STATISTICAL ANALYSIS

Mrs. Yasmin Sharma M.Sc (N) I Year

INTRODUCTIONThe word statistics conveys a variety of

meaning to people in different walks to life. To some it is an imposing from of mathematics; to other it could be simply tables, charts, and figures which one commonly find in newspaper, journals, books, reports, various reports and speeches, classroom lectures etc.

It is also used to refer to a body of knowledge known as statistical methods, developed for handling data in general, particularly in the field experimentation and research.

Definition Statistics can be defined as

‘numerical data involving variability and the treatment of such data’. According to croxton and cowden, ‘Statistics is defined as collection, presentation, analysis and interpretation of numerical data’.

Uses and Application of StatisticsIt presents facts in a definite

form.It facilitates comparisons.It simplifies the masses of

figures.It helps in formulating and

testing hypothesis.It helps in prediction.

Application of Biostatistics in health sciencesDefining normal and not normal

in context of various aspects related to health and illness

Establishing the accuracy of the diagnostic procedures.

Planning of experiments and analysis of results

CONT….Assessment of treatment

protocol and different interventions used for care and treatment of the patients.

Collections, analysis, and dissemination of various population health statistics.

SCALES OF MEASUREMENTS

Measurement is a central and essential to the process of obtaining data. The term ‘measure’ means to ascertain the dimensions of quantity or capacity to mark off, usually with reference to some unit of measurement.

Level of Measurements

NominalAttributes are only named; weakest

ordinal Attributes can be ordered

Interval

Distance is meaningful

Ratio Absolute zero

Example of Nominal Scale

GENDER CODE

MALE 1

FEMALE 2

Example of Ordinal ScaleCLIENT’S ABILITY TO PERFORM ACTIVITIES OF DAILY LIVING

RANKING

COMPLETELY DEPENDENT 1

NEEDS ANOTHER PERSON ASSISSTANCE

2

NEEDS MECHANICAL ASSISSTANCE

3

COMPLETELY INDEPENDENT 4

DESCRIPTIVE STATISTICS

Descriptive statistics are used to organize and summarize data to draw meaningful interpretations. Descriptive statistics also allow the researcher to interpret the data meaningfully, so that research questions can be answered completely and appropriately

Classification of the Descriptive Statistics Frequency distribution and

graphical presentationMeasures of central tendencyMeasures of dispersionMeasures of relationship

(Correlation coefficient).

FREQUENCY DISTRIBUTION

An appropriate presentation of data involves organization of data in such manner that meaningful conclusion and inferences can be drawn to answers the research questions. Unsorted and ungrouped records do not allow us to draw clear conclusions. Quantitative data are generally condensed and frequency distribution is presented through tables, charts, graphs and diagrams.

TABLES

A table presents data in a concise, systematic manner from masses of statistical data.

Tabulation is the first steps before data is used for further statistical analysis and interpretation.

Tabulation means a systematic presentation of information contained in the data in rows and columns in accordance with some features and characteristics. Rows are horizontal and columns are vertical arrangement.

GENERAL PRINCIPLES OF TABULATIONA table should be precise, understandable,

and self- explanatory.Every table should have title, which is

placed at the top of the table. The title must describe the content clearly and precisely.

Items should be arranged alphabetically or according to size, importance, or casual relationship to facilitate comparison.

Rows and columns to be compared with one another should be brought together.

CONT….The content of the table, as a whole as

well as the items in each column and row should be defined clearly and fully.

The unit of measurement must be clearly stated.

Percentage can be given in parenthesis or can be worked out to one decimal figure to drawn the reader attention to the fact that the figure is a percentage and not an absolute number.

Totals can be placed at the bottom of the columns.

CONT…..References symbols can be

directly placed beneath the table for any explanatory footnotes.

Two or three small tables should be preferred to one large one.

PARTS OF A TABLE

TABLE NUMBERTITLEHEAD NOTESCAPTION AND STUBSBODY OF TABLEFOOTNOTESSOURCE NOTES

TABLE NO. 9KNOWLEGDE SCORE OF DIABETIC PATIENT

(SD & t value)

*** = p< 0.0001 SD(2.59) & t value 17.56 shows a a highly

significant improvement in the knowledge score

TABLE NO.

