basic statistical concepts and methods

Basic Statistical Concepts and Methods

Ahmed-Refat AG RefatFOM-ZU

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Definition of StatisticsStatistics is the science of dealing with numbers.It is used for collection, summarization, presentation and analysis of data. Statistics provides a way of organizing data to get information on a wider and more formal (objective) basis than relying on personal experience (subjective).

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Uses of medical statisticsMedical statistics are used in1-Planning, monitoring and evaluating community health care programs.2-Epidemiological research studies.3-Diagnosis of community health problems.4-Comparison of health status and diseases in different countries and in one country over years. 5-To form standards for the different biological measurements as weight, height.6-To differentiate between diseased and normal groups.

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Types of data Any aspect of an individual that is measured, is called variable. Variables are either 1-Quantitative or 2-Qualitative.1- Quantitative data: it is numerical data.Discrete data: are usually whole numbers, such as number of cases of certain disease, number of hospital beds (no decimal fraction).Continuous data: it implies the measurement on a continuous scale e.g. height, weight, age (a decimal fraction can be present).

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1- Quantitative data. Quantitative data: it is numerical data.Tow TypesA- Discrete data: are usually whole numbers, such as number of cases of certain disease, number of hospital beds (no decimal fraction).B- Continuous data: it implies the measurement on a continuous scale e.g. height, weight, age (a decimal fraction can be present).

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2- Qualitative data Qualitative data: It is non numerical data and is subdivided into Two Types: A- Categorical : data are purely descriptive and imply no ordering of any kind such as sex, area of residence. B- Ordinal data: are those which imply some kind of ordering like - Level of education: - Socio-economic status: - Degree of severity of disease:

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Presentation Of DataThe first step in statistical analysis is to present data in an easy way to be understood. The two basic ways for data presentation are: Tabular presentation.Graphical presentation

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TabulationSome rules for the construction tables: 1- The table must be self-explanatory. 2- Title: written at the top of table to define precisely the content, the place and the time.3- Clear heading of the columns and rows and units of measurements 4- The size of the table depends on the number of classes. Usually lie between 2 and 10 rows or classes. Its selection depends on the form of data and the requirement of the distribution. Too small may obscure some information and too long will not differ from raw data.

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Types of tablesFor Qualitative data, draw a simple table eg., List Table : count the number of observations ( frequencies) in each category. For Quantitative data, we have to form a frequency distribution TableList tables (2 columns- one value for each measured variable) Frequency Distribution Tables

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Types of tables:List: A table consisting of two columns, the first giving an identification of the observational unit and the second giving the value of variable for that unit.Example : number of patients in each hospital department are Medicine 100 patients Surgery 80 ENT 28 Ophthalmology 30

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Frequency Distribution tablesFDTs are used for presentation of qualitative ( and quantitative Discrete) data, By recording the number of observations in each category. These counts are called frequencies. .No Classes .. No Intervals

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Frequency Distribution tablesFDT for Quantitative Continuous Data consists of a series of classes (intervals) together with the number of observations ( frequency) whose values fall within the interval of each class.

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Frequency Distribution tablesEXAMPLE (1) Assume we have a group of 20 individuals whose blood groups were as followed : A , AB, AB, O, B, A, A, B, B, AB, O, AB, AB, A, B, B, B, A, O, A. We want to present these data by table.????? Type of data >>>>>>

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How to Construct a Frequency Distribution tablesFour Steps Title, Table, No , %1-Put a title2- Draw Columns & Rows3-Enumerate the individuals in each category 4-Calculate The relative frequency (%)

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How to Construct a Frequency Distribution tablesFour Steps 1-Put a title eg., Distribution of the studied individuals according to their blood group. 2- Draw a table (Columns & Rows),First column > Studied Variable Blood Group, 2nd column heading >Frequency-Number3rd column heading > Percentage %

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Frequency Distribution tables3-Enumerate the individuals in each blood group , i.e. individuals with blood group A are 6 and those with blood group B are 6 , AB are 5 and blood group O are 3. Make sure that the total number of individuals in all blood groups is 20 (the number of the studied group).

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Frequency Distribution tables4-Calculate The relative frequency (%) of each blood group by dividing the frequency of that group over the total number of individuals and multiplied by 100 i.e. the percentage of group A = 6 / 20 x 100, and the same for group AB = 5 / 20 x 100 and group O = 3 / 20 x 100. The final table will be :

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Frequency Distribution tables What is Your Conclusion?

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Frequency Distribution tablesWe can conclude from this table that blood groups A & B are the most common groups and the rarest is group O (depending on the percentage of each group).So presenting data in table is beneficial in deducing facts and simplify information than raw data.

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Frequency Distribution tablesEXAMPLE (3) : The Following data are Systolic Blood Pressure measurements (mmHg) of 30 patients with hypertension. Present these data in frequency table:

150, 155, 160, 154, 162, 170, 165, 155, 190, 186, 180, 178, 195, 200, 180,156, 173, 188, 173, 189, 190, 177, 186, 177, 174, 155, 164, 163, 172, 160.??????? Type of Data

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Frequency Distribution tablesFour Steps 1-Put a title eg., Frequency distribution of blood pressure measurements (mmHg) among a group of hypertensive patients. 2- Draw a table (Columns & Rows),First column > Studied Variable Blood Pressure-mm Hg, 2nd column heading >Frequency-Number3rd column heading > Percentage %

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Frequency Distribution tables3-In the first column we have to classify blood pressure into categories or classes because we have a large sample (N=30) and the measured variable is of continuous type (not discrete as in the previous examples).

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Frequency Distribution tablesconstruction of classesCalculate the Range of observation: subtract the lowest value of blood pressures from the highest value (the highest was 200 and the lowest was 150) the difference is 50.Determine the number of classes and the width class intervals Let class interval be 10 , so we will have 50/10 = 5 classes. Enumerate the Frequency By Tally MethodsCalculate the Exact Frequncy & Relative frequency

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Frequency Distribution tablesconstruction of classesDetermine the the number of classes You want to display ( not too few ~2 and too frequent >8. it is a matter of trial and sense !!!Let class interval= 10 mmHg , we will have 5 classes. If we choose 5 mmHg as a class interval-width we will obtain 10 classes (too long table). We must maintain constant width for all intervals. Choose the upper and lower limits of the class start with the lowest value i.e 150 List the intervals in order every 10

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2-Graphical PresentationThe diagram should be: Simple Easy to understandSave a lot of words Self explanatory Has a clear title indicating its content Fully labeledThe y axis (vertical) is usually used for frequency

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2-Graphical PresentationGraphic presentations used to illustrate and clarify information. Tables are essential in presentation of scientific data and diagrams are complementary to summarize these tables in an easy, attractive and simple way.

