Chapter 5The Normal Curve and

Standard Scores

EPS 525Introduction to Statistics

X

z Score: -3 -2 -1 0 1 2 3

34.1334.13 13.5913.592.15 2.15

68.26

95.54

99.74

47.72 47.72

49.8749.87

XX

z Score: -3 -2 -1 0 1 2 3

34.1334.13 13.5913.592.15 2.15

68.26

95.54

99.74

47.72 47.72

49.8749.87

0.13 0.13

X

z Score: -3 -2 -1 0 1 2 3

34.1334.13 13.5913.592.15 2.15

68.26

95.54

99.74

47.72 47.72

49.8749.87

XX

z Score: -3 -2 -1 0 1 2 3

34.1334.13 13.5913.592.15 2.15

68.26

95.54

99.74

47.72 47.72

49.8749.87

0.130.13

Raw Score (IQ): 52 68 84 100 116 132 148

X s 100 16,

X s 0 1,

zX X

s

z

100 100

16

0

160z

68 100

16

32

162 0. z

116 100

16

16

1610.

Table C.1Areas under Standard Normal Curve for Values of z

z

Areabetween

X and z

Areabeyond

z Ordinate z

Areabetween

X and z

Areabeyond

Z Ordinate

0.00 .0000 .5000 .3989 0.40 .1554 .3446 .3683

0.01 .0040 .4960 .3989 0.41 .1591 .3409 .3668

0.02 .0080 .4920 .3989 0.42 .1628 .3372 .3653

0.03 .0120 .4880 .3988 0.43 .1664 .3336 .3637

0.04 .0160 .4840 .3986 0.44 .1700 .3300 .3621

0.05 .0199 .4801 .3984 0.45 .1736 .3264 .3605

0.06 .0239 .4761 .3982 0.46 .1772 .3228 .3589

0.07 .0279 .4721 .3980 0.47 .1808 .3192 .3572

0.08 .0319 .4681 .3977 0.48 .1844 .3156 .3555

0.09 .0359 .4641 .3973 0.49 .1879 .3121 .3538

0.10 .0398 .4602 .3970 0.50 .1915 .3085 .3521

0.11 .0438 .4562 .3965 0.51 .1950 .3050 .3503. . . . . . . .. . . . . . . .. . . . . . . .

Beginning on Page 529

Table C.1Areas under Standard Normal Curve for Values of z

z

Areabetween

X and z

Areabeyond

z Ordinate z

Areabetween

X and z

Areabeyond

Z Ordinate

0.00 .0000 .5000 .3989 0.40 .1554 .3446 .3683

0.01 .0040 .4960 .3989 0.41 .1591 .3409 .3668

0.02 .0080 .4920 .3989 0.42 .1628 .3372 .3653

0.03 .0120 .4880 .3988 0.43 .1664 .3336 .3637

0.04 .0160 .4840 .3986 0.44 .1700 .3300 .3621

0.05 .0199 .4801 .3984 0.45 .1736 .3264 .3605

0.06 .0239 .4761 .3982 0.46 .1772 .3228 .3589

0.07 .0279 .4721 .3980 0.47 .1808 .3192 .3572

0.08 .0319 .4681 .3977 0.48 .1844 .3156 .3555

0.09 .0359 .4641 .3973 0.49 .1879 .3121 .3538

0.10 .0398 .4602 .3970 0.50 .1915 .3085 .3521

0.11 .0438 .4562 .3965 0.51 .1950 .3050 .3503. . . . . . . .. . . . . . . .. . . . . . . .

z Score: -3 -2 -1 0 1 2 3

Table C.1Areas under Standard Normal Curve for Values of z

z

Areabetween

X and z

Areabeyond

z Ordinate z

Areabetween

X and z

Areabeyond

Z Ordinate

0.00 .0000 .5000 .3989 0.40 .1554 .3446 .3683

0.01 .0040 .4960 .3989 0.41 .1591 .3409 .3668

0.02 .0080 .4920 .3989 0.42 .1628 .3372 .3653

0.03 .0120 .4880 .3988 0.43 .1664 .3336 .3637

0.04 .0160 .4840 .3986 0.44 .1700 .3300 .3621

0.05 .0199 .4801 .3984 0.45 .1736 .3264 .3605

0.06 .0239 .4761 .3982 0.46 .1772 .3228 .3589

0.07 .0279 .4721 .3980 0.47 .1808 .3192 .3572

0.08 .0319 .4681 .3977 0.48 .1844 .3156 .3555

0.09 .0359 .4641 .3973 0.49 .1879 .3121 .3538

0.10 .0398 .4602 .3970 0.50 .1915 .3085 .3521

0.11 .0438 .4562 .3965 0.51 .1950 .3050 .3503. . . . . . . .. . . . . . . .. . . . . . . .

