Chapter 6 Part 1 Using the Mean and Standard Deviation Together
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Transcript of Chapter 6 Part 1 Using the Mean and Standard Deviation Together
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Chapter 6 Part 1Using the Mean and Standard
Deviation Together
z-scores
68-95-99.7 rule
Changing units (shifting and rescaling data)
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Z-scores: Standardized Data Values
Measures the distance of a number from the mean in units of
the standard deviation
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z-score corresponding to y
where
original data value
the sample mean
s the sample standard deviation
the z-score corresponding to
y yz
s
y
y
z y
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Exam 1: y1 = 88, s1 = 6; exam 1 score: 91
Exam 2: y2 = 88, s2 = 10; exam 2 score: 92
Which score is better?
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2
91 88 3z .5
6 692 88 4
z .410 10
91 on exam 1 is better than 92 on exam 2
If data has mean and standard deviation ,
then standardizing a particular value of
indicates how many standard deviations
is above or below the mean .
y s
y
y
y
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Comparing SAT and ACT Scores
SAT Math: Eleanor’s score 680
SAT mean =500 sd=100 ACT Math: Gerald’s score 27
ACT mean=18 sd=6 Eleanor’s z-score: z=(680-500)/100=1.8 Gerald’s z-score: z=(27-18)/6=1.5 Eleanor’s score is better.
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Z-scores add to zeroStudent/Institutional Support to Athletic Depts For the 9 Public ACC
Schools: 2013 ($ millions)
School Support y - ybar Z-score
Maryland 15.5 6.4 1.79
UVA 13.1 4.0 1.12
Louisville 10.9 1.8 0.50
UNC 9.2 0.1 0.03
VaTech 7.9 -1.2 -0.34
FSU 7.9 -1.2 -0.34
GaTech 7.1 -2.0 -0.56
NCSU 6.5 -2.6 -0.73
Clemson 3.8 -5.3 -1.47
Mean=9.1000, s=3.5697
Sum = 0 Sum = 0
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In a recent year the mean tuition at 4-yr public colleges/universities in the U.S. was $6185 with a standard deviation of $1804. In NC the mean tuition was $4320. What is NC’s z-score?
1. 2. 3. 4 5
37%
55%
0%2%7%
1. 1.03
2. -1.03
3. 2.39
4. 1865
5. -1865
Changing Units of Measurement
How shifting and rescaling data affect data summaries
Shifting and rescaling: linear transformations
Original data x1, x2, . . . xn
Linear transformation:x* = a + bx, (intercept a, slope b)
x
x*
0
aShifts data by a
Changes scale
Linear Transformationsx* = a+ b x
Examples: Changing1. from feet (x) to inches (x*): x*=12x2. from dollars (x) to cents (x*):
x*=100x3. from degrees celsius (x) to degrees
fahrenheit (x*): x* = 32 + (9/5)x 4. from ACT (x) to SAT (x*): x*=150+40x5. from inches (x) to centimeters (x*):
x* = 2.54x
0 120 10032 9/5150 400 2.54
Shifting data only: b = 1x* = a + x
Adding the same value a to each value in the data set: changes the mean, median, Q1 and Q3
by a The standard deviation, IQR and
variance are NOT CHANGED. Everything shifts together. Spread of the items does not change.
Shifting data only: b = 1x* = a + x (cont.)
weights of 80 men age 19 to 24 of average height (5'8" to 5'10") x = 82.36 kg
NIH recommends maximum healthy weight of 74 kg. To compare their weights to the recommended maximum, subtract 74 kg from each weight; x* = x – 74 (a=-74, b=1)
x* = x – 74 = 8.36 kg
1. No change in shape
2. No change in spread
3. Shift by 74
Shifting and Rescaling data: x* = a + bx, b > 0
Original x data:x1, x2, x3, . . ., xn
Summary statistics:mean xmedian m1st quartile Q1
3rd quartile Q3
stand dev svariance s2
IQR
x* data: x* = a + bxx1*, x2*, x3*, . . ., xn*
Summary statistics:new mean x* = a + bxnew median m* = a+bmnew 1st quart Q1*= a+bQ1
new 3rd quart Q3* = a+bQ3
new stand dev s* = b snew variance s*2 = b2 s2
new IQR* = b IQR
Rescaling data: x* = a + bx, b > 0 (cont.)