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Graphical Presentation 1-Bar chartIt is used for presenting discrete or qualitative data. It represent the measured value (or %) by separated rectangles of constant width and its lengths proportional to the frequencyType: >>>Simple , >>> Multiple, >>>Components

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Graphical Presentation 1-Bar chart- Simple

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Chart2

25.8

25.3

26.5

mean maternal age

The studied groups

Mean age in years

Mean maternal age of three studied groups

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Group IGroup IIGroup III

birth weight279526752731

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0

0

0

&R&"Arial,Italic"Results - 160

birth weight

groups

mean birth weight/gms

Classification of patients according to mean birth weight in gms

Sheet8

group Igroup IIgroup III

NVD877377

CS132723

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00

00

00


NVD

CS

percentage of cases

Classifcation of patients according to mode of delivery (cesarean section CS, , normal vaginal delivery NVD)

Sheet7


True + ve02711

False + ve0319

True - ve2600

False - ve400


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0000

0000

0000

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True + ve

False + ve

True - ve

False - ve

groups

Comparison between groups regarding cervicovaginal fetal fibronectin FFN

Sheet6


sample delivery interval21.24515.0521.203


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0

0

0

&R&"Times New Roman,Italic"&12Results - 147

sample delivery interval

groups

delivery interval

Sample delivery interval between groups

Sheet1


mean maternal age25.825.326.5

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0

0

0


mean maternal age

The studied groups

Mean age in years

Mean maternal age of three studied groups

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mean gestational age34.234.133.5

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0

0

0


mean gestational age

Mean gestational age in weeks

Mean gestational age at presentation of the studied groups

Sheet5

Group Igroup IIgroup III

closed56.7806.6

123.316.716.7

> 2203.376.7

07063.316.7

< 3023.336.730

>306.7053.3


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000

000

000

000

000

000


Group I

group II

group III

cervical dilatation cervical effacement

Percentage of cases

Cervical state among the studied groups

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delivery age37.23536.2537.123

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0

0

0


delivery age

Mean gestational age in weeks

Mean gestational age at delivery

Sheet3


primigravida606757

multigravada403343

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00

00

00

&R&"Arial,Italic"Results- 140

primigravida

multigravada

Percent

Classification of the studied patients accoding to parity

Graphical Presentation 1-Bar chartMultiple bar chart: Each observation has more than one value represented, by a group of bars. Percentage of males and females in different countries, percentage of deaths from heart diseases in old and young age, mode of delivery (cesarean or vaginal) in different female age groups.

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Graphical Presentation 1-Bar chart-MultipleMultiple bar chart:CancerAnemiaMalesFemales

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Graphical Presentation 1-Bar chart Component bar chart : subdivision of a single bar to indicate the composition of the total divided into sections according to their relative proportion.

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Graphical Presentation 1-Bar chart Component bar chart : For example two countries are compared in their socio-economic standard of living, each bar represent one country, the height of the bar is 100, it is divided horizontally into 3 components (low, moderate and high classes) of socio-economic classes (SE), each class is represented by different color or shape.

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Graphical Presentation 1-Bar chart- Component

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Chart2

603010

106030

low

moderate

high

percentage of population

Comparison between Egypt and USA in socio-economic standard of living

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EgyptUSA

low6010

moderate3060

high1030

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603010

106030

low

moderate

high



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Graphical Presentation 2-Pie diagram:Consist of a circle whose area represents the total frequency (100%) which is divided into segments. Each segment represents a proportional composition of the total frequency.

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Graphical Presentation 2-Pie diagram:

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Chart1

50

30

10

10

Percentage of causes of child death in Egypt

Sheet1

EgyptUSA

low6010

moderate3060

high1030

group 1group 2

10155

202530

304025

401510

>50530

diarrhea50

chest infection30

congenital10

accident10

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603010

106030

low

moderate

high



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50

30

10

10


Graphical Presentation 3- Histogram:It is very similar to the bar chart with the difference that the rectangles or bars are adherent (without gaps). It is used for presenting class frequency table (continuous data). Each bar represents a class and its height represents the frequency (number of cases), its width represent the class interval.

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Graphical Presentation 3- Histogram:

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Chart1

10

14

25

18

15

8

number

height in cm

number of individuals

Distribution of studied group according to their height

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EgyptUSAnumber

low6010100-10

moderate3060110-14

high1030120-25

group 1group 2130-18

10155140-15

202530150-8

304025

401510

>50530

diarrhea50

chest infection30

congenital10

accident10

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Sheet2

603010

106030

low

moderate

high



Sheet3

10

14

25

18

15

8

number

height in cm

number of individuals

Distribution of studied group according to their height

50

30

10

10


Graphical Presentation 4 -Frequency PolygonDerived from a histogram by connecting the mid points of the tops of the rectangles in the histogram.The line connecting the centers of histogram rectangles is called frequency polygon. We can draw polygon without rectangles so we will get simpler form of line graph.A special type of frequency polygon is the Normal Distribution Curve.

