Berghold, IMI, MUG

Exploring, summarizing and presenting data

Berghold, IMI, MUG

Example

Patient Nr Gender Age Weight Height PAVK-Grade Walking Distance

Physical Functioning Scale Total Cholesterol Triglycerides

01 m 65 90 185 II b 200 70 179 8402 m 70 75 170 II b 100 45 185 5903 m 98 110 186 II b 150 75 175 8704 f 50 75 162 II b 20 10 215 19605 m 79 78 163 IV 20 00 221 33006 f 68 92 164 III 200 55 200 18907 f 56 68 161 II b 50 25 185 3908 m 63 82 168 IV 10 00 196 7509 m 70 72 177 III 50 15 187 17410 f 79 60 155 III 100 30 177 10511 m 51 48 180 II b 200 50 239 8812 m 63 72 166 II b 100 10 184 15313 f 70 74 158 II b 200 45 137 29414 m 55 85 181 II b 50 25 183 10115 m 46 98 174 II b 100 80 124 16016 f 62 67 151 IV 100 20 183 8617 f 60 77 158 II b 100 15 189 12018 f 85 68 159 II b 30 25 195 7619 m 67 87 173 II b 20 10 211 12120 m 80 95 181 III 5 00 201 15821 f 54 90 160 III 10 00 216 17322 m 61 75 179 II b 100 50 219 4723 f 57 62 160 IV 40 25 208 9224 m 68 79 178 III 50 25 190 14925 m 81 92 170 II b 50 55 248 369

Berghold, IMI, MUG

Scales

• Nominal scale

• Ordinal scale

• Numerical scale

Berghold, IMI, MUG

Nominal Scale

The values of any two study units can be classified either as identical or non identical

• hair colour• place of birth• blood group• Binary (dichotomous) variables: gender, rhesus factor, ...

Berghold, IMI, MUG

Ordinal Scale

Observation are still classified but some observations have "more" orare "greater than" other observations.

• school grades• stage of breast cancer• side effect of a drug

(mild, average, severe)• pain-scores ...

Berghold, IMI, MUG

Numerical Scale

continuous (e.g. age, height - measurements) discrete (e.g. number of fractures, number of children - counts)

• weight• body temperature• blood pressure• serum cholesterol ...

Berghold, IMI, MUG

Types of Data

• Qualitative data – categorical variable

• Nominal scale

• Ordinal scale

• Quantitative data

• Discrete variables

• Continuous variables

Berghold, IMI, MUG

Examples

• Protein measured in urine

• Spontaneous urine – using test strips(neg., pos.: +,++,+++)

• 24 hours sample of urineprotein g/24hours

• Smoking

• Consumed tobacco g/day

• Number of smoked cigarettes per day

• Non-smoker, smoker

Berghold, IMI, MUG

Criteria - measurements

• Reliability

• Validity

• Ease of Use

Berghold, IMI, MUG

Reliability

reliable unreliable

●●●

● ●

●

●●●● ●

●●

●

●

●

● ●●

●

Berghold, IMI, MUG

Validity

Valid Not valid

●●●

● ●

●

●●●●

●●●

● ●

●

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Berghold, IMI, MUG

Descriptive Statistics

• Exploring and presenting data in form of graphs

• Summarizing - data reduction (mean, variance etc.)

• Presenting data in form of tables

Berghold, IMI, MUG

Frequency

• Qualitative data

• absolute and relative frequency

• Quantitative data – define class intervals

• Determine the number of class intervals

• There should be enough class intervals to show the shapeof the distribution but not too many that minor fluctuationsare noticeable.

Berghold, IMI, MUG

Graphs

• Barchart

• Piechart

• Histogram

• Box-and-whisker plot

• Scatterplot

• Time series plot

• ...

Berghold, IMI, MUG

Barchart

number of decayed teeth in pupils

25 33,3 33,3

26 34,7 68,0

9 12,0 80,0

7 9,3 89,3

2 2,7 92,0

4 5,3 97,3

1 1,3 98,7

1 1,3 100,0

75 100,0

0

1

2

3

4

5

6

7

total

frequencies percentagecumulativepercentage

decayed teeth in pupils

number of decayed teeth in pupils

76543210

abso

lute

freq

uenc

y

30

20

10

0

Berghold, IMI, MUG

Piechart

PAVK-Grade

IV24%

III26%

II b50%

Berghold, IMI, MUG

Histogram and cumulative distribution

FT3

1-1,5 1,5-2 2-2,5 2,5-3 3-3,5 3,5-4 4-4,5 4,5-5 5-5,5 5,5-6

rel.

frequ

ency

0,00

0,05

0,10

0,15

0,20

0,25

0,30

0,35

FT3

1-1,5 1,5-2 2-2,5 2,5-3 3-3,5 3,5-4 4-4,5 4,5-5 5-5,5 5,5-6

F(x)

0,0

0,2

0,4

0,6

0,8

1,0

Berghold, IMI, MUG

Histogram 1

TRIGLYCERIDES (mg / 100 ml)

240

230

220

210

200

190

180

170

160

150

140

130

120

110

100908070

frequ

ency

12

10

8

6

4

2

0

Std.dev. = 38,83 Mean = 129

N = 80,00

Berghold, IMI, MUG

Histogram 2

TOTAL CHOLESTEROL (mg / 100 ml)

800

750

700

650

600

550

500

450

400

350

300

250

200

150

100

frequ

ency

50

40

30

20

10

0

Std.dev. = 92,46 Mean = 220

N = 80,00

Berghold, IMI, MUG

Histogram 3

SYSTOLIC BLOOD PRESSURE (mmHg)

220200180160140120100

frequ

ency

30

25

20

15

10

5

0

Std.dev. = 21,97 Mean = 162

N = 80,00

Berghold, IMI, MUG

a) unimodal b) skewed positively c) skewed negatively

c) bimodal e) trapezoid f) truncated

g) L- shaped h) J - shaped i) U - shaped

TypesTypes of Distributionof Distribution

Berghold, IMI, MUG

Scatterplot

LDL

250200150100500

HD

L

200

150

100

50

0

Berghold, IMI, MUG

Summarizing Data

Common statistics used to summarize data and describe certainattributes of a set of data.

