Practice

Descriptive statisticsTablesGraphs

2920 4050 2300 3400 3912 3234 3110 4045 3310 2594 2187 2700 3300 4250 3350 2750 2342 2350 4680 4123 3470 2084 2211 2450 4070 4360 3700 2495 3203 2950 3250 2920 2630 2466 3100 2780 4265 1588 3700 4990 3510 3020 4174 2950 3380 2607 3225 2410 3450 1800 3110 3860 4100 2540 2960 1790 3610 2225 3232 2625

Birthweights of 60 infants are given below:

Find

Mean, median and mode

Q1, Q3

Standard deviation and coefficient of variation

60

2625254031102920

60n

60

1 i

n

1 ixx ii

2.313660

188167x

Average birthweight of infants is approximately 3136 gr.

Arithmetic Mean

x

1588 2410 2750 3110 3400 4050 1790 2450 2780 3203 3450 4070 1800 2466 2920 3225 3470 4100 2084 2495 2920 3232 3510 4123 2187 2540 2950 3234 3610 4174 2211 2594 2950 3250 3700 4250 2225 2607 2960 3300 3700 4265 2300 2625 3020 3310 3860 4360 2342 2630 3100 3350 3912 4680 2350 2700 3110 3380 4045 4990

Ordered data are:

Median

30th value

31st value

1588 2410 2750 3110 3400 4050 1790 2450 2780 3203 3450 4070 1800 2466 2920 3225 3470 4100 2084 2495 2920 3232 3510 4123 2187 2540 2950 3234 3610 4174 2211 2594 2950 3250 3700 4250 2225 2607 2960 3300 3700 4265 2300 2625 3020 3310 3860 4360 2342 2630 3100 3350 3912 4680 2350 2700 3110 3380 4045 4990

n=60

5.30261

2160

21

n

i Median is 3110

Multiple modes exist.

Mode

valuethQ 25.154

1601

1588 2410 2750 3110 3400 4050 1790 2450 2780 3203 3450 4070 1800 2466 2920 3225 3470 4100 2084 2495 2920 3232 3510 4123 2187 2540 2950 3234 3610 4174 2211 2594 2950 3250 3700 4250 2225 2607 2960 3300 3700 4265 2300 2625 3020 3310 3860 4360 2342 2630 3100 3350 3912 4680 2350 2700 3110 3380 4045 4990

The difference between 15th and 16th observations is 54.

54/4=13.5

Q1=2540+13.5=2553.5

We add 13.5 to 15th observation to find Q1

valuethQ 75.454

)160(33

1588 2410 2750 3110 3400 4050 1790 2450 2780 3203 3450 4070 1800 2466 2920 3225 3470 4100 2084 2495 2920 3232 3510 4123 2187 2540 2950 3234 3610 4174 2211 2594 2950 3250 3700 4250 2225 2607 2960 3300 3700 4265 2300 2625 3020 3310 3860 4360 2342 2630 3100 3350 3912 4680 2350 2700 3110 3380 4045 4990

Q3=3610+67.5=3677.5

The difference between 45th and 46th observations is 90.

3*(90/4)=67.5

We add 67.5 to 45th observation to find Q3

1

2

2

nn

xx

s

ii

3623,689,082625254031102920 22222 ix

188,1672625254031102920 ix

,88935,406,819)( 2 ix

grs 754.37160

60/889,819,406,35083,689,623

Average distance of observations from the mean is 754.37 gr.

If infant weight were given as kg instead of gr, what would be the result of the mean and standart deviation.

Find the result without using any formulations.

0.75437(1,000)754.37/s754.37

1362.3000,1/2.136,32.136,3

2

s

xx

%241003136

37.754100

x

sCV

The amount variation between observations in this distribution is 24% in terms of the mean.

L=1588

H=4990

Range=H-L=4990-1588=3402

#of classes=8

Class Int.=3402/8=425.25426

To find lower limits:

1588

1588+426=2014

2014+426=2440...4114+426=4570

To find upper limits:

2014-1=2013

2013+426=2439

2439+426=2865...4569+426=4995

Lower-upper limit

1588-2013

2014-2439

2440-2865 . . . 4570-4995

CLASSES Lower Limit Upper Limit Frequency bi

Relative Frequency

Class Mid-Point

True Class Limit

Less than ... Frequency

Less than ... Rel. Freq.