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Graphical Presentation 5 - Scatter diagram- It is useful to represent the relationship between two numeric measurements, each observation being represented by a point corresponding to its value on each axis

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This scatter diagram showed a positive or direct relationship between NAG and albumin/creatinine among diabetic patients

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Chart14

5.5

7.7

9.9

11.7

6.8

8.2

15.4

8.7

11.2

6.8

11.7

12.4

10.8

26.2

8.8

16.6

15.3

28.9

23.9

14.1

11.9

29.3

32.6

19.8

32.5

NAG

albumin creatinine ratio

NAG

Correlation between NAG and albumin creatinine ratio in group of early diabetics

Chart1

8.29

10.33

9.86

5.85

HgA1c

mean level of HgA1c

Fig.(1) : Comparison between diabetic cases and the control group in HgA1c

Chart3

181.36246.76

165.88252.28

241.72317.76

78.2796.53

FBS

RBS

mean blood sugar

Fig.(2) : Comparison between diabetic cases and the control group in fasting and random blood sugar levels

Chart2

Chart4

51.3

55.25

64.32

46.37

SA

mean level of SA

Fig.(3) : Comparison between diabetic cases and the control group in the level of serum sailic acid (SA)

Chart5

2.43

2.78

2.65

1.87

PRA

mean level of PRA

Fig.(4) : Comparison between diabetic cases and the control group in serum prorenine (PRA)

Chart6

24.92

28.67

33.06

13.63

NO

mean level of NO

Fig.(5) : Comparison between diabetic cases and the control group in serum nitric oxide (NO)

Chart7

0.18

0.2

0.22

0.11

alb/creat

mean level of alb/creatinine

Fig.(6) : Comparison between diabetic cases and the control group in albumin/creatinine ratio in urine

Chart8

15.47

18.81

19.96

7.77

NAG

mean level of NAG

Fig.(7) : Comparison between diabetic cases and the control group in N-acetyl-glucosamidase (NAG)

Chart9

19.28

25.04

29.2

12.6

serum urea

mean level of serum urea

Fig.(8) : Comparison between diabetic cases and the control group in serum urea

Chart10

0.76

0.85

0.82

0.67

serum creatinine

mean serum creat. level

Fig.(9) : Comparison between diabetic cases and the control group in serum creatinine level

Chart11

15.3

6.9

6.9

16.2

30.2

21.2

15.3

10

14.6

18.8

7.8

16.5

10.7

11.2

19.9

39.2

15.8

33.8

52.3

33.8

39.6

49.2

42.1

44.5

51.2

NO

albumin/creatinine ratio

NO

Fig.(10) : Correaltion between nitric oxide (NO) and albumin/creatinine ratio in group A (early diabetics)

Chart12

25.9

49.2

41.9

52.3

33.9

38.8

40.8

32.9

37.7

33.9

37.2

50.8

51.3

49.4

41.2

52.8

60.2

71.2

81.2

69.3

59.9

62.4

65.2

72.3

70.8

SA

albumin/creatinine ratio

SA

Fig.(11): Correlation between seru sailic acid (SA) and albumin/creatinine ratio in group A (early diabetics)

Chart13

1.45

1.41

1.6

1.92

2.1

1.56

1.3

1.41

2.12

1.5

1.92

2.12

1.49

1.44

1.6

2.7

1.72

3.4

3.1

2.92

5.2

3

6.1

4.2

3.4

PRO

albumin /creatinine ratio

PRO

Fig.(12) : Correlation between prorenine level (PRO) and albumin /creatinine ratio in group A (early diabetics)

Chart15

6

7.2

9.1

8.4

7.6

6

10.3

5

10.5

7.8

7.8

8

7.8

9

10.3

7.5

8.9

11

7

6.9

10.2

9

11

5.9

8

HBA1C


HgA1c

Fig.(14) : Correlation between HBA1C and albumin creatinine ratio in group A (early diabetics)

Sheet1

HgA1cFBSRBSSAPRA

group A8.29group A181.36246.76group A51.3group A2.43

group B10.33group B165.88252.28group B55.25group B2.78

group C9.86group C241.72317.76group C64.32group C2.65

group D5.85group D78.2796.53group D46.37group D1.87

NOalb/creatNAGserum urea

group A24.92group A0.18group A15.47group A19.28

group B28.67group B0.2group B18.81group B25.04

group C33.06group C0.22group C19.96group C29.2

group D13.63group D0.11group D7.77group D12.6

serum creatinine

group A0.76PROALB/CREAT

group B0.851.410.1

group C0.821.450.08

group D0.671.920.14

2.120.14

1.60.1

1.410.12

1.490.15

1.720.2

3.40.25

2.120.12

1.440.15

1.920.1

2.10.1

1.560.1

6.10.31

1.30.1

1.50.12

5.20.28

4.20.33

3.10.25

1.60.15

30.3

2.920.26

2.70.18

3.40.33

ALB/CREATNOALB/CREATSAALB/CREATPROALB/CREATNAGALB/CREATHBA1C

0.0815.30.0825.90.081.450.085.50.086

0.16.90.149.20.11.410.17.70.17.2

0.16.90.141.90.11.60.19.90.19.1

0.116.20.152.30.11.920.111.70.18.4

0.130.20.133.90.12.10.16.80.17.6

0.121.20.138.80.11.560.18.20.16

0.115.30.140.80.11.30.115.40.110.3

0.12100.1232.90.121.410.128.70.125

0.1214.60.1237.70.122.120.1211.20.1210.5

0.1218.80.1233.90.121.50.126.80.127.8

0.147.80.1437.20.141.920.1411.70.147.8

0.1416.50.1450.80.142.120.1412.40.148

0.1510.70.1551.30.151.490.1510.80.157.8

0.1511.20.1549.40.151.440.1526.20.159

0.1519.90.1541.20.151.60.158.80.1510.3

0.1839.20.1852.80.182.70.1816.60.187.5

0.215.80.260.20.21.720.215.30.28.9

0.2533.80.2571.20.253.40.2528.90.2511

0.2552.30.2581.20.253.10.2523.90.257

0.2633.80.2669.30.262.920.2614.10.266.9

0.2839.60.2859.90.285.20.2811.90.2810.2

0.349.20.362.40.330.329.30.39

0.3142.10.3165.20.316.10.3132.60.3111

0.3344.50.3372.30.334.20.3319.80.335.9

0.3351.20.3370.80.333.40.3332.50.338

Sheet2

Sheet3

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5.5

7.7

9.9

11.7

6.8

8.2

15.4

8.7

11.2

6.8

11.7

12.4

10.8

26.2

8.8

16.6

15.3

28.9

23.9

14.1

11.9

29.3

32.6

19.8

32.5

NAG


NAG

Correlation between NAG and albumin creatinine ratio in group of early diabetics

In negative correlation, the points will be scattered in downward direction, meaning that the relation between the two studied measurements is controversial i.e. if one measure increases the other decreases. As shown in the following graph