TITLEHead Notes

KNOWLEDGE AREA

SD t VALUE

PRE-TEST2.59 17.56***POST-TEST

CAPTION

STUBS

FOOT NOTES

SOURCE NOTE

BODY OF TABLE

TYPES OF TABLES

FREQUENCY DISTRIBUTION TABLE

CONTINGENCY TABLE

MULTI – RESPONSE TABLE

MISCELLANEOUS TABLE

Example of Frequency Distribution Table

Sociodemographic variables

frequency percentage

Age (in years)20-4041-60

1842

3070

Gender Male Female

3921

6535

Education statusIlliterate MiddleSenior secondaryGraduate & above

08172015

13.328.333.325.0

N = 60

Example of Contingency Table

Bowel movement

Mode of ventilation

Total f

X2

value

Spontaneous ventilationf (%)

Mechanical ventilationf (%)

45.87*df=1Present 391 64 32 29.4 423

Absent 220 36 77 70.6 297

Total 611 109

720

Examples of Multi – Response TableFactors

N=60f (%)

Blood sampling 35 58.3Diagnostic tests 33 55.0Medication 33 55.0Vital signs monitoring

32 53.0

Noise 32 53.0Bright lights 30 50.0

GRAPHICAL PRESENTATION OF DATAThey are the most convenient and

appealing ways in which statistical results may be presented.

They given an overall view of entire data.They are visually more attractive than

others ways of representing data.It is easier to understand and memorize

data through graphical representation.They facilitate comparison of data

relating to different periods of time of different origins.

CONSTRUCTING DIAGRAMS/GRAPHSThey must have a title, and indexThe proportion between width and

height be balancedThe selection of scale must appropriateFootnotes may be included wherever it

is needed.Principal of simplicity must be kept in

mind.Neatness and cleanliness in

construction of graphs must be ensured.

TYPES OF DIAGRAM AND GRAPHS

BAR DIAGRAMit is a convenient, graphical device that

is particularly useful for displaying nominal or ordinal data. It is an easy method adopted for visual comparison of the magnitude of different frequencies. Length of vertical bar diagrams or horizontal diagrams indicates the frequency of a character the bar chart are called vertical bar charts (or column charts) if the bars are placed horizontally, we get horizontal bar chart.

Types of Bar DiagramSIMPLE BAR DIAGRAMSMULTIPLE BAR DIGRAMSPROPORTION BAR DIAGRAMS

SIMPLE BAR DIAGRAMS

SIMPLE BAR DIGRAM SHOWING DIETARY PATTERN OF PEOPLE

0

10

20

30

40

50

60

70

80

VEGETERIAN NONVEGETERIAN

72

27

NO

.OF

PEO

PLE

TYPE OF DIETARY PATTERN

MULTIPLE BAR DIGRAMS

MULTIPLE BAR DIAGRAMS SHOWING DISTURBING FACTORS

0 20 40 60 80

COUGHING

MOBILE PHONE

SNORING

WHISPERING

ARRIVING LATE

SNEEZING

FLASH PHOTOGRAPHY

NUMBER OF THEATER PERFORMERS

DIS

TURB

IMG

FAC

TORS

PROPORTION BAR DIAGRAMS

PROPORTIONATE BAR GRAPH

0

10

20

30

40

50

60

ASIA AFRICA EUROPE NORTH AMERICA

60

1420

9

20 22

8

22

PERC

ENTA

GE

OF

TOTA

L W

ORL

D

POPU

LATI

ON

& L

AND

ARE

A

REGION

POPULATION

LAND

PIE DIAGRAM/SECTOR DIAGRAMit is another useful pictorial

device for presenting discrete data of qualitative characteristics such as age groups, genders and occupational groups in a population. The total area of the circle represents the entire data under consideration. Researcher must remember that only percentage data must be used to prepare pie diagrams

EXAMPLE OF PIE DIAGRAM

26%

17%

2%

34%

10%

11%

BASKETBALL

SOCCER

READING

COMPUTER GAMES

LISTENING

NETBALL

HISTOGRAMit is most commonly used graphical

representation of grouped frequency distribution. Variable characters of the different group are indicated on the horizontal line (x-axis) and frequencies (no. of observation) are indicated on the vertical on the vertical line (y-axis). Frequency of each group forms a column or rectangle. Such diagram is called Histogram.

