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Material Type: Notes; Professor: Martineau; Class: Statistics I-WR; Subject: Mathematics (MTH); University: Monroe Community College; Term: Unknown 1996;
Typology: Study notes
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Brigitte Martineau Chapter 2
2.2 Graphs, Pareto Diagrams, (and Stem-and-Leaf Displays)
Why graphing?
Graphs for qualitative data: Circle graph and Bar graph
1. Circle Graph (also known as _________ _____________ )
n y
Category
y 14, 25.9%
n 40, 74.1%
Pie Chart of Smokers?
In Minitab: Graph > Pie Chart > select a variable….you can also select various options within the pie chart menu.
Brigitte Martineau Chapter 2
Pareto Diagram (a special type of bar graph)
Example: The final daily inspection defect report for a cabinet manufacturer is given in the table below: Defect Number Defect Number Dent 5 Chip 25 Stain 12 Scratch 40 Blemish 43 Others 10
Management has given the cabinet production line the goal for reducing their defects by 50%. What defects should they give special attention to in working toward this goal?
In Minitab: Stat > Quality tools > Pareto Chart > select a variable
140 120 100 80 60 40 20 0
100
80
60
40
20
0
Count (^) Percent
Defec Blemi Scrat Chi Stai Othe De Count (^) 43 40 25 12 10 5 Percent (^) 31.9 29.6 18.5 8.9 7.4 3. Cum% (^) 31.9 61.5 80.0 88.9 96.3 100.
Daily Defect Inspection Report
Brigitte Martineau Chapter 2
Graphs for quantitative data: dotplots (and stem-and-leaf)
Graphs for quantitative data are useful to display the distribution.
But what is a distribution?
1. Dotplot
Height
Gender 60 63 66 69 72 75 78
f m
Dotplot of Height vs Gender
In Minitab: Graph > Dotplot > select a variable
Height 60 63 66 69 72 75 78
Dotplot of Height
Brigitte Martineau Chapter 2
Example: A video store has computed the number of movies rented for every day of the last month: 74 142 179 127 198 105 98 87 189 154 189 207 76 95 108 163 205 96 149 174 123 147 108 101 185 125 87 119 138 162
Classes Frequency Midpoint
Total
What is a histogram?
Brigitte Martineau Chapter 2
Histograms can have different shapes:
Symmetrical Uniform (rectangular)
Skewed to right Skewed to left
J-Shaped Bimodal
Brigitte Martineau Chapter 2
What is it?
Symbol
How to find the median?
Examples: Find the median of 4, 8, 3, 8, 2, 9, 2, 11, 3
Find the median of 4, 8, 3, 8, 2, 9, 2, 11, 3, 15
What is it?
Symbol
Bimodal or no mode
Example:
Brigitte Martineau Chapter 2
What is it?
Symbol
Formula:
lowest value + highest value midrange = 2
Examples
Consider the following 2 sets of data. Calculate the Averages for each.
Data Set Mean Median Mode Midrange Best Measure
The results of 3 tests of MTH 160 are shown in the following table. Explain what is going on?
Mean Median Explanation
Test 1 74 73
Test 2 73 80
Test 3 65 60
Brigitte Martineau Chapter 2
Example Find the range, the variance and the standard deviation of the following set of numbers
Range =
Variance =
Standard Deviation =
Step 1: Find the mean X
Step 2: Fill in the table below:
2 2 2 1
x x s n n
Data Set (^) x X
2 x X
2 5 4 8 4 3 6 8
Sum =
Sum = (^) ( X X )^2 =
s =
n - 1
Brigitte Martineau Chapter 2
Find the range, the variance and the standard deviation of the following data: 8, 8, 12, 14, 6, 6
Range: Variance: Standard Deviation:
2 2 2 1
x x s n n
The results of 2 tests of MTH 160 are shown in the following table. Explain.
Mean Standard Deviation Explanation
Test 1 74 4
Test 2 73 22
Data Set (^) x X
2 x X
Sum =
Sum = (^) ( X X )^2 =
2
1
s n
1
s n
Brigitte Martineau Chapter 2
How to find percentiles and quartiles
nk A
If A is a fraction or a decimal The position of the percentile is at the next larger integer after A. The percentile is the value of the data at that position mentioned above.
Examples
The following data represents the pH levels of a random sample of swimming pools in a
California town
5.6 5.6 5.8 5.9 6.0 6.0 6.1 6.2 6.3 6. 6.7 6.8 6.8 6.8 6.9 7.0 7.3 7.4 7.4 7.
Find the 34 th^ and the 60th^ percentile as well as the 1st^ and 3rd^ quartile.
Brigitte Martineau Chapter 2
The midquartile is another measure of ________________________________
Formula:
The mean, median, mode, midrange and midquartile are all measures of central tendency.
Are they all equal in value?
Can you find an example where they would be?
The 5-number summary indicates how much the data are spread in each _______________
The Box-and-Whisker display, also called boxplot, displays the 5-number summary.
Vertical or horizontal The box is used to depict….
The whiskers are line segments used to depict…..
The line through the box represents ______________________ One line segment represents…
The other line segment represents…
The outlier….
Shoe Size
Boxplot of Shoe Size
Brigitte Martineau Chapter 2
A certain data set has mean 76 and standard deviation 10. Find the z-scores for 90 and 60.
Bill and Joe both got 79% on their statistics test. Bill is in section 1 where the mean was 75 and the standard deviation was 10. Joe is in section 5 where the mean was 77 and standard deviation was 21. Who has the best relative score?
Brigitte Martineau Chapter 2
2.7 - Interpreting and Understanding Standard Deviation
Standard deviation is a measure of ______________________
There are 2 rules to describe data that rely on the standard deviation:
If a variable is normally distributed:
Note:
x 3 s x 2 s x s x x s x 2 s x 3 s
68%
95%
99.7%