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Chart5

0.15

0.37

0.38

0.39

0.4

0.41

0.42

0.43

0.44

0.45

0.46

0.47

0.48

0.49

0.5

0.5

0.52

0.53

0.54

0.55

0.56

0.57

0.66

0.66

0.67

0.67

0.68

0.68

0.69

0.7

0.7

0.71

0.71

0.72

0.73

0.78

0.87

0.88

0.89

0.9

0.9

0.9

0.91

0.91

0.92

RI

baby weight in kg

RI

Correlation between Doppler velocimetry (RI) and baby birth weight

Chart1

0

1

0

1

0

0

1

0

1

0

0

0

0

2

0

0

1

0

0

1

2

2

2

1

2

1

2

2

2

2

1

1

1

0

1

1

1

1

1

0

0

0

1

0

2

oedema

Inhibin level

Presence of oedema

Fig. ( ) : Correlation between inhibin level and presence of oedema (0 = no oedema , 1 = tibial oedema ; 2=generalized oedema)

Chart2

0

0

0

0

0

0

0

0

0

0

0

0

0

3

3

0

3

2

3

2

3

2

4

4

3

4

4

3

4

4

3

4

4

3

4

3

3

2

3

2

4

3

4

2

4

protein

inhibin level

degree of proteinuria

Fig.( ) : Correlation between inhibin level and proteinuria (0=no proteinuria ,1=+ ;2=++ ,3=+++,4=++++)

Chart3

0.49

0.48

0.45

0.44

0.39

0.5

0.41

0.43

0.37

0.47

0.42

0.38

0.5

0.9

0.87

0.4

0.91

0.46

0.7

0.9

0.89

0.91

0.88

0.9

0.7

0.92

0.71

0.67

0.68

0.69

0.15

0.66

0.73

0.56

0.53

0.67

0.52

0.68

0.57

0.66

0.78

0.55

0.71

0.54

0.72

RI

inhibin level

RI

Fig.( ) : Correlation between inhibin level and Doppler velocimetry (RI)

Chart4

3.1

3.9

3.9

3.9

3.9

3.9

3

3.3

3.4

3.9

3.9

3.9

3.5

2.5

2.5

3.9

2.2

3.2

2.8

2.1

2.3

3.3

2.6

3.1

3.3

2.8

2.1

2.5

2.7

2.5

2.9

2.1

3.6

3.2

3.4

2.9

3.6

3

2.9

2.8

3

3.1

2.9

3.5

3

baby wt

inhibin level

baby weight

Fig. ( ) : Correlation between inhibin level and baby birth weight

Chart6

0.15

0.37

0.38

0.39

0.4

0.41

0.42

0.43

0.44

0.45

0.46

0.47

0.48

0.49

0.5

0.5

0.52

0.53

0.54

0.55

0.56

0.57

0.66

0.66

0.67

0.67

0.68

0.68

0.69

0.7

0.7

0.71

0.71

0.72

0.73

0.78

0.87

0.88

0.89

0.9

0.9

0.9

0.91

0.91

0.92

RI

Apgar score 1

RI

Fig.( ) : Correlation between Doppler velocimetry (RI) and Apgar score at 1 minute

Chart7

0.15

0.37

0.38

0.39

0.4

0.41

0.42

0.43

0.44

0.45

0.46

0.47

0.48

0.49

0.5

0.5

0.52

0.53

0.54

0.55

0.56

0.57

0.66

0.66

0.67

0.67

0.68

0.68

0.69

0.7

0.7

0.71

0.71

0.72

0.73

0.78

0.87

0.88

0.89

0.9

0.9

0.9

0.91

0.91

0.92

I.U.GR

RI

Fig.( ) : Correlation between Doppler velocimetry (RI) and intrauterine GR (o= -VE IUGR , 1 = +VE IUGR)

