Download 5-Minute Check on Activity 7-8

5-Minute Check on Activity 7-8
1. What type of an experiment is it when neither the patient nor the
doctor knows what type of pill is being given?
Double-blind experiment
2. List the three major components of any experimental design
Randomization, replication, and control
Placebo
3. A “sugar pill” is also known as a ________________.
4. What is the only thing that can establish cause and effect?
Well designed experiment
5. What do we call a group in the experiment which treatments are
measured against?
Control group
Click the mouse button or press the Space Bar to display the answers.
Submarine interior (unspecified class) at the Royal Naval Museum, Copenhagen, Denmark
Activity 7 - 9
A Switch Decision
Objectives
• Measure the variability of a frequency distribution
Vocabulary
• Standard Deviation – measures how much the data
deviates from the mean
• Boxplot – statistical graph that helps visualize the
variability of a distribution
• Five-number Summary – the min, quartile 1, 2 and 3
and the max of the data set
Activity
The following sets of data are the result of testing two
different switches that can be used in the life-support
system on a submarine. Two hundred of each type of
switch were placed under continuous stress until they
failed, the recorded in hours. Switch A and B have
approximately the same means and medians, as
displayed by the following histograms.
Activity cont
1. What does the means and medians being the same tell
us about the distributions?
Distributions are symmetric
Activity cont
2. Which distribution is most spread out?
Switch B is more spread out
Activity cont
3. Which distribution is packed more closely together
around its center?)
Switch A is tighter
Activity cont
4. Which of these two switches would you choose and
why?
Switch A because is varies less
Activity cont
5. Determine the range of the two switches:
Switch A: 87.52 – 76.68 = 10.84
Switch B: 94.03 – 65.87 = 28.16
Activity cont
6. Determine the IQR (interquartile range),
which is Q3 – Q1 for each switch
Switch A: 83.47 – 80.53 = 2.94
Switch B: 85.45 – 78.99 = 6.46
Activity cont
7. Write down sx for each switch (this is something we
will call the standard deviation)
Switch A: 2.03
Switch B: 5.00
Measures of Spread
Variability is the key to Statistics. Without
variability, there would be no need for the
subject.
When describing data, never rely on center alone.
Measures of Spread:
Range - {rarely used ... why?}
Quartiles - InterQuartile Range {IQR=Q3-Q1}
Variance and Standard Deviation {var and sx}
Like Measures of Center, you must choose
the most appropriate measure of spread.
Standard Deviation
Another common measure of spread is the
Standard Deviation: a measure of the
“average” deviation of all observations from
the mean.
To calculate Standard Deviation:
Calculate the mean.
Determine each observation’s deviation (x - xbar).
“Average” the squared-deviations by dividing the
total squared deviation by (n-1).
This quantity is the Variance.
Square root the result to determine the Standard
Deviation.
Standard Deviation Properties
s measures spread about the mean and should be
used only when the mean is used as the measure of
center
s = 0 only when there is no spread/variability. This
happens only when all observations have the same
value. Otherwise, s > 0. As the observations
become more spread out about their mean, s gets
larger
s, like the mean x-bar, is not resistant. A few
outliers can make s very large
Standard Deviation
Variance:
(x1  x ) 2  (x2  x ) 2  ... (xn  x ) 2
var 
n 1
Standard Deviation:

