − Q1 = 52.5− 30 = 22.5
and then we consider the new endpoints (fences).
Q1 − 1.5IQR = 30− 1.5(22.5) = 30− 33.75 = −3.75
Q3 + 1.5IQR = 52.5 + 1.5(22.5) = 52.5 + 33.75 = 86.25
Since 110 is larger than this upper threshhold, we would say it is anoutlier.
Using the IQR Criterion
So, basically what we are doing is saying that any values no furtheraway from the middle 50% than 1.5 times the range of the middle50% are acceptable. Anything outside that range is an outlier.
ExampleAre there any outliers in the previous data set?
First we find the IQR, which is
Q3 − Q1 = 52.5− 30 = 22.5
and then we consider the new endpoints (fences).
Q1 − 1.5IQR = 30− 1.5(22.5) = 30− 33.75 = −3.75
Q3 + 1.5IQR = 52.5 + 1.5(22.5) = 52.5 + 33.75 = 86.25
Since 110 is larger than this upper threshhold, we would say it is anoutlier.
Using the IQR Criterion
So, basically what we are doing is saying that any values no furtheraway from the middle 50% than 1.5 times the range of the middle50% are acceptable. Anything outside that range is an outlier.
ExampleAre there any outliers in the previous data set?
First we find the IQR, which is
Q3 − Q1 = 52.5− 30 = 22.5
and then we consider the new endpoints (fences).
Q1 − 1.5IQR = 30− 1.5(22.5) = 30− 33.75 = −3.75
Q3 + 1.5IQR = 52.5 + 1.5(22.5) = 52.5 + 33.75 = 86.25
Since 110 is larger than this upper threshhold, we would say it is anoutlier.
Standard Deviation
The standard deviation measures the variation in data by measuringthe distance that the observations are from the mean. The standarddeviation tells us how far we can expect the average observation to befrom the mean.
Absolute Deviation
1n
n∑i=1
|xi − x|
Standard Deviation
s =
√∑(x− xi)2
n− 1
Standard Deviation
The standard deviation measures the variation in data by measuringthe distance that the observations are from the mean. The standarddeviation tells us how far we can expect the average observation to befrom the mean.
Absolute Deviation
1n
n∑i=1
|xi − x|
Standard Deviation
s =
√∑(x− xi)2
n− 1
Standard Deviation
The standard deviation measures the variation in data by measuringthe distance that the observations are from the mean. The standarddeviation tells us how far we can expect the average observation to befrom the mean.
Absolute Deviation
1n
n∑i=1
|xi − x|
Standard Deviation
s =
√∑(x− xi)2
n− 1
Standard Deviation and Variance
Standard Deviation
s =
√∑(xi − x)2
n− 1
Whereas it won’t have a lot of use for our purposes
Variance
s2 =
∑(xi − x)2
n− 1
Finding the Standard Deviation
ExampleFind the standard deviation of the daily caloric intake for a personover the course of a week.
{1792, 1666, 1362, 1614, 1460, 1867, 1439}
First we find the mean.
x =11200
7= 1600
Finding the Standard Deviation
ExampleFind the standard deviation of the daily caloric intake for a personover the course of a week.
{1792, 1666, 1362, 1614, 1460, 1867, 1439}
First we find the mean.
x =11200
7= 1600
Finding the Standard Deviation
Then, we need to find the difference between each of these values andthe mean, then square that differences and then sum them.
xi (xi − x)2 square contribution1792 (1792− 1600)2 1922 36864
1666 (1666− 1600)2 662 43561362 (1362− 1600)2 (−238)2 566441614 (1614− 1600)2 142 1961460 (1460− 1600)2 (−140)2 196001867 (1867− 1600)2 2672 712891439 (1439− 1600)2 (−161)2 25921
Finding the Standard Deviation
Then, we need to find the difference between each of these values andthe mean, then square that differences and then sum them.
xi (xi − x)2 square contribution1792 (1792− 1600)2 1922 368641666 (1666− 1600)2 662 43561362 (1362− 1600)2 (−238)2 566441614 (1614− 1600)2 142 1961460 (1460− 1600)2 (−140)2 196001867 (1867− 1600)2 2672 712891439 (1439− 1600)2 (−161)2 25921
Finding the Standard Deviation
Then, we need to find the difference between each of these values andthe mean, then square that differences and then sum them.
xi (xi − x)2 square contribution1792 (1792− 1600)2 1922 368641666 (1666− 1600)2 662 43561362 (1362− 1600)2 (−238)2 566441614 (1614− 1600)2 142 1961460 (1460− 1600)2 (−140)2 196001867 (1867− 1600)2 2672 712891439 (1439− 1600)2 (−161)2 25921
sum 214870
Finding the Standard Deviation
Next, we divide by 6.
s2 =214870
6≈ 35811.67
s2 is the ...
variance.