Sheet1

inhibinoedemaproteinRIbaby wtGRAPGA1inhibinoedema

7100.413182.20

6.8000.53.9063.61

9.5000.43.9185.80

2.2000.493.1086.21

3.6100.483.9076.30

5.8000.453.9066.80

6.3000.393.90871

11.1000.463.21870

6.2100.443.90871

8000.383.9077.40

7000.433.3067.60

7.4000.473.90880

7100.373.41880

8000.53.5088.22

7.6000.423.9079.20

20.1010.563.2169.50

18.4130.152.90710.21

15.9220.73.31811.10

20.9120.523.60312.10

24.1010.662.81713.21

32.3120.712.90613.62

20.7130.533.40413.62

20.7120.672.917142

45220.7231815.71

32.3020.543.50715.92

22.8110.6830816.61

19.9120.733.61717.62

27.9030.553.103182

23.2120.572.906182

27.5030.78317182

8.2240.92.51218.41

9.2030.872.51318.91

10.2140.912.20419.91

14240.882.61620.10

13.6240.892.31520.71

18.9140.662.11720.71

18230.672.51420.91

18240.682.71322.81

13.2140.92.11423.21

18240.692.50824.10

13.6240.913.31627.50

12.1040.72.81627.90

16.6140.922.81432.31

15.7140.93.11232.30

17.6240.712.114452

inhibinproteininhibinRIinhibinbaby wtAPGA1RIGRRI

2.202.20.492.23.170.1500.15

3.603.60.483.63.980.3710.37

5.805.80.455.83.970.3800.38

6.206.20.446.23.980.3900.39

6.306.30.396.33.980.410.4

6.806.80.56.83.980.4110.41

7070.417370.4200.42

7070.4373.360.4300.43

7070.3773.480.4400.44

7.407.40.477.43.960.4500.45

7.607.60.427.63.980.4610.46

8080.3883.980.4700.47

8080.583.570.4800.48

8.238.20.98.22.580.4900.49

9.239.20.879.22.560.500.5

9.509.50.49.53.980.500.5

10.2310.20.9110.22.230.5200.52

11.1211.10.4611.13.240.5300.53

12.1312.10.712.12.870.5400.54

13.2213.20.913.22.130.5500.55

13.6313.60.8913.62.360.5610.56

13.6213.60.9113.63.360.5700.57

144140.88142.670.6610.66

15.7415.70.915.73.170.6610.66

15.9315.90.715.93.370.6710.67

16.6416.60.9216.62.840.6710.67

17.6417.60.7117.62.180.6800.68

18.43180.67182.530.6810.68

18.54180.68182.780.6900.69

18.64180.69182.580.710.7

19.5318.40.1518.42.960.710.7

19.6418.90.6618.92.160.7100.71

20.1419.90.7319.93.640.7110.71

20.1320.10.5620.13.280.7210.72

20.7420.70.5320.73.470.7310.73

20.7320.70.6720.72.970.7810.78

20.9320.90.5220.93.630.8710.87

22.8222.80.6822.8360.8810.88

23.2323.20.5723.22.950.8910.89

24.1224.10.6624.12.820.910.9

27.5427.50.7827.5340.910.9

27.9327.90.5527.93.120.910.9

32.3432.30.7132.32.940.9100.91

32.3232.30.5432.33.560.9110.91

454450.7245340.9210.92

baby wtRI

2.90.15

3.40.37

3.90.38

3.90.39

3.90.4

30.41

3.90.42

3.30.43

3.90.44

3.90.45

3.20.46

3.90.47

3.90.48

3.10.49

3.90.5

3.50.5

3.60.52

3.40.53

3.50.54

3.10.55

3.20.56

2.90.57

2.80.66

2.10.66

2.90.67

2.50.67

30.68

2.70.68

2.50.69

3.30.7

2.80.7

2.90.71

2.10.71

30.72

3.60.73

30.78

2.50.87

2.60.88

2.30.89

2.50.9

2.10.9

3.10.9

2.20.91

3.30.91

2.80.92

Sheet2

Sheet2

0.15

0.37

0.38

0.39

0.4

0.41

0.42

0.43

0.44

0.45

0.46

0.47

0.48

0.49

0.5

0.5

0.52

0.53

0.54

0.55

0.56

0.57

0.66

0.66

0.67

0.67

0.68

0.68

0.69

0.7

0.7

0.71

0.71

0.72

0.73

0.78

0.87

0.88

0.89

0.9

0.9

0.9

0.91

0.91

0.92

RI

baby weight in kg

RI


Sheet3

Graphical Presentation 6- Line graph: it is diagram showing the relationship between two numeric variables (as the scatter) but the points are joined together to form a line (either broken line or smooth curve)

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Normal Distribution Curve

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Normal Distribution curveNDC is a Graphical Presentation of any Quantitative Biologic Variables

The Normal Distribution Curve is the frequency polygon of a quantitative variable measured in large number. It is a form of presentation of frequency distribution of biologic variables such as weights, heights, hemoglobin level and blood pressure or any continuous data.

It occupies a major role in the techniques of statistical analysis.

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Characteristics of Normal Distribution curve1- It is bell shaped, continuous curve.2-It is symmetrical i.e. can be divided into two equal halves vertically.3-The tails never touch the base line but extended to infinity in either direction.4-The mean, median and mode values coincide5-It is described by two parameters: arithmetic mean determine the location of the center of the curve and standard deviation represents the scatter around the mean.

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Areas under the normal curveX 1 SD = 68% of the area on each side of the mean.X 2 SD = 95% of area on each side of the mean.X 3 SD = 99% of area on each side of the mean.

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Skewed dataIf we represent a collected data by a frequency polygon graph and the resulted curve does not simulate the normal distribution curve (with all its characteristics)

then these data are not normally distributed

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Causes of Skewed CurveNot Normally Distributed DataThe curve may be skewed to the right or to the left sideThis is because The data collected are from: certain heterogeneous group or from diseased or abnormal population

therefore the results obtained from these data can not be applied or generalized on the whole population.

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NDC can be used in distinguishing between normal from abnormal measurements. Example:If we have NDC for hemoglobin levels for a population of normal adult males with mean SD = 11 1.5

If we obtain a hemoglobin reading for an individual = 8.1 and we want to know if he/she is normal or anemic. If this reading lies within the area under the curve at 95% of normal (i.e. mean 2 SD)he /she will be considered normal. If his reading is less then he is anemic.

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The normal range for hemoglobin in this example will be: the higher level of hemoglobin: 11 + 2 ( 1.5 ) =14. The lower hemoglobin level 11 2 ( 1.5 ) = 8.i.e the normal range of hemoglobin of adult males is from 8 to 14. our sample (8.1 ) lies within the 95% of his population. therefore this individual is normal because his reading lies within the 95% of his population.

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Data SummarizationTo summarize data, we need to use one or two parameters that can describe the data. Measures of Central tendency which describes the center of the data and the Measures of Dispersion, which show how the data are scattered around its center.

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Measures of central tendencyVariable usually has a point (center) around which the observed values lie. These averages are also called measures of central tendency. The three most commonly used averages are:The arithmetic mean:The MedianThe Mode

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1- The arithmetic mean:the sum of observation divided by the number of observations: x = x nWhere : x = mean denotes the (sum of)x the values of observation n the number of observation

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1- The arithmetic mean:Example: In a study the age of 5 students were: 12 , 15, 10, 17, 13Mean = sum of observations / number of observationsThen the mean X = (12 + 15 + 10 + 17 + 13) / 5 =13.4 years

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Calculation of Mean For frequency Distribution DataIn case of frequency distribution data we calculate the mean by this equation:

x = fx nwhere f = frequency for example : we want to calculate the mean incubation period of this group.

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Calculation of Mean For frequency Distribution Data

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Calculation of Mean For frequency Distribution Data with class intervals If data is presented in frequency table with class intervals we calculate mean by the same equation summation of f x1 /n , x1 denotes the midpoint of class interval. Example : calculate the mean of blood pressure of the following group :

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2- MedianIt is the middle observation in a series of observation after arranging them in an ascending or descending manner.The rank of median for is (n + 1)/2 if the number of observation is odd and n/2 if the number is even

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2- Median Calculate the median of the following data 5, 6, 8, 9, 11 n = 5~ Odd!!-The rank of the median = n + 1 / 2 i.e. (5+ 1)/ 2 = 3 The median is the third value in these groups when data are arranged in ascending (or descending) manner. - So the median is 8 (the third value)

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2- Median- If the number of observation is even, the median will be calculated as follows:e.g. 5, 6, 8, 9 n = 4 - The rank of median = n / 2 i.e. 4 / 2 = 2 .The median is the second value of that group. If data are arranged ascendingly then the median will be 6 and if arranged descendingly the median will be 8 therefore the median will be the mean of both observations i.e. (6 + 8)/2 =7.