weights of 80 men age 19 to 24, of average height (5'8" to 5'10")
x = 82.36 kg min=54.30 kg max=161.50 kg range=107.20 kg s = 18.35 kg
Change from kilograms to pounds:x* = 2.2x (a = 0, b = 2.2)
x* = 2.2(82.36)=181.19 pounds min* = 2.2(54.30)=119.46 pounds max* = 2.2(161.50)=355.3 pounds range*= 2.2(107.20)=235.84 pounds s* = 18.35 * 2.2 = 40.37 pounds
Example of x* = a + bx
4 student heights in inches
(x data)62, 64, 74, 72x = 68 inchess = 5.89 inches
Suppose we wantcentimeters instead:x* = 2.54x(a = 0, b = 2.54)
4 student heights in centimeters:
157.48 = 2.54(62)162.56 = 2.54(64)187.96 = 2.54(74)182.88 = 2.54(72)x* = 172.72 centimeterss* = 14.9606 centimeters
Note thatx* = 2.54x = 2.54(68)=172.2s* = 2.54s =
2.54(5.89)=14.9606
not necessary!UNC method
Go directly to this. NCSU method
Example of x* = a + bxx data:Percent returns from 4investments during2003:5%, 4%, 3%, 6%x = 4.5%s = 1.29%Inflation during 2003:2%x* data:Inflation-adjusted returns.x* = x – 2%(a=-2, b=1)
x* data:
3% = 5% - 2%2% = 4% - 2%1% = 3% - 2%4% = 6% - 2%x* = 10%/4 = 2.5%s* = s = 1.29%
x* = x – 2% = 4.5% –2%s* = s = 1.29% (note!
thats* ≠ s – 2%) !!
not necessary!
Go directly to this
Example Original data x: Jim Bob’s jumbo watermelons from
his garden have the following weights (lbs):
23, 34, 38, 44, 48, 55, 55, 68, 72, 75s = 17.12; Q1=37, Q3 =69; IQR = 69 – 37 = 32
Melons over 50 lbs are priced differently; the amount each melon is over (or under) 50 lbs is:
x* = x 50 (x* = a + bx, a=-50, b=1)-27, -16, -12, -6, -2, 5, 5, 18, 22, 25
s* = 17.12; Q*1 = 37 - 50 =-13, Q*3 = 69 - 50 = 19
IQR* = 19 – (-13) = 32 NOTE: s* = s, IQR*= IQR
Z-scores: a special linear transformation a + bx
1 1where ,
x x x xz x a bx a b
s s s s s
Example. At a community college, if a student takes x credit hours the tuition is x* = $250 + $35x. The credit hours taken by students in an Intro Stats class have mean x = 15.7 hrs and standard deviation s = 2.7 hrs.
Question 1. A student’s tuition charge is $941.25. What is the z-score of this tuition?
x* = $250+$35(15.7) = $799.50; s* = $35(2.7) = $94.50
941.25 799.50 141.75 1.594.50 94.50
z
Z-scores: a special linear transformation a + bx (cont.)Example. At a community college, if a student takes x credit hours the tuition is x* = $250 + $35x. The credit hours taken by students in an Intro Stats class have mean x = 15.7 hrs and standard deviation s = 2.7 hrs.
Question 2. Roger is a student in the Intro Stats class who has a course load of x = 13 credit hours. The z-score isz = (13 – 15.7)/2.7 = -2.7/2.7 = -1.What is the z-score of Roger’s tuition?
Roger’s tuition is x* = $250 + $35(13) = $705
Since x* = $250+$35(15.7) = $799.50; s* = $35(2.7) = $94.50
705-799.50 -94.50z= = =-194.50 94.50
This is why z-scores are so useful!!
