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Let’s Practice! x: { 3, 8, 1}

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x

3

8

1

m

4

4

4

(x-m)

-1

4

-3

S(x-m)2 = SS

(x-m)2

1

16

9

Find s:

S(x-m)2 = 26

S(x-m)2

N

=√(26/3)

√

= 2.94

OR

(Sx)2

__

__

SS = Sx2

Sx = 12

x

3

8

1

x2

9

64

1

Sx2 = 74

N

74 - (144/3) = 26

Then √(26/3) = 2.94

Frequency

1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8

31500 31600 31700 31800 31900 32000 32100 32200 32300 32400 32500

GPA

Proportion

GPA

The Philosophy of Statistics & Standard Deviation

Proportion

1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8

GPA

Standard Deviation and Distribution Shape

IQ

x - x

z =

s

With some simple

calculation we find:

x =140.33

s = 27.91

z(144) = [ 144 – 140.33]/ 27.91

z(198) = [ 198 – 140.33]/ 27.91

z(94) = [ 94 – 140.33]/ 27.91

ID IQ

1128

2155

3135

4134

5144

6101

7167

8198

994

10128

11155

12145

= +0.13, “normal”

= +2.07, abnormally high

= -1.66, low side of normal

“forward”

x - m

z = s

Forward and reverse transforms

“reverse”

x = m+z s

Z- score Raw Score

Raw score Z-score

population

x = x+z s

x - x

z = s

sample

Example: If population μ = 120 and σ =20

Find the raw score associated with a z-score of 2.5

x = 120 + 2.5(20)

x = 120 + 50

x = 170

- z-scores can be used to describe how normal/abnormal scores within a distribution are
- With a normal distribution, there are certain relationships between z-scores and the proportion of scores contained in the distribution that are ALWAYS true.
- 1. The entire distribution contains 100% of the scores
- 2. 68% of the scores are contained within 1 standard deviation below and above the mean
- 3. 95% of the scores are contained within 2 standard deviations below and above the mean

m= 128

s = 32

95%

68%

Z-score -4 -3 -2 -1 0 1 2 3 4

- What percentage of scores are contained between 96 and 160?
- What percentage of scores are between 128 and 160?
- If I have a total of 200 scores, how many of them are less than 128?

m= 128

s = 32

What proportion of people got a z score of 1.5 or higher?

But how do we find areas associated with z-scores that are not simply

0, 1, or 2?

Table A in appendix D contains the areas under the normal curve indexed by Z-score.

Z-score -4 -3 -2 -1 0 1 2 3 4

From these tables you can determine the number

of individuals on either side of any z-score.

1.5

z-score

Examples of AREA C

2.3

-1.7

What percentage of people have a z-score of 0 or greater?

50%

What percentage of people have a z-score of 1 or greater?

15.87%

What percentage of people have a z-score of -2.5 or less?

.62%

What percentage of people have a z-score of 2.3 or greater?

1.07%

What percentage of people have a z-score of -1.7 or less?

4.46%

Examples of AREA B

What percentage of people have a z-score between 0 and 1?

34.13%

What percentage of people have a z-score between 0 and 2.3?

48.93%

What percentage of people have a z-score between 0 and -2.4?

49.18%

What percentage of people have a z-score between 0 and 1.27?

39.80%

What percentage of people have a z-score between 0 and 1.79?

46.33%

What percentage of people have a z-score between 0 and -3.24?

49.94%

Areas which require a COMBINATION of z-scores

What percentage of people have a z-score of 1 or less?

84.13%

Areas which require a COMBINATION of z-scores

What percentage of people have a z-score between -1 and 2.3?

84.13%

Areas which require a COMBINATION of z-scores

What percentage of people have a z-score of 1 or less?

84.13%

Areas which require a COMBINATION of z-scores

What percentage of people have a z-score of 1 or less?

84.13%

Areas which require a COMBINATION of z-scores

What percentage of people have a z-score of 1 or less?

84.13%

Areas which require a COMBINATION of z-scores

What percentage of people have a z-score of 1 or less?

84.13%

m= 128

s = 32

Raw Score 0 32 64 96 128 160 192 224 256

What percentage of people have a z-score of -1.7 or less?

4.46%

What percentage of people have a score of 73.6 or less? 4.46%?

What z-score is required for someone to be in the bottom 4.46%?

-1.7

What score is required for someone to be in the bottom 4.46%?

128 + (-1.7)32

128 - 54.4

73.6 or below

What z-score is required for someone to be in the top 25%?

.68

What z-score is required for someone to be in the top 5%?

1.65

What z-score is required for someone to be in the bottom 10%?

-1.29

What z-score is required for someone to be in the bottom 70%?

.52

What z-score is required for someone to be in the top 50%?

0

What z-score is required for someone to be in the bottom 30%?

-.53

m= 128

s = 32

What percentage of scores fall between the mean and a score of 132?

Here, we must first convert this raw score to a z-score in order to be able to use what we know about the normal distribution. (132-128)/32 = 0.125, or rounded, 0.13.

Area B in the z-table indicates that the area contained between the mean and a z-score of .13 is .0517, which is 5.17%

m= 128

s = 32

What percentage of scores fall between a z-score of -1 and 1.5?

If we refer to the illustration above, it will require two separate areas added together in order to obtain the total area:

Area B for a z-score of -1: .3413

Area B for a z-score of 1.5: .4332

Added together, we get .7745, or 77.45%

m= 128

s = 32

What percentage of scores fall between a z-score of 1.2 and 2.4?

Notice that this area is not directly defined in the z-table. Again, we must use two different areas to come up with the area we need. This time, however, we will use subtraction.

Area B for a z-score of 2.4: .4918

Area B for a z-score of 1.2: .3849

When we subtract, we get .1069, which is 10.69%

m= 128

s = 32

If my population has 200 people in it, how many people have an IQ

below a 65?

First, we must convert 65 into a z-score: (65-128)/32 = -1.96875, rounded = -1.97

Since we want the proportion BELOW -1.97, we are looking for Area C of a z-score of 1.97 (remember, the distribution is symmetrical!) : .0244 = 2.44%

Last step: What is 2.44% of 200?

200(.0244) = 4.88

m= 128

s = 32

What IQ score would I need to have in order to make it to the top 5%?

Since we’re interested in the ‘top’ or the high end of the distribution, we want to find an Area C that is closest to .0500, then find the z-score associated with it.

The closest we can come is .0495 (always better to go under). The z-score associated with this area is 1.65.

Let’s turn this z-score into a raw score: 128 + 1.65(32) = 180.8

A possible type of test question:

80 - 65

A class of 30 students takes a difficult statistics exam. The average grade turns out to be 65. Michael is a student in this class. His grade on the exam is 80. The following is known:

9.80

SS = 2883.2

Assuming that these 30 students make up the population of interest, what is the approximate number of people that did better than Michael on the exam?

SS= 2883.2

2883.2

30

SS

N

m = 65

√

√

N = 30

s =

=√96.11 = 9.80

=

z(80) = (80-65)/9.80 = 1.53

Area C for a z score of 1.53 = .0630, so about 6.3%, or 1.89 people, about 2.