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2- Median For simplicity we can apply the same equation used for odd numbers i.e. n + 1 / 2. The median rank will be 4 + 1 /2 = 2 i.e. the median will be the second and the third values i.e. 6 and 8, take their mean = 7.

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3- ModeThe most frequent occurring value in the data is the mode and is calculated as follows: Example: 5, 6, 7, 5, 10. The mode in this data is 5 since number 5 is repeated twice. Sometimes, there is more than one mode and sometimes there is no mode especially in small set of observations.

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3- ModeExample : 20 , 18 , 14, 20, 13, 14, 30, 19. There are two modes 14 and 20.Example : 300, 280 , 130, 125 , 240 , 270 . Has no mode.

Unimodal Bimodal Nomodal

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Advantages and disadvantages of the measures of central Tendency:

- Mean: is the preferred CTM since it takes into account each individual observation but its main disadvantage is that it is affected by the extreme valus of observations.

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Advantages and disadvantages of the measures of central Tendency:

Median: it is a useful descriptive measure if there are one or two extremely high or low values.

-Mode: is seldom used.

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Measures of DispersionThe measure of dispersion describes the degree of variations or scatter or dispersion of the data around its central values: (dispersion = variation = spread = scatter).Range - RVariance -V Standard Deviation - SD Coefficient of Variation -COV

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1- Range:is the difference between the largest and smallest values. is the simplest measure of variation.disadvantages, it is based only on two of the observations and gives no idea of how the other observations are arranged between these two. Also, it tends to be large when the size of the sample increases

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2- Variance If we want to get the average of differences between the mean and each observation in the data,we have to reduce each value from the mean and then sum these differences and divide it by the number of observation. V = (mean xi) / n

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2- Variance Variance V = (mean x) / nThe value of this equation will be equal to zero

because the differences between each value and the mean will have negative and positive signs that will equalize zero on algebraic summation.

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2- VarianceTo overcome this zero we square the difference between the mean and each value so the sign will be always positive . Thus we get: V = (mean x)2 / n - 1

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3- Standard Deviation SDThe main disadvantage of the variance is that it is the square of the units used. So, it is more convenient to express the variation in the original units by taking the square root of the variance. This is called the standard deviation (SD). Therefore SD = V i.e. SD = (mean x)2 / n - 1

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4- Coefficient of variation CoVThe coefficient of variation expresses the standard deviation as a percentage of the sample mean.C. V = SD / mean * 100 C.V is useful when, we are interested in the relative size of the variability in the data. Example : if we have observations 5, 7, 10, 12 and 16. Their mean will be 50/5=10. SD = (25+9 +0 + 4 + 36 ) / (5-1) = 74 / 4 = 4.3C.V. = 4.3 / 10 x 100 = 43%

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ExampleCalculate the mean, variance, SD and CV From the following measurements 5, 7, 10, 12 and 16. Mean= 5+7+10+12+16/5=10. SD = (25+9 +0 + 4 + 36 ) / (5-1) = 74 / 4 = 4.3C.V. = 4.3 / 10 x 100 = 43%

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ExampleAnother observations are 2, 2, 5, 10, and 11. Their mean = 30 / 5 = 6SD = (16 + 16 + 1 + 16 + 25)/(5 1) = 74 / 4 = 4.3 C.V = 4.3 /6 x 100 = 71.6 %Both observations have the same SD but they are different in C.V. because data in the first group is homogenous (so C.V. is not high), while data in the second observations is heterogenous (so C.V. is high).

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ExampleExample: In a study where age was recorded the following were the observed values: 6, 8, 9, 7, 6. and the number of observations were 5.Calculate the mean, SD and range, mode and median.- The mean = sum of observation / their number

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ExamplesThe variance = Sum of the squared differences (mean minus observation) / number of observations. (7.2 6)2 + (7.2 8)2 + (7.2 9)2 + (7.2 7)2 + (7.2 6)2 / 5 1. which is equal to (1.2)2 + (- 0.8)2 + (- 1.8) 2 +(0.2)2 + (1.2)2 / 4 = 1.7 - So the variance = 1.7

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Examples- The S.D. = 1.7 = 1.3- Range = 9 6 = 3 - The mode is 6- The median is : first we have to arrange data ascendingly i.e. 6 6 7 8 9.The rank of median = n + 1 / 2 i.e. 5 + 1 / 2 = 3 therefore the median is the third value i.e. median = 7

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Inferential statisticsInference involves making a Generalization about a larger group of individuals on the basis of a subset or sample.

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Inferential statisticsHypothesis Testing In hypothesis testing we want to find out whether the observed variation among sampling is explained by chance alone ???? (i.e., the chance of random sampling variations ), or due to a real difference ???? between groups.

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Hypothesis TestingIt involves conducting a test of statistical significance quantifying the chance of random sampling variations that may account for observed results. In hypotheses testing, we are asking whether the sample mean for example is consistent with a certain hypothesis value for the population mean.

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Hypothesis TestingThe method of assessing the hypotheses testing is known as significance test. The significance testing is a method for assessing whether a result is likely to be due to chance or due to a real effect.

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Hypothesis Testing Steps >>> Formulate Hypothesis >>> Collect the Data>>>> Test Your Hypothesis >>> Accept of Reject Your Hypothesis

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Null and alternative hypothesesIn hypotheses testing, a specific hypothesis ( Null and alternative Hypothesis ) are formulated and tested. The null hypotheses H0 means : X1=X 2 Or X1-X 2=0this means that there is no difference between x1 and x2

The alternative hypotheses H1 means X1>X2 or X1< X2

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Null and alternative hypothesesThe alternative hypotheses H1 means X1>X2 or X1< X2this means that there is no difference between x1 and x2. If we reject the null hypothesis, i.e there is a difference between the two readings, it is either H1 : x1 < x2 or H2 : x1> x2in other words the null hypothesis is rejected because x1 is different from x2.