EXAMPLE OF HISTOGRAM

FREQUENCY POLYGONit is the curve obtaining by joining the

midpoints of the tops of the rectangle in a histogram by straight line. It gives a polygon i.e. figure with many angles. In this, the two end points of the line drawn are joined to the horizontal axis at the midpoint of the empty class- interval at both ends of the frequency distribution. Frequency polygons are simple and sketch an outline of data pattern more clearly than histograms

A frequency polygon can be drawn by using following steps:

Draw the histogram of given data.Join the midpoint of upper horizontal

sides of each rectangle with adjacent one by a straight line.

Close the polygon at the both ends of distribution by extending them to base line

Hypothetical classes at the each end would have to be included each end with a frequency of zero

EXAMPLE OF FREQUENCY POLYGON

LINE DIGRAMS Variables in the frequency polygon are

designed by line. It is mostly used where data collected over a long period of time. On x-axis, values of independent variables are taken and values of dependent variables are taken on y-axis. Vertical axis may not start from zero, but at some point from where the frequency starts. With reference to x-axis and y-axis, he given data may be plotted and these consecutive points or data are then joined by straight lines.

EXAMPLE OF LINE DIAGRAMS

0

100

200

300

400

500

600

2001 2002 2003 2004 2005 2006 2007

no o

f car

s sol

d

in delhi

in mumbai

CUMULATIVE FREQUENCY CURVE OR OGIVEThis graph represents the data of

a cumulative frequency distribution. For drawing ogive, an ordinary frequency distribution table is converted into cumulative frequency table. the cumulative frequencies are then plotted corresponding to the upper limit of the classes are joined by a free hand curve. The diagram made is called ogive.

Example of Cumulative Frequency Curve

SCATTERED OR DOTTED DIAGRAMSit is a graphic presentation, made

to show the nature of correlation between two variables characters x and y on the similar features or characteristics such as height and weight in men aged 20 years. Therefore, it is also called correlation diagram.

Example of Scattered Diagram

PICTOGRAMS OR PICTUIRE DIAGRAMthis method is used to impress

the frequency of the occurrence of events to common people such as, attacks, deaths, number operations, admission, accidents, and discharge in a population

Example of Pictogram

Pictogram showing proportion of people of respective economic classes

High class

Middle class

Below poverty line

Developing countries

Developed countries

MAP DIAGRAM OR SPOT MAP

These maps are prepared to show geographic distribution of frequencies of characteristics.

Example OF Map Diagram

LIMITATIONS OF GRAPHSConfusing (may be false or true)Present only quantitative aspect

under studyThey can present only

approximate valuesGetting limited information on

only one or two aspect or characteristic

MEASURES OF CENTRAL TENDENCY

Definition According to croxton and cowden

‘An average value is a single value within the range of the data that is used to represent all of the value of series. Since an average is somewhere within the range of data, it is also called a measures of central value’.

MEASURES OF CENTRAL TENDENCY

Arithmetic mean

MedianMode

ARITHMATIC MEAN

The mean or average is probably the most commonly used method of describing central tendency.

CONT…..

Arithmetic mean is represented by X

  X = SUM OF THE VALUES (Σx)

NUMBER OF VALUES(n)

ExampleQ. The heamoglobin of ten women is

12.5, 13, 10, 11.5, 11, 14, 9, 7.5,10 and 12. Calculate the mean heamoglobin among this sample of women.

X = 110.5 = 11.5 10 SO, The mean Hb of for ten women's are

11.5  

CALCULATING MEAN FORM DISCRETE FREQUENCY TABLE

In discrete frequency table the mean is calculated using following formula:

  X = Σxf Σf

example

Q. following data gives the age of 100 adolescent girls. Find the mean ageAge in year

(x)No. of students (f)

Xf

16 35 560

17 31 527

18 20 360

19 14 266

Σf = 100

Σxf = 1713

= 1713 = 17.13 100

SO, The mean age of 100 adolescent girls are 17.13

CALULATING ARITHMATIC MEAN FROM COTINOUS FREQUENCY TABLE

formula: X = Σfm ΣfMidpoint of the class interval is

calculated by following formula: Midpoint = lower limit + upper

limit 2

EXAMPLE

Q.Calculate the mean age of the following group of people:

Class interval of age

No. of people (f)

Midpoint (m)

Fm

15-20 15 17.5 262.5

20-25 40 22.5 450

25-30 40 27.5 1,100

30-35 60 32.5 1,950

Σf = 135

Σfm = 3,762.5

= 3726.5 = 27.87 135 Therefore the mean age of the

people in this group will be 27.87 years.

MERITS OF ARITHMATIC MEAN

Simple to understand and easiest to compute

It is affected by the value of every item in a series.