Now we take the square root.
s =√
35811.67 ≈ 189.24
So, the average value of the caloric intake is approximately 189calories from the mean. Notice that we only care about magnitude andnot whether we are above or below the mean.
Finding the Standard Deviation
Next, we divide by 6.
s2 =214870
6≈ 35811.67
s2 is the ...variance.
Now we take the square root.
s =√
35811.67 ≈ 189.24
So, the average value of the caloric intake is approximately 189calories from the mean. Notice that we only care about magnitude andnot whether we are above or below the mean.
Finding the Standard Deviation
Next, we divide by 6.
s2 =214870
6≈ 35811.67
s2 is the ...variance.
Now we take the square root.
s =√
35811.67 ≈ 189.24
So, the average value of the caloric intake is approximately 189calories from the mean. Notice that we only care about magnitude andnot whether we are above or below the mean.
Mean and Standard Deviation
So what can we do with mean and standard deviation?
We can use them to relate individuals within our data set to thedistribution of the sample.
This is related to probability.
The total area underneath a distribution curve is always 1, so thearea under the curve is the same as the percent of observationsfalling in the region.
We will see this better when we get to Normal distributions.
Mean and Standard Deviation
So what can we do with mean and standard deviation?
We can use them to relate individuals within our data set to thedistribution of the sample.
This is related to probability.
The total area underneath a distribution curve is always 1, so thearea under the curve is the same as the percent of observationsfalling in the region.
We will see this better when we get to Normal distributions.
Mean and Standard Deviation
So what can we do with mean and standard deviation?
We can use them to relate individuals within our data set to thedistribution of the sample.
This is related to probability.
The total area underneath a distribution curve is always 1, so thearea under the curve is the same as the percent of observationsfalling in the region.
We will see this better when we get to Normal distributions.
Mean and Standard Deviation
So what can we do with mean and standard deviation?
We can use them to relate individuals within our data set to thedistribution of the sample.
This is related to probability.
The total area underneath a distribution curve is always 1, so thearea under the curve is the same as the percent of observationsfalling in the region.
We will see this better when we get to Normal distributions.
Uniform Distributions
For these, all values have the same probability of occurring. So, theshape is that of a rectangle.
Random Number Between 0 and 2
x
f (x) 12
1 2
Uniform Distributions
ExampleIf we have a uniform distribution for a random number to be chosenbetween 0 and 2, what is the probability that the number selected isbetween .5 and 1.1?
Random Number Between 0 and 2
x
f (x) 12
1 2
Uniform Distributions
ExampleIf we have a uniform distribution for a random number to be chosenbetween 0 and 2, what is the probability that the number selected isbetween .5 and 1.1?
Random Number Between 0 and 2
x
f (x) 12
1 2
Uniform Distributions
Random Number Between 0 and 2
x
f (x) 12
1 2
What is the area of a rectangle? length× width.What are our dimensions? .5× .6 = .3.So, there is a 30% chance that the number randomly selected falls inthis region.
Uniform Distributions
Random Number Between 0 and 2
x
f (x) 12
1 2
What is the area of a rectangle?
length× width.What are our dimensions? .5× .6 = .3.So, there is a 30% chance that the number randomly selected falls inthis region.
Uniform Distributions
Random Number Between 0 and 2
x
f (x) 12
1 2
What is the area of a rectangle? length× width.
What are our dimensions? .5× .6 = .3.So, there is a 30% chance that the number randomly selected falls inthis region.
Uniform Distributions
Random Number Between 0 and 2
x
f (x) 12
1 2
What is the area of a rectangle? length× width.What are our dimensions?
.5× .6 = .3.So, there is a 30% chance that the number randomly selected falls inthis region.
Uniform Distributions
Random Number Between 0 and 2
x
f (x) 12
1 2
What is the area of a rectangle? length× width.What are our dimensions? .5× .6 = .3.
So, there is a 30% chance that the number randomly selected falls inthis region.
Uniform Distributions
Random Number Between 0 and 2
x
f (x) 12
1 2
What is the area of a rectangle? length× width.What are our dimensions? .5× .6 = .3.So, there is a 30% chance that the number randomly selected falls inthis region.
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