SUMMARY: Linear Transformations x* = a + bx
Assembly Time (seconds)
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10
15
20
25
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Fre
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Linear transformations do not affect the shape of the distribution of the data-for example, if the original data is right-skewed, the transformed data is right-skewed
Assembly Time (minutes)
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20
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SUMMARY: Shifting and Rescaling data, x* = a + bx, b > 0
* * *1 2 3 1 2 3
*
*
*1 1 1
*3 3 3
original data , , ,... transformed data , , ,...
summary statistics summary statistics
mean new mean
median new median
1st new
3rd new
st dev
x x x x x x
x x a bx
m m a bm
Q Q a bQ
Q Q a bQ
*
2 2 2 2
new st dev
var. new var. *
new *
s s bs
s s b s
IQR IQR bIQR
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68-95-99.7 rule
Mean andStandard Deviation
(numerical)
Histogram(graphical)
68-95-99.7 rule
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The 68-95-99.7 rule; applies only to mound-shaped data
approximately 68% of the measurements
are within 1 standard deviation of the mean,
that is, in ( , )
approx. 95% of the measurements are within
2 stand. dev. of the mean, i.e., in ( 2 , 2 )
almos
y s y s
y s y s
t all the measurements are within 3 stan.
dev of the mean, i.e., in ( 3 , 3 )y s y s
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68-95-99.7 rule: 68% within 1 stan. dev. of the mean
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
-5
-4.5 -4
-3.5 -3
-2.5 -2
-1.5 -1
-0.5 0
0.5 1
1.5 2
2.5 3
3.5 4
4.5 5
68%
34%34%
y-s y y+s
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68-95-99.7 rule: 95% within 2 stan. dev. of the mean
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
-5
-4.5 -4
-3.5 -3
-2.5 -2
-1.5 -1
-0.5 0
0.5 1
1.5 2
2.5 3
3.5 4
4.5 5
95%
47.5% 47.5%
y-2s y y+2s
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Example: textbook costs
375.48
42.72
50
y
s
n
286 291 307 308 315 316 327 328340 342 346 347 348 348 349 354355 355 360 361 364 367 369 371373 377 380 381 382 385 385 387390 390 397 398 409 409 410 418422 424 425 426 428 433 434 437440 480
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Example: textbook costs (cont.)286 291 307 308 315 316 327 328340 342 346 347 348 348 349 354355 355 360 361 364 367 369 371373 377 380 381 382 385 385 387390 390 397 398 409 409 410 418422 424 425 426 428 433 434 437440 480
375.48 42.72
( , ) (332.76, 418.20)
32percentage of data values in this interval 64%;
5068-95-99.7 rule: 68%
y s
y s y s
1 standard deviation interval about the mean
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Example: textbook costs (cont.)286 291 307 308 315 316 327 328340 342 346 347 348 348 349 354355 355 360 361 364 367 369 371373 377 380 381 382 385 385 387390 390 397 398 409 409 410 418422 424 425 426 428 433 434 437440 480
375.48 42.72
( 2 , 2 ) (290.04, 460.92)
48percentage of data values in this interval 96%;
5068-95-99.7 rule: 95%
y s
y s y s
2 standard deviation interval about the mean
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Example: textbook costs (cont.)286 291 307 308 315 316 327 328340 342 346 347 348 348 349 354355 355 360 361 364 367 369 371373 377 380 381 382 385 385 387390 390 397 398 409 409 410 418422 424 425 426 428 433 434 437440 480
375.48 42.72
( 3 , 3 ) (247.32, 503.64)
50percentage of data values in this interval 100%;
5068-95-99.7 rule: 99.7%
y s
y s y s
3 standard deviation interval about the mean
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The best estimate of the standard deviation of the men’s weights
displayed in this dotplot is
1 2 3 4
4%9%
71%
16%
1. 10
2. 15
3. 20
4. 40
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End of Chapter 6 Part 1.Next: Part 2 Normal Models
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Z-scores add to zeroStudent/Institutional Support to Athletic Depts For the 9 Public ACC
Schools: 2013 ($ millions)
School Support y - ybar Z-score
Maryland 15.5 6.4 1.79
UVA 13.1 4.0 1.12
Louisville 10.9 1.8 0.50
UNC 9.2 0.1 0.03
VaTech 7.9 -1.2 -0.34
FSU 7.9 -1.2 -0.34
GaTech 7.1 -2.0 -0.56
NCSU 6.5 -2.6 -0.73
Clemson 3.8 -5.3 -1.47
Mean=9.1000, s=3.5697
Sum = 0 Sum = 0