It is defined by rigid mathematical formula, with the result that irrespective of whoever computers the average, he or she will get the same answer.

CONT….It can be treated algebraically

It is a reliable method of calculating average

Calculated value is not based on the position of series.

DEMERITS OF ARITHMETIC MEAN

Very small and every items usually affect the values of average

In the distribution with open- end classes, values of mean cannot be computed without making assumption.

Not always a good measures of central tendency.

MEDIAN

A median of sets of value is the middle – most value when the data is arranged in ascending order of magnitude. The middle value will divide the number of observation in the data into two equal parts.

The median is denoted by M.

CALCULATING OF MEDIAN FOR THE INDIVIDUAL DATA

Formula M = n + 1 2

Example

Q. Following data gives the weight of the seven people in pounds; calculate the median of given data.

158, 167, 143, 169, 172, 146, 151

Solution First arrange the data in to

ascending or descending order.143, 146, 151, 158, 167, 169, 172 = 7 + 1 = 8 = 4 2 2Therefore the fourth observation in

the data will be the median. i.e 158

Calculating of median for the discrete frequency table

Q. Calculate the median for the following frequency table data

Income/ day (x)

100 150 200 250 300 350

No. of households (f)

05 19 03 11 06 09

Cumulative frequency (cf)

5 24 27 38 44 53

M = 53 + 1 = 54 = 27 2 2

The cumulative frequency has been calculated as the 27th observation in array, which means that the x value in front of 27th cumulative frequency that is 200 is the median.

CALCULATING OF MEDIAN FOR THE CONTINOUSN FREQUENCY TABLE

formula: M = l + (N/2 – cf) X i F

EXAMPLE

Q. Calculate the median for the following frequency table

Income/day(x)

100-150

150-200

200-250

250-300

300-350

350-400

No. of households(f)

05 19 03 11 06 09

Cumulative frequency(cf)

5 24 27 38 44 53

In case of the continuous frequency table, median can be calculated by using following formula:

M = l + (N/2 – cf) X i FWhere, l = the lowest limit = 200 i = class interval = 50 f = frequency of the median class = 03

cf = cumulative frequency of a class just before the median class = 24

N = Σf = 53M = 200 + (26.5 – 24) 50 =

200+41.66 03

So, the median income/day

according to no. of house holds is 241.66

MERITS OF MEDIANUses in case of open – ended

classes and unequal classes.Extreme values do not affect the

median.Most appropriate average in

dealing with qualitative data.

The value of median can be determine graphically

DEMERITS OF MEDIAN

For calculating median, it is necessary to arrange the data.

Since it is the position averages, the values is not determined by each and every observation.

Median is not calculated for qualitative data.

MODE

It is the value which has the highest frequency. That means mode is the most frequently occurring value in the data. In others words, the mode of the distribution value at the point around which the items tend to be most heavily concentrated.

It is denoted as Z

CALCULATING OF MODE FOR INDIVIDUAL FREQUENCY

For the data in Individual frequency table the most frequently occurring value is considered as mode.

For example 3, 1, 7, 4, 1, 2, 5, 3, 4, 6, 5, 5, 4, 4, 5, 2, 4

In this, mode or Z = 4

CALCULATING OF MODE FOR DISCREET FREQUENCY

Z = 140

Income/day(x)

110 120 130 140 150 160

No. of households(f)

2 4 8 10 5 3

Calculating of mode for continuous frequency data

Z = l1 + (f1-f0) X i

2f1 – f0– f2

Where l = lowest limit of the model class f1= frequency of the model class i= class interval f0 = frequency of class just before the

model class f2 = frequency of class just after the model

class

Example

Q. Calculate the mode for following data

Income/day (x)

100-150

150-200

200-250

250-300

300-350

350-400

No. of households (f)

05 10 03 11 06 09

Solution

Z = 250 + (11 -03) x 50

2 X 11 – 03 – 06

 Z = 250 + 30.7 = 280.7

So, the mode = 280. 7

Merits of mode

It is not affected by extremes value.

It can be used to describe quantitative phenomenon.

Values of mode can be determined graphically.

Demerits of mode

The value of mode cannot always be determined.

It is not capable of algebraic manipulation.

It is not based on all the values.

