2.4 (cont.)
Using the Mean and Standard
Deviation Together
68-95-99.7 rule
z-scores
1
68-95-99.7 rule
Mean and
Standard Deviation
(numerical)
Histogram
(graphical)
68-95-99.7 rule
2
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 ( y  s, y  s)

approx. 95% of the measurements are within
2 stand. dev. of the mean, i.e., in ( y  2 s, y  2 s )

almost all the measurements are within 3 stan.
dev of the mean, i.e., in ( y  3s, y  3s)
3
68-95-99.7 rule: 68% within 1
stan. dev. of the mean
0.4
0.35
0.3
0.25
68%
0.2
0.15
0.1
34%
34%
0.05
-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
0
y-s
y
y+s
4
68-95-99.7 rule: 95% within 2
stan. dev. of the mean
0.4
0.35
0.3
0.25
95%
0.2
0.15
0.1
47.5% 47.5%
0.05
-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
0
y-2s
y
y+2s
5
Example: textbook costs
286
328
349
367
382
398
425
480
291
340
354
369
385
409
426
307
342
355
371
385
409
428
308
346
355
373
387
410
433
315
347
360
377
390
418
434
316
348
361
380
390
422
437
327
348
364
381
397
424
440
n  50
y  375.48
s  42.72
6
Example: textbook costs (cont.)
286
340
355
373
390
422
440
291
342
355
377
390
424
480
307
346
360
380
397
425
308
347
361
381
398
426
315
348
364
382
409
428
316
348
367
385
409
433
327
349
369
385
410
434
328
354
371
387
418
437
1 standard deviation interval about the mean
y  375.48 s  42.72
( y  s, y  s )  (332.76, 418.20)
32
percentage of data values in this interval
 64%;
50 7
68-95-99.7 rule:  68%
Example: textbook costs (cont.)
286
340
355
373
390
422
440
291
342
355
377
390
424
480
307
346
360
380
397
425
308
347
361
381
398
426
315
348
364
382
409
428
316
348
367
385
409
433
327
349
369
385
410
434
328
354
371
387
418
437
2 standard deviation interval about the mean
y  375.48 s  42.72
( y  2 s, y  2 s )  (290.04, 460.92)
48
percentage of data values in this interval
 96%;
50 8
68-95-99.7 rule:  95%
Example: textbook costs (cont.)
286
340
355
373
390
422
440
291
342
355
377
390
424
480
307
346
360
380
397
425
308
347
361
381
398
426
315
348
364
382
409
428
316
348
367
385
409
433
327
349
369
385
410
434
328
354
371
387
418
437
3 standard deviation interval about the mean
y  375.48 s  42.72
( y  3s, y  3s )  (247.32, 503.64)
50
percentage of data values in this interval
 100%;
50
9
68-95-99.7 rule:  99.7%
The best estimate of the standard
deviation of the men’s weights
displayed in this dotplot is
1.
2.
3.
4.
10
15
20
40
0%
1
0%
2
0%
3
10
0%
4
Z-scores: Standardized Data
Values
Measures the distance of a
number from the mean in units of
the standard deviation
12
z-score corresponding to y
y y
z
s
where
y  original data value
y  the sample mean
s  the sample standard deviation
z  the z-score corresponding to y
13
If data has mean y and standard deviation s,
then standardizing a particular value of y
indicates how many standard deviations y
is above or below the mean y .

Exam 1: y1 = 88, s1 = 6; exam 1 score: 91
Exam 2: y2 = 88, s2 = 10; exam 2 score: 92
Which score is better?
z1 
91  88

3
 .5
6
6
92  88 4
z2 

 .4
10
10
91 on exam 1 is better than 92 on exam 2
14
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.

15
Z-scores add to zero
Student/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
16
In 2007-08 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.
1.03
-1.03
2.39
1865
-1865
0%
1.
0%
0%
2.
3.
0%
4
17
0%
5
End of Section 2.4
18