• Variance, standard deviation• Range• Interquartile range

• Mean• Median, quantile• Mode

• Measures of dispersion:the spread of data

• Measures of location:the central tendency

Berghold, IMI, MUG

Mean

Mean = arithmetic mean

Note:

• The mean is sensitive to extreme values

∑=

=n

iix

nx

1

1

Berghold, IMI, MUG

Example

Values: 1, 2, 30

3021

11:

=xmean

( ) 113

3021=

++=x

Berghold, IMI, MUG

Variance, standard deviation

The variance of a data set is the arithmetic mean of the squareddifferences between the observations and the mean.

The standard deviation is primarily used to describe data. It is thesquare root of the variance.

In many circumstances the large majority (about 95%) of a set of observations will be within two standard deviations of the mean(depends on the shape of the distribution – normal distribution) – normal range

( )∑=

−−

=n

ii xx

ns

1

22

11

2ss =

Berghold, IMI, MUG

Example

The number of cows 4 farmers own in 3 villages

standard deviation

mean

observations

s = 10.0s = 0s = 1.8

0, 0, 0, 205, 5, 5, 53, 6, 7, 4

village 3village 2village 1

5=x 5=x 5=x

Berghold, IMI, MUG

Time Series Plot

R-TCIInduction of Anaesthesia

Time Course (min)

-2 0 2 5 10 15

0

2

4

6

20

40

60

80

100

120

140

all data points: n = 30

Berghold, IMI, MUG

Geometric mean

Geometric mean

The geometric mean is generally used with data measured on a logarithmic scale

The logarithm of the geometric mean is equal to the mean of thelogarithms of the observations

nnxxxG ...21=

∑=

=n

i

i

nxG

1

loglog

Berghold, IMI, MUG

Median

Median

The median is the central value of the distribution

• if n is odd

• if n is even

( )( )2/1~

+= nxx

( ) ( )( )12/2/21~

++= nn xxx

Berghold, IMI, MUG

Mean - Median

Example: n = 3values: 1, 2, 30

3021

2~:

=xmedian

11=xmean:

Berghold, IMI, MUG

skewed negatively

MoMo xxMeMe < < <<

skewed positively

MoMoxx MeMe <<<<

Skewness by mean, median and mode

Berghold, IMI, MUG

Quantiles

The α-Quantile

The median is only a special case that is based on rank order.

α-quantile xα: that at least α % of measurements are smaller orequal than the value xα.

• 1st quartile (α = 0.25)• 2nd quartile or median• 3rd quartile (α = 0.75)

• Percentiles (centiles)

Berghold, IMI, MUG

Quantiles

The α-Quantile xα

Calculation:

α*n, rankorder m

• if α*n is not an integer, than m is the next integer following α*nand xα= x(m).

• if α*n is an integer, than m = α*n and

21++

= mm xxxα

Berghold, IMI, MUG

Quantiles

Berghold, IMI, MUG

Data:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

5, 2, 2, 6, 7, 2, -40, 2, 3, 2, 1, 1, 12, 3, 4, 0

-40, 0, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 5, 6, 7, 12

Interquartile range = Q3 – Q1 = 3

Me = 2.0 Q3 = 4.5Q1 = 1.5

Quantiles

Berghold, IMI, MUG

Interquartile Range

Interquartile range

The 50% central range is sometimes used to describe variability

IQR = 3rd quartile - 1st quartile

Berghold, IMI, MUG

Box-and-Whisker Plot

minimum

1st quartile

median

3rd quartile

maximum

Berghold, IMI, MUG

Example Box-and-Whisker Plot

51170100 49152104N =

age groups

13-16 yrs9-12 yrs5-8 yrs

one-

seco

nd-c

apac

ity(L

)

6

5

4

3

2

1

0

Gender

female

male

Berghold, IMI, MUG

In bunten Bildernwenig Klarheit,viel Irrtum undein wenig Wahrheit.

J. W. v. Goethe

15

0

5

20

25

0

546

213

Berghold, IMI, MUG

Numerical Presentation

Data summary should not be by the mean (median) alone, but someindication of variability should also be provided.

E.g.: "... the mean diastolic blood pressure was 102.3 mm Hg (SD 11.9)."

mean: quote it to one extra decimal place compared withthe raw data (depending on amount of data)

standard deviation: display with same precision as meanor with one more decimal place.

Presentation of Results

Berghold, IMI, MUG

Tables

Mean (SD)

Age 67,8 (10,8) Total Cholesterol 213,3 (41,1)

Triglycerides 129,4 (72,0)

frequency % Gender f 35 (46)

m 41 (54)

PAVK-Grade II b 38 (50) III 20 (26) IV 18 (24)