MEASURES OF DISPERSION

The observation deviating from the central value is different in different sets of values of character. In some distribution, the difference may be less, whereas in other it may be more. This property of deviation of the values from the average is called variations or dispersion. The degree of variation indicated by measure of dispersion

Various Measures of the Dispersion RangeMean deviationStandard deviationQuartile deviation

RANGE

It is the difference between highest and lowest value in the data. If ‘H’ is the highest and ‘L’ is the lowest value.

Range (R) = H –L

EXAMPLE

Q. Calculate the range for the following data

3, 5, 7, 8, 9, 12, 15, 17, 19, 20,

21, 23, 26

R = H –L = 26-3 = 23

MERITS OF RANGE

Range is very simple to understand.

It is also easy to calculate.

DEMERITS OF RANGE

It is not suitable for deep analysis.

It is not suitable in case of extreme values.

STANDARD DEVIATION

Standard deviation is the positive square root of mean of the square deviations of values from the arithmetic mean.

. It is denoted by ‘SD’ or ‘σ’

If the Standard Deviation is large it means the numbers are spread out from their mean. If the Standard Deviation is small , it means the numbers are close to their mean.

Here are the scores on the math quiz for Team A:

72768080818384858589

Average: 81.5

The Standard Deviation measures how far away each number in a set of data is from their mean.

For example, start with the lowest score, 72. How far away is 72 from the mean of 81.5?

72 - 81.5 = - 9.5

Cont….Or, start with the highest score,

89. How far away is 89 from the mean of 81.5?

89 - 81.5 = 7.5

So, the first step to finding the Standard Deviation is to find all the distances from the mean.

72768080818384858589

-9.5

7.5

Distance from Mean

So, the first step to finding the Standard Deviation is to find all the distances from the mean.

72768080818384858589

- 9.5

- 5.5

- 1.5

- 1.5

- 0.5

1.5

2.5

3.5

3.5

7.5

Distance from Mean

Next, you need to square each of the distances to turn them all into positive numbers

72768080818384858589

- 9.5

- 5.5

- 1.5

- 1.5

- 0.5

1.5

2.5

3.5

3.5

7.5

Distance from Mean

90.25

30.25

Distances

Squared

Next, you need to square each of the distances to turn them all into positive numbers

72768080818384858589

- 9.5

- 5.5

- 1.5

- 1.5

- 0.5

1.5

2.5

3.5

3.5

7.5

Distance from Mean

90.25

30.25

2.25

2.25

0.25

2.25

6.25

12.25

12.25

56.25

Distances

Squared

Add up all of the distance

72768080818384858589

- 9.5

- 5.5

- 1.5

- 1.5

- 0.5

1.5

2.5

3.5

3.5

7.5

Distance from Mean

90.25

30.25

2.25

2.25

0.25

2.25

6.25

12.25

12.25

56.25

Distances

Squared

Sum:214.5

Divide by (n) where n represents the amount of numbers you have.

72768080818384858589

- 9.5

- 5.5

- 1.5

- 1.5

- 0.5

1.5

2.5

3.5

3.5

7.5

Distance from Mean

90.25

30.25

2.25

2.25

0.25

2.25

6.25

12.25

12.25

56.25

Distances

Squared

Sum:214.5

(10 )

= 21.45

Finally, take the Square Root of the average distance

72768080818384858589

- 9.5

- 5.5

- 1.5

- 1.5

- 0.5

1.5

2.5

3.5

3.5

7.5

Distance from Mean

90.25

30.25

2.25

2.25

0.25

2.25

6.25

12.25

12.25

56.25

Distances

Squared

Sum:214.5

(10)

= 21.45

= 4.63

This is the Standard Deviation

72768080818384858589

- 9.5

- 5.5

- 1.5

- 1.5

- 0.5

1.5

2.5

3.5

3.5

7.5

Distance from Mean

90.25

30.25

2.25

2.25

0.25

2.25

6.25

12.25

12.25

56.25

Distances

Squared

Sum:214.5

(10 )

= 21.45

= 4.63

Now find the Standard Deviation for the other class grades

57658394959698937163

- 24.5

- 16.5

1.5

12.5

13.5

14.5

16.5

11.5

- 10.5

-18.5

Distance from Mean

600.25

272.25

2.25

156.25

182.25

210.25

272.25

132.25

110.25

342.25

Distances

Squared

Sum:2280.5

(10 )

= 228.05

= 15.10

Now, lets compare the two classes again

Team A Team B

Average on the Quiz

Standard Deviation

81.5 81.5

4.63 15.10

calculating SD from individual data

SD = Σ (x- x)2

N OR

Σ(dx )2

N

CALCULATION OF THE SD FROM DISCRETE & CONTINOUS DATA

SD = Σ (dx)2 f Σf

CORRELATION COEFFICENT

Sometimes two continuous characters are measured in a series or in a similar pattern such as weight and cholesterol, weight and height etc. this relationship or association between two quantitatively measured or continuously variables is called correlation.