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General principles of significance testsset up a null hypothesis and its alternative.find the value of the test statistic.refer the value of the test statistic to a known distribution which it would follow if the null hypothesis was true.

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General principles of significance tests 4-conclude that the data are consistent or inconsistent with the null hypothesis.If the data are not consistent with the null hypotheses, the difference is said to be statistically significant. If the data are consistent with the null hypotheses it is said that we accept it i.e. statistically insignificant.

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General principles of significance tests P

Tests of significanceThe selection of test of significance depends essentially on the type of data that we have.1-Quantitative Data ( Means & SD): t test ,paired t test and ,ANOVA2-Qualitative Data>>> Chi, and Z test.

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Tests of significanceComparison of means:1-comparing two means of large samples using the normal distribution:(z test or SND standard normal deviate)If we have a large sample size i.e. 60 or more and it follows a normal distribution then we have to use the z-test. z = (population mean sample mean) / SD. If the result of z >2 then there is significant difference.

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Tests of significanceSince the normal range for any biological reading lies between the mean value of the population reading 2 SD. (this range includes 95% of the area under the normal distribution curve).

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Students t-test2-Comparing two means of small samples using t-test:If we have a small sample size (less than 60), we can use the t distribution instead of the normal distribution.T = mean1 mean2 / (SD1 2 / n1) + (SD22 / n2)

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t-test The value of t will be compared to values in the specific table of "t distribution test" at the value of the degree of freedom. If the value of t is less than that in the table , then the difference between samples is insignificant. If the t value is larger than that in the table so the difference is significant i.e. the null hypothesis is rejected.

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t-test2-Comparing two means of small samples using t-test:If we have a small sample size (less than 60), we can use the t distribution instead of the normal distribution.T = mean1 mean2 / (SD1 2 / n1) + (SD22 / n2)

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Paired t-test3-paired t-test:If we are comparing repeated observation in the same individual or difference between paired data, we have to use paired t-test where the analysis is carried out using the mean and standard deviation of the difference between each pair.

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ANOVA4-comparing several means:Sometimes we need to compare more than two means, this can be done by the use of several t-test which is not only tedious but can lead to spurious significant results. Therefore we have to use what we call analysis of variance or ANOVA.

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ANOVA4-comparing several means:There are two main types: one-way analysis of variance and two-way analysis of variance. One-way analysis of variance is appropriate when the subgroups to be compared are defined by just one factor, for example comparison between means of different socio-economic classes. The two-way analysis of variables is used when the subdivision is based upon more than one factor

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ANOVAThe main idea in the analysis of variance is that we have to take into account the variability within the groups and between the groups and value of F is equal to the ratio between the means sum square of between the groups and within the groups.F = between-groups MS / within-groups MS

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Chi-Squared Testb-Qualitative variables:1)Chi -squared test:Qualitative data are arranged in table formed by rows and columns. One variable define the rows and the categories of the other variable define the column.

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Chi-Squared TestA chi-squared test is used to test whether there is an association between the row variable and the column variable or, in other words whether the distribution of individuals among the categories of one variable is independent of their distribution among the categories of the other. X2=(O-E)2 / E

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Chi-Squared Test1)Chi -squared test:degree of freedom = (row - 1) (column - 1)O = observed value in the table E = expected value calculated as follows: E= Rt x Ct / GT total of row x total of column / grand total

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Chi-Squared TestFrom tables of X2 significance at degree of freedom (row 3-1)x(column 3-1) = 2x 2=4. The level of significance at 0.05 level, d.f.=4 is 9.48. therefore we conclude that there is significant relation between socioeconomic level and the degree of intelligence (because the value of X2 > that of the table).

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Z Test2) Z test for comparing two percentages: z = p1 p2 /p1q1/n1 + p2q2/n2. where p1=percentage in the 1st group. P2 = percentage in the 2nd group, q1=100-p1, q2=100-p2, n1= sample size of group 1, n2=sample size of group2.Z test is significant(at 0.05 level)if the result>2.

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Chi-Squared TestExample: if the number of anemic patients in group 1 which includes 50 patients is 5 and the number of anemic patients in group 2 which contains 60 patients is 20. To find if groups 1 & 2 are statistically different in prevalence of anemia we calculate z test.P1=5/50=10% p2=20/60=33% q1=100-10=90 q2=100-33=67

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Chi-Squared TestZ=10 33/ 10x90/50 + 33x67/60 Z= 23 / 18 + 36.85 z= 23/ 7.4 z= 3.1Therefore there is statistical significant difference between percentages of anemia in the studied groups (because z >2).

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Correlation & regressionc-Correlation and regression:Correlation measures the closeness of the association between two continuous variables, while linear regression gives the equation of the straight line that best describes and enables the prediction of one variable from the other.

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Correlation & regression1-Correlation:In the correlation, the closeness of the association is measured by the correlation coefficient, r. The values of r ranges between + 1 and 1. One means perfect correlation while 0 means no correlation. If r value is near the zero, it means weak correlation while near the one it means strong correlation. The sign and + denotes the direction of correlation,

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Correlation1-Correlation:the +ve correlation means that if one variable increases the other one increases similarly while for the ve correlation means that when one variable increases the other one decreases

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Linear regression2- Linear regression:Similar to correlation, linear regression is used to determine the relation and prediction of the change in a variable due to changes in other variable. For linear regression, the independent factor has to be specified from the dependent variable.

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Linear regression2- Linear regression:The linear regression, not only allow assessment of the presence of association between the independent and dependent variable but also allows the prediction of dependent variable for a particular independent variable. However, regression for prediction should not be used outside the range of original data. a t-test is also used for the assessment of the level of significance. The dependent variable in linear regression must be a continuous one.

