3 Descriptive Stats

8/9/2019 3 Descriptive Stats

1/27

Numeric Summaries and

Descriptive Statistics


2/27

populations vs. samples

we want to describe both samples and

populations

the latter is a matter of inference


3/27

outliers

minority cases, so different from the majoritythat they merit separate consideration

are they errors?

are they indicative of a different pattern? think about possible outliers with care, but

beware of mechanical treatments

significance of outliers depends on yourresearch interests


4/27


5/27

summaries of distributions

graphic vs. numeric

graphic may be better for visualization

numeric are better for statistical/inferential

purposes

resistance to outliers is usually an advantage

in either case


6/27

general characteristics

kurtosis

leptokurtic platykurtic

[peakedness]


7/27

right

(positive)skew

left

(negative)

skew

skew (skewness)


8/27


9/27

central tendency

measures of central tendency

provide a sense of the value expressed by

multiple cases, over all

mean

median

mode


10/27

mean

center of gravity

evenly partitions the sum of all

measurement among all cases; average of

all measures

n

x

x

n

i

i== 1


11/27

crucial for inferential statistics

mean is not very resistant to outliers

a trimmed mean may be better fordescriptive purposes

mean pro and con


12/27

meanrim diameter (cm)

unit 1 unit 2

12.6 16.211.6 16.4

16.3 13.8

13.1 13.2

12.1 11.3

26.9 14.0

9.7 9.0

11.5 12.5

14.8 15.613.5 11.2

12.4 12.2

13.6 15.5

11.7

n 12 13

total 168.1 172.6

total/n 14.0 13.3

unit 1 unit 2

9 26

2524

23

22

21

20

19

18

17

3 16 24

15 56

14.0== 8 14 0

651 13 28 ==13.3

641 12 25

65 11 237

107 9 0

R: mean(x)


13/27

trimmed meanrim diameter (cm)

unit 1 unit 29.7 9.0

11.5 11.2

11.6 11.3

12.1 11.7

12.4 12.2

12.6 12.5

13.1 13.213.5 13.8

13.6 14.0

14.8 15.5

16.3 15.6

26.9 16.2

16.4

n 10 11

total 131.5 147.2

total/n 13.2 13.4

unit 1 unit 29 26

25

24

23

22

21

2019

18

17

3 16 24

15 56

8 14 0

13.2== 651 13 28 ==13.4

641 12 25

65 11 237

10

7 9 0

R: mean(x, trim=.1)


14/27

median

50th percentile

less useful for inferential purposes

more resistant to effects of outliers


15/27

median

rim diameter (cm)

unit 1 unit 2

9.7 9.0

11.5 11.2

11.6 11.312.1 11.7

12.4 12.2

12.6 12.5

12.9


16/27

mode

the most numerous category

for ratio data, often implies that data have

been grouped in some way can be more or less created by the grouping

procedure

for theoretical distributionssimply thelocation of the peak on the frequencydistribution


17/27

i

solate

dscatte

rs

hamle

ts

villag

es

r

egion

alcen te

rs

r

egionalcen te

rs

modal class = hamlets

1.0 1.5 2.0 2.5


18/27

dispersion

measures of dispersion

summarize degree of clustering of cases, esp.

with respect to central tendency

range

variance

standard deviation


19/27

range

unit 1 unit 2

9.7 9.0

11.5 11.2

11.6 11.3

12.1 11.7

12.4 12.2

12.6 12.513.1 13.2

13.5 13.8

13.6 14.0

14.8 15.5

16.3 15.6

26.9 16.216.4

unit 1 unit 2

* 9 26

| 25

| 24

| 23

| 22

| 21

| 20

| 19| 18

| 17

| 3 16 24 *

| 15 56 |

| 8 14 0 |

| 651 13 28 |

| 641 12 25 |

| 65 11 237 |

| 10 |

* 7 9 0 *

would be better to use midspread

R: range(x)


20/27

variance

analogous to average deviation of cases from

mean

in fact, based on sum of squared deviations fromthe meansum-of-squares

( )

1

1

2

2

==

n

xx

s

n

i

i

R: var(x)


21/27

variance

computational form:

1

/

2

11

2

2

=

==

n

nxx

s

n

i

i

n

i

i


22/27

note: units of variance are squared

this makes variance hard to interpret

ex.: projectile point sample:

mean = 22.6 mm

variance = 38 mm2

what does this mean???


23/27

standard deviation

square root of variance:

( )

1

1

2

==

n

xx

s

n

i

i

1

/

1

2

1

2

=

= =

n

nxx

s

n

i

n

i

ii


24/27

standard deviation

units are in same units as base measurements

ex.: projectile point sample:mean = 22.6 mm

standard deviation = 6.2 mm

mean +/- sd (16.428.8 mm) should give at least some intuitive sense of where most

of the cases lie, barring major effects of outliers


25/27

rim diameter (cm)

unit 1 unit 2

12.6 16.2

11.6 16.4

16.3 13.8

13.1 13.2

12.1 11.3

26.9 14.0

9.7 9.011.5 12.5

14.8 15.6

13.5 11.2

12.4 12.2

13.6 15.5

11.7

mean: 14.0 13.3

n: 12 13

unit 1 unit 2

-1.4 2.9

-2.4 3.1

2.3 0.5

-0.9 -0.1

-1.9 -2.0

12.9 0.7

-4.3 -4.3

-2.5 -0.8

0.8 2.3

-0.5 -2.1

-1.6 -1.1

-0.4 2.2

-1.6

unit 1 unit 2

1.98 8.54

5.80 9.75

5.25 0.27

0.83 0.01

3.64 3.91

166.20 0.52

18.56 18.29

6.29 0.60

0.63 5.40

0.26 4.31

2.59 1.16

0.17 4.94

2.49

sum of sq.: 212.19 60.20

variance: 19.29 5.02

stand. dev.: 4.39 2.24


26/27

trimmed dispersion measures

variance and sd are even more sensitive to

extreme values (outliers) than the mean

why??

you can calculate a trimmed version of the

variance simply by eliminating cases from the

tails, and calculating the variance in the normalway


27/27

trimmed standard deviation

trimmed sd is calculated differently

sT = trimmed standard deviation

n = number of cases in untrimmed batchs2w = variance of trimmed (winsorized) batch

nT = number of cases in the trimmed batch

1

)1( 2

=T

W

T

n

sn

s

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3 Descriptive Stats