?

0 +2

Table C.1Areas under Standard Normal Curve for Values of z

z

Areabetween

X and z

Areabeyond

z Ordinate z

Areabetween

X and z

Areabeyond

Z Ordinate

0.00 .0000 .5000 .3989 0.40 .1554 .3446 .3683

0.01 .0040 .4960 .3989 0.41 .1591 .3409 .3668

0.02 .0080 .4920 .3989 0.42 .1628 .3372 .3653

0.03 .0120 .4880 .3988 0.43 .1664 .3336 .3637

0.04 .0160 .4840 .3986 0.44 .1700 .3300 .3621

0.05 .0199 .4801 .3984 0.45 .1736 .3264 .3605

0.06 .0239 .4761 .3982 0.46 .1772 .3228 .3589

0.07 .0279 .4721 .3980 0.47 .1808 .3192 .3572

0.08 .0319 .4681 .3977 0.48 .1844 .3156 .3555

0.09 .0359 .4641 .3973 0.49 .1879 .3121 .3538

0.10 .0398 .4602 .3970 0.50 .1915 .3085 .3521

0.11 .0438 .4562 .3965 0.51 .1950 .3050 .3503. . . . . . . .. . . . . . . .. . . . . . . .

?

0 +2

What is the proportion of scoresin a normal distribution

between the mean and z = +0.52?

f

z 0 0.52

f

z 0 0.52

Answer: 19.85%

What is the proportion of scoresin a normal distribution

between the mean and z = -1.89?

f

z -1.89 0

f

z -1.89 0

Answer: 47.06%

What is the proportion of scoresin a normal distribution

between z = -0.19 and z = +3.02?

f

z -.19 3.02

A B

0

f

z -.19 3.02

A B

0

Therefore, the Area of Interest = 7.53 + 49.87 = 57.40%

Area A = 7.53% (and) Area B = 49.87%

What is the proportion of scoresin a normal distribution

between z = +0.19 and z = +1.12?

f

z .19 1.12

AB* (Area of Interest)

0

f

z .19 1.12

AB* (Area of Interest)

0

Therefore, the Area of Interest* = 36.86 - 7.53 = 29.33%

Area A = 7.53% (and) Area B (total) = 36.86%

f

z .19 1 .12

7.53: A rea A

29.33: A rea B

0

36.86

A B

f

z .19 1 .12

7.53: A rea A

29.33: A rea B

0

36.86

A B

What is the proportion of scoresin a normal distribution

between z = -1.09 and z = -3.02?

f

z -3.02 -1.09

A

B*

0

f

z -3.02 -1.09

A

B*

0

Therefore, the Area of Interest* = 49.87 – 36.21 = 13.66%

Area A = 36.21% (and) Area B (total) = 49.87%

What is the proportion of scoresin a normal distribution above z = +0.87?

f

z .87

A

B

0

f

z .87

A

B

0

Area A (total beyond z = 0.00) = 50.00% (and) Area B = 30.78%

Or – use “Area Beyond z” Column

Locate z = 0.87 and you will find 19.22%

Therefore, the Area of Interest = 50.00 – 30.78 = 19.22%

What is the proportion of scoresin a normal distribution below z = +1.28?

f

z 1.28

AB

0

f

z 1.28

AB

0

We know that Area A (total beyond z = 0.00) = 50.00%We find Area B = 39.97%

Therefore, the Area of Interest = 50.00 + 39.97 = 89.97%

What is the proportion of scoresin a normal distribution above z = -2.00?

f

z -2.00

AB

0

f

z -2.00

AB

0

We know that Area B (total beyond z = 0.00) = 50.00%We find Area A = 47.72%

Therefore, the Area of Interest = 50.00 + 47.72 = 97.72%

What is the proportion of scoresin a normal distribution below z = -0.52?

Area A = 19.85% (and) Area B (total beyond z = 0.00) = 50.00%

Or – use “Area Beyond z” Column

Locate z = 0.52 and you will find 30.15%

Therefore, the Area of Interest = 50.00 – 19.85 = 30.15%

f

z -.52

A

B

0

f

z -.52

A

B

0

Knowing that the mean is 50 and s = 3, what are the rawscores that bound the middle 39% of the normal distribution?

z Score: -3 -2 -1 0 1 2 3

Using Table A – we find thatz = + .51 will bound the middle 39% of the scores.

Area Between Mean and z = .1950 (39/2 = .1950)

-.51 .51

Solve for the lower X:

47.48)53.1(50

)51.0(350)(

X

XzsXX

Solve for the upper X:

53.5153.150

)51.0(350)(

X

XzsXX

Therefore, the two raw scores that bound the middle 39%are 48.47 and 51.53

48.47 51.53