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Chart5

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RI

baby weight in kg

RI


Chart1

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oedema

Inhibin level

Presence of oedema

Fig. ( ) : Correlation between inhibin level and presence of oedema (0 = no oedema , 1 = tibial oedema ; 2=generalized oedema)

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protein

inhibin level

degree of proteinuria

Fig.( ) : Correlation between inhibin level and proteinuria (0=no proteinuria ,1=+ ;2=++ ,3=+++,4=++++)

Chart3

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0.39

0.5

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inhibin level

RI

Fig.( ) : Correlation between inhibin level and Doppler velocimetry (RI)

Chart4

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3

3.1

2.9

3.5

3

baby wt

inhibin level

baby weight

Fig. ( ) : Correlation between inhibin level and baby birth weight

Chart6

0.15

0.37

0.38

0.39

0.4

0.41

0.42

0.43

0.44

0.45

0.46

0.47

0.48

0.49

0.5

0.5

0.52

0.53

0.54

0.55

0.56

0.57

0.66

0.66

0.67

0.67

0.68

0.68

0.69

0.7

0.7

0.71

0.71

0.72

0.73

0.78

0.87

0.88

0.89

0.9

0.9

0.9

0.91

0.91

0.92

RI

Apgar score 1

RI

Fig.( ) : Correlation between Doppler velocimetry (RI) and Apgar score at 1 minute

Chart7

0.15

0.37

0.38

0.39

0.4

0.41

0.42

0.43

0.44

0.45

0.46

0.47

0.48

0.49

0.5

0.5

0.52

0.53

0.54

0.55

0.56

0.57

0.66

0.66

0.67

0.67

0.68

0.68

0.69

0.7

0.7

0.71

0.71

0.72

0.73

0.78

0.87

0.88

0.89

0.9

0.9

0.9

0.91

0.91

0.92

I.U.GR

RI

Fig.( ) : Correlation between Doppler velocimetry (RI) and intrauterine GR (o= -VE IUGR , 1 = +VE IUGR)

Sheet1

inhibinoedemaproteinRIbaby wtGRAPGA1inhibinoedema

7100.413182.20

6.8000.53.9063.61

9.5000.43.9185.80

2.2000.493.1086.21

3.6100.483.9076.30

5.8000.453.9066.80

6.3000.393.90871

11.1000.463.21870

6.2100.443.90871

8000.383.9077.40

7000.433.3067.60

7.4000.473.90880

7100.373.41880

8000.53.5088.22

7.6000.423.9079.20

20.1010.563.2169.50

18.4130.152.90710.21

15.9220.73.31811.10

20.9120.523.60312.10

24.1010.662.81713.21

32.3120.712.90613.62

20.7130.533.40413.62

20.7120.672.917142

45220.7231815.71

32.3020.543.50715.92

22.8110.6830816.61

19.9120.733.61717.62

27.9030.553.103182

23.2120.572.906182

27.5030.78317182

8.2240.92.51218.41

9.2030.872.51318.91

10.2140.912.20419.91

14240.882.61620.10

13.6240.892.31520.71

18.9140.662.11720.71

18230.672.51420.91

18240.682.71322.81

13.2140.92.11423.21

18240.692.50824.10

13.6240.913.31627.50

12.1040.72.81627.90

16.6140.922.81432.31

15.7140.93.11232.30

17.6240.712.114452

inhibinproteininhibinRIinhibinbaby wtAPGA1RIGRRI

2.202.20.492.23.170.1500.15

3.603.60.483.63.980.3710.37

5.805.80.455.83.970.3800.38

6.206.20.446.23.980.3900.39

6.306.30.396.33.980.410.4

6.806.80.56.83.980.4110.41

7070.417370.4200.42

7070.4373.360.4300.43

7070.3773.480.4400.44

7.407.40.477.43.960.4500.45

7.607.60.427.63.980.4610.46

8080.3883.980.4700.47

8080.583.570.4800.48

8.238.20.98.22.580.4900.49

9.239.20.879.22.560.500.5

9.509.50.49.53.980.500.5

10.2310.20.9110.22.230.5200.52

11.1211.10.4611.13.240.5300.53

12.1312.10.712.12.870.5400.54

13.2213.20.913.22.130.5500.55

13.6313.60.8913.62.360.5610.56

13.6213.60.9113.63.360.5700.57

144140.88142.670.6610.66

15.7415.70.915.73.170.6610.66

15.9315.90.715.93.370.6710.67

16.6416.60.9216.62.840.6710.67

17.6417.60.7117.62.180.6800.68

18.43180.67182.530.6810.68

18.54180.68182.780.6900.69

18.64180.69182.580.710.7

19.5318.40.1518.42.960.710.7

19.6418.90.6618.92.160.7100.71

20.1419.90.7319.93.640.7110.71

20.1320.10.5620.13.280.7210.72

20.7420.70.5320.73.470.7310.73

20.7320.70.6720.72.970.7810.78

20.9320.90.5220.93.630.8710.87

22.8222.80.6822.8360.8810.88

23.2323.20.5723.22.950.8910.89

24.1224.10.6624.12.820.910.9

27.5427.50.7827.5340.910.9

27.9327.90.5527.93.120.910.9

32.3432.30.7132.32.940.9100.91

32.3232.30.5432.33.560.9110.91

454450.7245340.9210.92

baby wtRI

2.90.15

3.40.37

3.90.38

3.90.39

3.90.4

30.41

3.90.42

3.30.43

3.90.44

3.90.45

3.20.46

3.90.47

3.90.48

3.10.49

3.90.5

3.50.5

3.60.52

3.40.53

3.50.54

3.10.55

3.20.56

2.90.57

2.80.66

2.10.66

2.90.67

2.50.67

30.68

2.70.68

2.50.69

3.30.7

2.80.7

2.90.71

2.10.71

30.72

3.60.73

30.78

2.50.87

2.60.88

2.30.89

2.50.9

2.10.9

3.10.9

2.20.91

3.30.91

2.80.92

Sheet2

Sheet2

0.15

0.37

0.38

0.39

0.4

0.41

0.42

0.43

0.44

0.45

0.46

0.47

0.48

0.49

0.5

0.5

0.52

0.53

0.54

0.55

0.56

0.57

0.66

0.66

0.67

0.67

0.68

0.68

0.69

0.7

0.7

0.71

0.71

0.72

0.73

0.78

0.87

0.88

0.89

0.9

0.9

0.9

0.91

0.91

0.92

RI

baby weight in kg

RI


Sheet3

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