sx 
2
(x

x
)
 i
n 1
Example 1.16 (p.85 of YMS): Metabolic Rates
1792
1666
1362

1614
1460
1867
1439
Standard Deviation
1792
1666
1362
1614
1460
1867
1439
Metabolic Rates: mean=1600
x
(x - x)
(x - x)2
1792
192
36864
1666
66
4356
1362
-238
56644
1614
14
196
1460
-140
19600
1867
267
71289
1439
-161
25921
Totals:
0
214870
Total
Squared
Deviation
214870
Variance
var=214870/6
var=35811.66
Standard
Deviation
s=√35811.66
s=189.24 cal
What does this value, s, mean?
Example 1
Which of the following measures of spread are
resistant?
1. Range
Not Resistant
2. Variance
Not Resistant
3. Standard Deviation
Not Resistant
Standard Deviation Using the TI-83
• Enter the test data into List, L1
– STAT, EDIT enter data into L1
• Calculate Standard Deviation
– Hit STAT go over to CALC
and select 1-Var Stats and hit 2nd 1 (L1)
– Read sx to get standard deviation
– Square sx to get variance
– x is population standard deviation
(and won’t be used by AFDA)
• Don’t worry about the formula we just went over
Example 2
Given the following set of data:
19
22
23
23
23
26
What is the range?
26
27
29
30
32 - 19 = 13
What is the standard deviation?
What is the variance?
28
3.751
(3.751)2 = 14.070
31
32
Quartiles
Quartiles Q1 and Q3 represent the 25th and
75th percentiles.
To find them, order data from min to max.
Determine the median - average if necessary.
The first quartile is the middle of the ‘bottom half’.
The third quartile is the middle of the ‘top half’.
19
22
23
23
23
68
74
Q1
26
27
28
med
Q1=23
45
26
75
76
29
30
31
32
Q3=29.5
82
med=79
82
91
Q3
93
98
5-Number Summary, Boxplots
The 5 Number Summary provides a reasonably complete
description of the center and spread of distribution
MIN
Q1
MED
Q3
MAX
We can visualize the 5 Number Summary with a boxplot.
min=45
45
50
Q1=74
55
Outlier?
60
med=79
65
70
75
Q3=91
80
Quiz Scores
85
max=98
90
95 100
Box Plots Using the TI-83
• Enter the test data into List, L1
– STAT, EDIT enter data into L1
• Calculate 5 Number Summary
– Hit STAT go over to CALC
and select 1-Var Stats and hit 2nd 1 (L1)
• Use 2nd Y= (STAT PLOT) to graph the box plot
–
–
–
–
–
Turn plot1 ON
Select BOX PLOT (4th option, first in second row)
Xlist: L1
Freq: 1
Hit ZOOM 9:ZoomStat to graph the box plot
• Copy graph with appropriate labels and titles
Determining Outliers
“1.5 • IQR Rule”
InterQuartile Range “IQR”: Distance between Q1 and
Q3. Resistant measure of spread...only measures
middle 50% of data.
IQR = Q3 - Q1 {width of the “box” in a boxplot}
1.5 IQR Rule: If an observation falls more than 1.5
IQRs above Q3 or below Q1, it is an outlier.
Why 1.5? According to John Tukey, 1 IQR seemed like too little and 2 IQRs
seemed like too much...
Outliers: 1.5 • IQR Rule
To determine outliers:
1. Find 5 Number Summary
2. Determine IQR (Q3 – Q1)
3. Multiply 1.5xIQR
4. Set up “fences”
A. Lower Fence: Q1-(1.5∙IQR)
B. Upper Fence: Q3+(1.5∙IQR)
5. Observations “outside” the fences are outliers.
Outlier Example
IQR=45.72-19.06
IQR=26.66
fence: 19.06-39.99
= -20.93
1.5IQR=1.5(26.66)
1.5IQR=39.99
fence: 45.72+39.99
= 85.71
{
}
0
10
20
30
40
50 60 70
Spending ($)
80
outliers
90
100
Example 3
Consumer Reports did a study of ice cream bars (sigh, only
vanilla flavored) in their August 1989 issue. Twenty-seven bars
having a taste-test rating of at least “fair” were listed, and calories
per bar was included. Calories vary quite a bit partly because
bars are not of uniform size. Just how many calories should an
ice cream bar contain?
342
377
319
353
295
234
294
286
377
182
310
439
111
201
182
197
209
147
190
151
131
151
Construct a boxplot for the data above.
Example 3 - Answer
Q1 = 182
Min = 111
IQR = 137
Q2 = 221.5
Max = 439
UF = 524.5
Q3 = 319
Range = 328
LF = -23.5
100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500
Calories
Example 4
The weights of 20 randomly selected juniors at MSHS
are recorded below:
121
126
130
132
143
137
141
144
148
205
125
128
131
133
135
139
141
147
153
213
a) Construct a boxplot of the data
b) Determine if there are any outliers
c) Comment on the distribution
Example 4 - Answer
Q1 = 130.5
Min = 121
IQR = 15
Mean = 143.6
StDev = 23.91
Q2 = 138
Max = 213
UF = 168
Q3 = 145.5
Range = 92
LF = 108
Extreme Outliers
( > 3 IQR from Q3)
*
100
110
120
130
140
150
160
170
180
190
200
*
210
220
Weight (lbs)
Shape: somewhat symmetric
Center: Median = 138
Outliers: 2 extreme outliers
Spread: IQR = 15
Summary and Homework
• Summary
– Variability of a frequency distribution refers to how
spread out the data is, away from center
– Range is the max – the min of the data
– Deviation of a data value is how far away from the
mean it is
– Standard deviation is a measure of how spread out
all of the data is
– Boxplot is a graph of the 5-number summary
• Homework
– pg 861 – 863; problems 1, 4, 6, 7