Medical Statistics Medical Statistics (full English class)(full English class)

Ji-Qian Fang

School of Public Health

Sun Yat-Sen University

Chapter 9Chapter 9

Statistical Analysis

For

Measurement Data

profile 轮廓 Symmetric 对称 Skew 偏倚 Positive skew 正偏倚 negative skew 负偏倚 Central position 中心位置 Variability 变异性 Outliers 异常值 Arithmetic Mean 算术均数 Geometric mean 几何均数 Median 中位数 raw data 原始资料 Variance 方差 Standard Deviation 标准差

Numerical DescriptionNumerical Description

Central position (central tendency)

Variation (measure of dispersion)

2. Measures for Average2. Measures for Average

(1) Arithmetic Mean

Based on observed data

Example: Blood sugar

6.2, 5.4, 5.7, 5.3, 6.1, 6.0, 5.8, 5.9

n

X

n

X

n

XXXX

n

ii

n

121 ...

8.58

4.46

8

9.5...4.52.6

X

Based on frequency tableBased on frequency table

n

fX

n

Xf

fff

XfXfXfX

n

iii

n

nn

1

121

2211

...

...

Table 9-3 Frequency table of the urine acid for 20-29 years old males

Concentration (mol/L) Mid-value (xi) Frequency (fi) fi xi fi xi2

270- 277.5 2 550.0 154012.5

285- 292.5 9 2632.5 770006.3

300- 307.5 11 3382.5 1040119.0

315- 322.5 22 7095.0 2288138.0

… … … … …

420-435 427.5 1 427.5 182756.0

Total fi

=150

fi xi

=52470.0

fi xi2

=18518738.0

)/(8.349150

52470

2...32

5.4271...5.29295.2772LmolX

(2) Geometric mean

Example 9-4 See Table 9-4

)lg

(lg)lg...lglg

(lg 1211

21

n

X

n

XXXG

XXXG

n

nn

(3) Median Ranking the values of observation from the smallest to the largest, Median = the value in the middle

Based on raw dataBased on raw data Example 1: (7 values)

120,123,125,127,128,130,132

Median =127 

Example 2: (8 values)

118,120,123,125,127,128,130,132

Median=(125+127)/2=126

21.19)81%50308(95

25)%50(? Lfn

f

i

21.6921.1950?5050 P

Frequency 95Lf 154%50 n 176 ffL

Interval 50L ?5050 P 75 iL

Based on Frequency TableBased on Frequency TableTable 9-5 Frequency table of urine acid for children under 6 years old

Urine lead (mmol/L)

(1)

Frequency f

(2)

Cumulative frequency f

(3)

Cumulative relative frequency

(4)=(3)/n

0- 27 27 8.77

25- 54 81 26.30

50- 95 176 57.14

75- 55 231 75.00

… … … …

175- 5 308 100.0

Total 308 ( fi) -- --

SymmetricSymmetric

RBC (10 /L)of 130 normal male adults in a place

3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 RBC(10 /L)

0

5

10

15

20

25

30 Frequency

Positive skewPositive skew

Hair Mercury (ug/g) of 238 normal adults

0.5 0.9 1.3 1.7 2.1 2.5 2.9 3.3 3.7 4.1 Hg(ug/g)

0

10

20

30

40

50

60

70 Frequency

Think aboutThink about

How to calculate P25? based on raw data? based on frequency table?How to calculate P75? based on raw data? based on frequency table?

SummarySummary

1. Mean:

Suitable to symmetric distribution.

2. Geometric mean:

Suitable to positive skew distribution

3. Median:

Suitable to all kinds of data,

but poor attribute for further analysis

3. Measures for variability3. Measures for variability

(1) Range

Range= Maximum - Minimum

Based on only two observations, it ignores the

observations within the two extremes.

The greater the number of observations, the

greater the range is.

(2) Inter- quartile range

Lower Quartile: 25 percentile

Upper Quartile: 75 percentile

Difference between two Quartiles

= Upper Quartile - Lower Quartile

= 13.120 – 8.083 = 5.037

15.48)27%25308(54

252525 P

0.100)176%75308(55

257575 P

Table 9-5 Frequency table of urine acid for children under 6 years old Urine lead (mmol/L)

(1) Frequency f

(2) Cumulative frequency f

(3) Cumulative relative frequency

(4)=(3)/n 0- 27 27 8.77

25- 54 81 26.30 50- 95 176 57.14 75- 55 231 75.00 … … … …

175- 5 308 100.0 Total 308 ( fi) -- --

(3)Variance and Standard Deviation

The mean of squared deviation

Standard deviation (SD)

n

X

22 )(

1

)(

1

)(

22

22

n

n

XX

n

XXs

2 2ss

Example 9-8 The weight of male infant

2.85,2.90, 2.96, 3.00, 3.05, 3.18

(1)

117.001368.0

01368.05

0684.0

16

)99.218.3(...)99.290.2()99.285.2( 2222

s

s

1

)(1

2

n

XXs

n

ii

2

/)(1

2

1

2

n

nXXs

n

ii

n

ii

117.016

6/)94.17(709.53 2

s

(2)

(4)(4)Coefficient of VariationCoefficient of Variation

X

sCV

Example 9-10 Variation of height and variation of weight

Mean

(1)

Standard deviation

(2)

Coefficient of

Variation (%)

(3)=(2)/(1)

Height 171.21(cm) 5.34(cm) 3.12

Weight 59.72 (kg) 4.16 (kg) 6.97