Transcript Section 1.2 - MathSpace.com

Section 1.2
Describing Distributions with
Numbers
Types of Measures
• Measures of Center:
– Mean, Median, Mode
• Measures of Spread:
– Range (Max-Min), Standard Deviation,
Quartiles, IQR
Means and Medians
Consider the following sample of test scores
from one of Dr. X’s recent classes (max score =
100):
65, 65, 70, 75, 78, 80, 83, 87, 91, 94
What is the Average (or Mean) Test Score?
What is the Median Test Score?
Consider the following sample of test scores from one of Dr.
X.’s recent classes (max score = 100):
65, 65, 70, 75, 78, 80, 83, 87, 91, 94
• Draw a Stem and Leaf Plot (Shape, Center, Spread?)
• Find the Mean and the Median
• Let’s Use our TI-83 Calculators!
– Enter data into a list via Stat|Edit
– Stat|Calc|1-Var Stats
• What happens to the Mean and Median if the lowest score was 20
instead of 65?
• What happens to the Mean and Median if a low score of 20 is added to
the data set (so we would now have 11 data points?)
What can we say about the Mean versus the Median?
Quartiles: Measures of Position
A Graphical Representation of Position of Data
(It really gives us an indication of how the data is
spread among its values!)
Using Measures of Position to Get Measures of Spread
IQR is a measure of how the data deviates from the median
5 Number Summary, IQR, Box Plot, and where
Outliers would be for Test Score Data:
65, 65, 70, 75, 78, 80, 83, 87, 91, 94
5 Number Summary, IQR, Box Plot, and where
Outliers would be for Test Score Data:
65, 65, 70, 75, 78, 80, 83, 87, 91, 94
5-number summary: Min = 65, Q1 = 70, Median = 79, Q3 = 87, Max = 94
IQR = Q3 - Q1 = 17
Suspected Outliers:
• anything less than Q1 - 1.5*IQR = 44.5
• anything greater than Q3 + 1.5*IQR = 112.5
BoxPlot
(By MegaStat)
60
65
70
75
80
85
90
95
100
test score
Note: MegaStat uses a slightly different definition for the quartiles, so the box plot here
is not perfectly consistent with the definitions we’re using.
Histograms of Flower Lengths
Problem 1.58
Generated via Minitab
Histogram of Flower Length
36
bihai
39
42
45
48
51
red
48
36
Percent
24
12
0
yellow
48
36
24
12
0
36
39
42
45
48
51
length
Panel variable: variety
Box Plot and 5-Number
Summary for Flower Length Data
Generated via Box Plot Macro for Excel
Box Plots for Flower Lengths
Bihai
Red
Yellow
Lengths (in mm)
55
Median
47.12
39.16
36.11
45
Q1
46.71
38.07
35.45
40
Min or In
Fence
46.34
37.4
34.57
Max or In
Fence
50.26
43.09
38.13
Q3
48.24
5
41.69
36.82
50
35
30
Bihai
Red
Yellow
Flower Color
Outliers?
Remember this histogram
from the Service Call Length
Data on page 9? How do you
expect the Mean and Median
to compare for this data?
Mean 188.6, Median 115
Box Plot for Call Length Data
More on Measures of Spread
• Data Range (Max – Min)
• IQR (75% Quartile minus 25% Quartile = range of
middle 50% of data)
• Standard Deviation
– Measures how the data deviates from the mean
• Recall the Sample Test Score Data:
65, 65, 70, 75, 78, 80, 83, 87, 91, 94
Recall the Sample Mean (X bar) was 78.8…
Deviations from the Mean
Recall the Sample Test Score Data:
65, 65, 70, 75, 78, 80, 83, 87, 91, 94
Recall the Sample Mean (X bar) was 78.8
78.8
65
4.2
13.8
65
70
75
80
x
What does the
number 4.2 measure?
How about 13.8?
83
85
90
95
Effects of Outliers on the Standard Deviation
Consider (again!) the following sample of test scores from
one of Dr. X.’s recent classes (max score = 100):
65, 65, 70, 75, 78, 80, 83, 87, 91, 94
What happens to the standard deviation and the location of
the 1st and 3rd quartiles if the lowest score was 20 instead of
65?
What happens to the standard deviation and the location of
the 1st and 3rd quartiles if a low score of 20 is added to the
data set (so we would now have 11 data points?)
What can we say about the effect of outliers on the standard
deviation and the quartiles of a data set?
Example 1.18: Stemplots
of Annual Returns for
Stocks (a) and Treasury
bills (b) On page 53 of text.
What are the stem and leaf
units????
Effects of Linear Transformations on the Mean
And Standard Deviation
Consider (again!) the following sample of test scores from one of Dr. X’s
recent classes (max score = 100):
65, 65, 70, 75, 78, 80, 83, 87, 91, 94
Xbar = 78.8
s = 10.2 (rounded)
Suppose we “curve” the grades by adding 5 points to every test score (i.e.
Xnew=Xold+5). What will be new mean and standard deviation?
Suppose we “curve” the grades by multiplying every test score times 1.5
(i.e. Xnew=1.5*Xold). What will be the new mean and standard deviation?
Suppose we “curve” the grades by multiplying every test score times 1.5
and adding 5 points (i.e. Xnew=1.5*Xold+5). What will be the new mean
and standard deviation?
Box Plots for Problems 1.62-1.64