TYPES OF CORRELATION COEFFICIENT

PERFECT POSITIVE CORRELATION PERFECT NEGATIVE CORRELATION MODERATELY POSITIVE CORRELATION MODERATELY NEGATIVE CORRELATION ABSOLUTELY NO CORRELATION

Karl Pearson’s correlation coefficient

It is used to measure the degree of linear relationship between two variables. It is also called product moment correlation. It is denoted by ‘r’

Karl Pearson’s correlation coefficient r = Σ xy - Σx Σy n [Σx2-( Σx)2] [Σy2-

(Σy)2] n

n

Spearman’s rank correlation coefficient

It is the method of finding the correlation between two variables by taking their ranks.

This method of finding correlation is especially useful in dealing with qualitative data.

It can be used when the actual magnitude of characteristics under consideration is not known, but relative position or rank of magnitude is known. It is denoted by ‘p’.

SPEARMAN’S RANK CORRELATION COEFFICIENT

p = n (n21)

INFERENTIAL STATISTICS

Inferential statistics are concerned with population, use sample data to make an inference about the population or to test the hypotheses considered at the beginning of the research study.

Inferential statistics help the researcher to determine if the difference found between two more groups, such as an experimental and control group, is a real difference or only a chance difference that occurred because an unrepresentative sample was chosen from the population.

TYPE I AND TYPE II ERROR

Type I error occurs when null hypothesis is rejected, when it should have been accepted; also called alpha error.

Type II error occurs where null hypothesis is accepted, when it actually have been rejected. It is called as beta error. These errors generally occur due to unrepresentative samples drawn from the population.

TYPE I AND TYPE II ERROR

REALITY DECISION

ACCEPT H0 REJECT H0

H0(TRUE) CORRECT DECISION

TYPE I ERROR

H0(FALSE) TYPE II ERROR

CORRECT DESICION

TEST OF SIGNIFICANCE

PARAMETRIC

TESTS

• t-TEST• Z- TEST• ANOVA

NON PARAMETRIC TESTS

•CHI-SQUARE•McNEMAR TEST•MANN-WHITNEY TEST•WILCOXON SIGNED RANK TEST•FISHER’S EXACT TEST

PARAMETRIC TESTSThese tests are known as normal

distribution statistical tests. The statistical methods of inference make certain assumptions about the population from which the samples are drawn.

t – Test

It is applied to find the significant difference between two means. This test can be applied when following criteria are fulfilled:

Randomly selected homogeneous sample.

Quantitative data( numerical data not the frequency distribution)

CONT…..Variability normally distributed.

Sample size less than 30; if the sample size is more than 30, then Z- tests is applied.

TYPES Of t – TESTSUnpaired t – Test: it is applied when

we obtain data from subjects of two independent separated groups of people or samples drawn from different populations.

Paired t – Test: it is applied on paired data of independent observations made on same samples before and after the intervention. Paired test is most commonly used in nursing research studies.

STEPS OF APPLICATION OF t – TEST

t = mean difference of mean

SEFirstly calculate standard

deviationThen calculate standard error(SE)

CONT….Formula for calculating SE for

paired t –TEST: SE =

Formula for calculating SE for unpaired t – TEST:

SE = 1/n1+1/n2

CONT….Calculate observed difference

between two means.Observed difference = X1 –X2;

where X1 is mean of the first sample and x2 is the mean of the second sample.

Calculate t – value by using following formula:

t = mean difference of mean

SE

CONT….COMPUTE THE DEGREE OF

FREEDOM (df):

Df = n1 + n2 -2 (unpaired sample)

n – 1 (paired sample)

Z- TESTWhen a sample is larger than 30

subjects, and a researcher wants to compare the difference in population mean and a simple mean or the difference between two sample means, then Z test is applied.

Prerequisites for Application of the Z- Test

The sample or samples must be randomly selected.

The data must be quantitative in nature.

The variability is assumed to follow normal distribution in the population.

The sample size must be larger than 30.

