Formula Compute a standard deviation with the Raw-Score Method Previously learned the deviation...

Formula

Compute a standard deviation with the Raw-Score Method

• Previously learned the deviation formula– Good to see “what's going on”

• Raw score formula – Easier to calculate than the deviation formula– Not as intuitive as the deviation formula

• They are algebraically the same!!

Raw-Score Formula

Note: This is the formula for both and S

Step 1: Create a table

Coffee X

X2

4 10 22 2 6

Step 2: Square each value

CoffeeX

X2

4 1610 10022 4842 46 36

Step 3: Sum

CoffeeX

X2

4 1610 10022 4842 46 36

X = 44 X2 = 640

Step 4: Plug in values

N = 5

X = 44

X2 = 640


N = 5

X = 44

X2 = 640

55


N = 5

X = 44

X2 = 640

5544


N = 5

X = 44

X2 = 640

5544

640

Step 5: Solve!

5544

6401936

Step 5: Solve!

5544

6401936387.2

Step 5: Solve!

5544

6401936387.250.56

Answer = 7.11

Practice

• Use the raw score formula and find the standard deviation of:

6, 3, 4, 10, 8

X X2

6 363 94 16

10 1008 64

X = 31 X2 = 225

N = 5

X = 31

X2 = 225

5531

225

5544

2251936192.2

2.56 =

Ŝ

• What if we want to use a sample standard deviation to estimate the population ?

• We need to make one small change to the formula to do this

• You need to make the s an “unbiased estimator”

Ŝ

• To do that you use Ŝ– This provides an estimate of the populations

variability

Remember

Just “ - 1”

Ŝ

Remember

S =

Just “ - 1”

Ŝ =-1

Why?

• The first formula is biased -- its answer tends to be too small

• Don’t worry about why -- unless you want too!!

Practice!

• Below is data from 5 people in this class. What is the estimated standard deviation of all the students in this class? Use the Ŝ raw score formula.

• Neuroticism scores

12, 15, 22, 10, 9

X X2 12 144 15 225 22 484 10 100 9 81 X = 68 X2 = 1034

N = 5

X = 68

X2 = 1034

5 - 1568

1034

4544

10341936924.8

5.22 =

Variance

• The last step in calculating a standard deviation is to find the square root

• The number you are fining the square root of is the variance!

2 = population variance

Ŝ 2 = sample variance used to estimate 2

Variance

S 2, 2 =

Ŝ 2 =

Variance

- 1Ŝ 2 =

S 2, 2 =

There are 12 different formulas!

• Standard Deviation– Deviation Formula , S, Ŝ– Raw Formula , S, Ŝ

• Variance– Deviation Formula 2, S 2, Ŝ 2 – Raw Formula 2, S 2, Ŝ 2

Review -- Important Formulas

• Standard Deviation -- Deviation Formula

=

Ŝ =


• Standard Deviation -- Deviation Formula


• Variance -- Deviation Formula

2 =

Ŝ 2 =


• Variance -- Deviation Formula

2


• Standard Deviation -- Raw Formula

and S =

Ŝ =- 1


• Variance -- Raw Formula

2 and S2 =

Ŝ 2 =- 1

How to know which to use

• 1) Does the question want a standard deviation or a variance (most of the time standard deviations are used)

• 2) Is the group of scores a sample or population?

• 3) If it’s a sample, do you want to generalize the findings to a population?

Practice

• You are interested in how citizens of the US feel about the president. You asked 8 people to rate the president on a 10 point scale. Describe how the country feels about the president -- be sure to report a measure of central tendency and the standard deviation.

8, 4, 9, 10, 6, 5, 7, 9

Central Tendency

8, 4, 9, 10, 6, 5, 7, 9

4, 5, 6, 7, 8, 9, 9, 10

Mean = 7.25

Median = (4.5) = 7.5

Mode = 9

Standard Deviation• Want to use Ŝ

Standard Deviation X X2

8 64

4 16

9 81

10 100

6 36

5 25

7 49

9 81

= 58 = 452

• Want to use Ŝ

-1


8 64

4 16

9 81

10 100

6 36

5 25

7 49

9 81

= 58 = 452

• Want to use Ŝ

-1

45258 8

8 - 1


8 64

4 16

9 81

10 100

6 36

5 25

7 49

9 81

= 58 = 452

• Want to use Ŝ

-1

58452 8

8 - 1


8 64

4 16

9 81

10 100

6 36

5 25

7 49

9 81

= 58 = 452

• Want to use Ŝ

-1

452 420.5

7


8 64

4 16

9 81

10 100

6 36

5 25

7 49

9 81

= 58 = 452

• Want to use Ŝ

-1

2.12

Boxplots

• The boxplot graphically displays three different characteristics of the distribution– Extreme scores– Interquartile range– Median

Boxplot

100N =

NEUR

40

30

20

10

0

Boxplot

100N =

NEUR

40

30

20

10

0

Interquartile range

25th - 75th percentile

Boxplot

100N =

NEUR

40

30

20

10

0

Extreme Scores

Boxplot

100N =

NEUR

40

30

20

10

0

Median

Boxplot

100N =

NEUR

40

30

20

10

0

Skew -- Look at the “whiskers” to determine if the distribution is skewed

Create a boxplot

• Create a boxplot with this data set

2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99

Create a boxplot


2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99

Median =

25th =

75th =

Lowest =

Highest =

Create a boxplot


2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99

Median = 22.5

25th = 6

75th = 62

Lowest = 2

Highest = 99

12N =

VAR00004

120

100

80

60

40

20

0

-20

Neuroticism

130130N =

MNEURCNEUR

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

Extraversion

130130N =

MEXTRACEXTRA

5.0

4.5

4.0

3.5

3.0

2.5

2.0

106120

15272

Conscientiousness

128128N =

MCONCCON

5

4

3

2

1

0

99999N =

VAR00006VAR00005VAR00004VAR00003VAR00002

120

100

80

60

40

20

0

-20

A

B

C

D

E

Which distribution has a positive skew?

99999N =


120

100

80

60

40

20

0

-20

A

B

C

D

E

Which distribution has a negative skew?

99999N =


120

100

80

60

40

20

0

-20

A

B

C

D

E

Which distribution is most compact?

99999N =


120

100

80

60

40

20

0

-20

A

B

C

D

E

Which distribution has a median close to 25?

99999N =


120

100

80

60

40

20

0

-20

A

B

C

D

E

Which distribution is most symmetrical?

99999N =


120

100

80

60

40

20

0

-20

A

B

C

D

E

Which distribution has has the largest range?

Practice

• Make a box plot using the simple frequency distribution of Satisfaction with Life scores on page 27

Practice

• Page 67

• 3.25

• 3.28

Practice

• 3.25

• Use S “hat”

• 33.16

Practice

• 3.28

• Use S “hat”

• A = 1.90• B = 1.55

• B is more consistent

Formula Compute a standard deviation with the Raw-Score Method Previously learned the deviation...

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