1587.5 0 0.00

1588 2013 3 -3 5.0 1800.5 2013.5 3 5.00

2014 2439 8 -2 13.33 2226.5 2439.5 11 18.33

2440 2865 11 -1 18.33 2652.5 2865.5 22 36.67

2866 3291 14 0 23.33 3078.5 3291.5 36 60.00

3292 3717 11 1 18.33 3504.5 3717.5 47 78.33

3718 4143 7 2 11.67 3930.5 4143.5 54 90.00

4144 4569 4 3 6.67 4356.5 4569.5 58 96.67

4570 4995 2 4 3.33 4756.5 TOTAL 60 100.00 4995.5 60 100.00

Mean

4.314260

18854460

5.478225.222685.180038

1

n

mfx i

ii

Cn

bfAx i

ii

8

1

or

42660

)4(2...)2(8)3(35.3078

x

4.314242660

95.3078

Median

Cf

Cfn

LM

2

9.310842614

222

60

5.2865

M

Since the class midpoint of interval which has the largest frequency is 3078.5, mode is 3078.5

Measures of position

What are the 25th, 75th percentiles and the median?

TCLxTCLx

TCLxx

xx

xx

23

2

23

21

True

Class Limit Less than ... Frequency

Less than ... Rel. Freq.

1587.5 0 0.00

2013.5 3 5.00 2439.5 11 18.33 2865.5 22 36.67 3291.5 36 60.00 3717.5 47 78.33 4143.5 54 90.00 4569.5 58 96.67 4995.5 60 100.00

True

Class Limit Less than ... Frequency

Less than ... Rel. Freq.

1587.5 0 0.00

2013.5 3 5.00 2439.5 11 18.33 2865.5 22 36.67 3291.5 36 60.00 3717.5 47 78.33 4143.5 54 90.00 4569.5 58 96.67 4995.5 60 100.00

25=x1

36.67-18.33 25-18.33

x=?

x-2439.52865.5-2439.5

2439.5 18.33

X 25

2865.5 36.67

TCLxTCLx

TCLxx

xx

xx

23

2

23

21

5.24395.2865

5.2439

33.1867.36

33.1825

xX=2594,3

True

Class Limit Less than ... Frequency

Less than ... Rel. Freq.

1587.5 0 0.00

2013.5 3 5.00 2439.5 11 18.33 2865.5 22 36.67 3291.5 36 60.00 3717.5 47 78.33 4143.5 54 90.00 4569.5 58 96.67 4995.5 60 100.00

75=x1x=?

3291.5 60.0

X 75

3717.5 78.33

5.32915.3717

5.3291

0.6033.78

0.6075

x1.3640 x

P75=Q3=3640.1

75% of observations lie below 3640.1.

3291.5 60.0

X 75

3717.5 78.33

If the central %90 were considered as normal birthweight, what are the two limits for normality?

x1 x2

Normal b.w90%

5% 5%

True

Class Limit Less than ... Frequency

Less than ... Rel. Freq.

1587.5 0 0.00

2013.5

3

5.00 4143.5 54 90.00 4569.5 58 96.67 4995.5 60 100.00

.

.

.

.

.

.

.

.

.

x1 x2

Normal b.w90%

5% 5%

x1

95

x2

5.41435.4569

5.4143

0.9067.96

0.9095

xgr 8.4462 x

1

8

1

22

nn

mfmf

s i

iiii

624356925)5.4782(2)5.2226(8)5.1800(3 2228

1

2

i

iimf

63554883994)188544( 2

28

1

i

iimf

73559

6063554883994

624356925

s

Standart deviation

1

)(8

1

22

nn

bfbf

Cs i

iiii

or

177)4(2...)2(8)3(3 2222iibf

81)4(2...)2(8)3(3)( 22 iibf

gr 73559

)60/81(177426

s

Coefficient of variation

%4.231004.3142

735100

x

sCV

The amount variation between observations in this distribution is 23.4% in terms of the mean.

A study was conducted to analyze the relation between coronary heart disease (CHD) and body weight status (obese and normal). Observed values are as follows:

Body weight status

CHD

Total

Yes No

n % n %Normal 4 8 46 92 50

Obese 10 25 30 75 40

Total 14 16 76 84 90

8

25

92

75

01020304050

60708090

100

Normal Obese

Body Weight Status

%

Yes

No

CHD

Distribution of CHD according to body weight status

825

9275

01020304050

60708090

100

Normal Obese

Body Weight Status

%

No

Yes

CHD

Distribution of CHD according to body weight status

Example

In a goiter hyper-endemic area, thyroid gland and its hormones T3, T4 and TSH (Thyroid Stimulating Hormone) were studied in 225 subjects in three groups

• Hypothyroid • Euthyroid• Hyperthyroid

T4 levels of three study groups are observed as follows:

T4 levels of the study groups

Hypothyroid Group 1,1 1,7 2,1 2,7 3,41,1 1,7 2,1 2,7 3,41,1 1,7 2,1 2,8 3,41,1 1,7 2,4 2,9 3,41,1 1,9 2,4 2,9 3,41,2 1,9 2,4 2,9 3,41,2 2,0 2,4 2,9 3,41,4 2,0 2,4 2,9 3,51,5 2,0 2,5 3,0 3,51,5 2,0 2,5 3,1 3,71,5 2,0 2,5 3,1 3,71,5 2,0 2,6 3,1 3,91,5 2,0 2,6 3,4 3,91,6 2,1 2,7 3,4 3,91,6 2,1 2,7 3,4 4,0

Euthyroid Group1,6 4,5 5,5 6,9 8,42,1 4,8 5,6 7,1 8,62,4 4,8 5,7 7,1 8,62,4 4,8 5,7 7,1 8,72,8 4,8 6,0 7,1 9,63,0 4,8 6,1 7,2 9,63,0 4,9 6,1 7,5 9,63,4 5,0 6,1 7,6 9,63,4 5,0 6,4 8,0 9,73,8 5,0 6,4 8,0 9,94,0 5,1 6,4 8,1 9,94,0 5,1 6,6 8,1 10,44,2 5,1 6,7 8,1 10,94,2 5,5 6,7 8,4 11,44,3 5,5 6,9 8,4 12,5

Hyperthyroid Group 9,4 11,2 12,4 13,9 15,09,4 11,2 12,6 13,9 15,09,5 11,2 12,7 14,0 15,09,6 11,3 12,7 14,2 15,1

10,0 11,8 12,7 14,2 15,610,0 11,8 13,0 14,2 15,610,0 11,8 13,2 14,2 16,210,3 12,0 13,2 14,2 16,310,3 12,1 13,4 14,4 16,410,3 12,1 13,5 14,6 16,710,9 12,1 13,5 14,7 16,711,0 12,1 13,5 14,7 16,711,0 12,1 13,5 14,7 16,711,0 12,4 13,6 14,8 17,911,2 12,4 13,6 14,8 18,1

Descriptive Statistics of the study groups for T4

GROUPS Hypothyroid Euthyroid Hyperthyroid Mean 2,46 6,38 13,16 Median 2,40 6,10 13,20 Mode 3,40 4,80 12,1 & 14,2 Std. Deviation 0,82 2,37 2,14 Variance 0,68 5,62 4,58 CV 33,33 37,15 16,26 Minimum 1,10 1,60 9,40 Maximum 4,00 12,50 18,10 Range 2,90 10,90 8,70 Quartiles 1st (25%) 1,70 4,80 11,30 3rd (75%) 3,10 8,10 14,70

Classification of the T4 values in hypothyroid group

Min = 1,1

Max = 4,0

Range = 4,0 – 1,1 = 2,9

# of classes = 8

Class Int. = 2,9/8 = 0,36 0,4

4,

05

3,65

3,25

2,85

2,45

2,05

1,65

1,25

Nu

mb

er

of

Pa

tien

ts

16

14

12

10

8

6

4

2

0

Histogram of T4 values in hypothyroid group

Box plot of T4 values in three groups

HyperthyroidEuthyroidHypothyroid

T4

20

15

10

5

0

Error bars of T4 values in three groups

HyperthyroidEuthyroidHypothyroid

Me

an

+-

1 S

D T

4

18

16

14

12

10

8

6

4

2

0

Hypothyroid Euthyroid Hyperthyroid

T4

20

15

10

5

0

Scatter plot of T4 values in three groups