FORMULAZ = observed difference

between two sample means

SE of different between two sample means

Z = A - B

SE

ANALYSIS OF VARIANCE (ANOVA) TESTWhen a researcher wants to

compare the difference between more than two samples means; t- test will be not useful and a need of alternative test will be felt. This need can be fulfilled by test known as analysis of variance (ANOVA) test. Therefore, it is clear that ANOVA is used to compare the more than two samples means drawn from corresponding normal population

STEPS FOR APPLICATION OF ANOVACalculate the total of sum of all

the group of observations.Calculate the sum square of all

the observations.Calculate the total of sum of

square by using following formula:

= ΣX2 – (ΣX)2

N

CONT…

Calculate the sum of squares between the groups by using following formula:

(ΣX1) +(ΣX2) + (ΣX3) +(ΣX 4) +……

n1 n2 n3 n4

so on – (ΣX)2

N

CONT….Calculate the sum of squares

within the groups (error sum of squares) by using following formula:

= total sum of squares – sum of squares between the groups

Calculate the degree of freedom for between and within the groups. df for error is = N –1( no. of

groups- 1)

CONT….Calculate the mean of sum of

square by using following formula.

Mean of sum of squares between the groups = sum of square between

the groups df for between the

groups

CONT….Mean of sum of squares within the groups

= sum of squares within the groups

df for within the groups Finally compute the F – ratio by using following

formula:F-ratio of square

= Mean of sum of squares between the groups

Mean of sum of squares within the groups

NON PARAMETRIC TESTSResearcher in the field of health

sciences many times may not be aware about the nature of the distribution or other required population parameters. In addition, sample may too small to test the hypothesis and generalize the findings for the population from which the sample is drawn.

CHI –SQUARE TESTIt is used to find out the

association between two events in binomial or multibinomial samples. It represented by a symbol x2 and used to find association between discrete attributes.

This test is also used to find the significance of difference in two or more than two proportions.

PREREQUEST OF CHI – SQUARES TEST

A random sample Quantitative data (frequency

data, not the means)Sample size should be more than

30Lowest expected frequency not

less than 5.

STEPS OF CHI-SQUAREMake the contingency tables.Note the frequencies observed in each

class of one event row – wise and numbers in each group of other event column- vise.

Determine the expected number (E) in each cell of table assumption of null hypothesis.

E = Column or vertical total X row or horizontal total

Grand total

CONT….Find the difference between the

observed and the expected frequencies in each cell (O –E)

Calculate the chi – square value for each cell by formula; X2 = (O –E)2

E

Sum up the x2 values of all the cells to get the total chi – square value.

McNEMAR TEST

When the proportions being compared are from two paired groups(e.g. when a pretest-posttest design is used to compare changes in proportions on a dichotomous variable), the appropriate test is the McNEMAR TEST

MANN-WHITNEY TEST

It is another type of non-parametric test for testing the difference between two independent groups. The test involves assigning ranks to the two groups of measures. The sum of the rank for the two group can be compared by calculating the U statistics.

WILCOXON SIGNED RANK TEST

When the ordinal data are paired (dependent) the wilcoxon signed rank test can be used. It involves taking the difference between paired scores and ranking the absolute difference.

FISHER’S EXACT TESTIt should be used to test the

significance of differences in proportions

Interpretation of data

By interpretation of data we mean that task of drawing conclusion or interferences and of explaning their significances, after careful analysis of the collected data.

The interpretation of research data cannot be considered in the abstract. In view of the diversity of research methods use in education, and the corresponding diversity of data, they seek, the interpretation of such data is best considered within context of each method. The analysis and interpretation of historical data.

Cont….The process of interpretation is

essentially one of stating that what the findings show.

The findings of the study are the result, conclusion, implication, interpretation, recommendations, generalizations, future research and nursing practice.

Cont…Interpreting the findings of a

study involves a search for their meaning in relation to the problem, conceptual framework, purpose and all the research decision made in developing and implementing the empirical phase of the study

Types of validity

Explanatory validity

Ecology validity and

Methodological validity

Explanatory validity It refers to the extent to which

the concept is chosen to account for that study findings to do so. This requires examination of alternative, equally plausible explanation for the findings.

Ecological validity It refers to the extent to which

the sample of observations in the study represents the substantive domain, the adequacy of the relationship between the study design and substance being studied

Methodological validityIt refers to the degree of which

the findings are a function of a set or method used to test the theory

Errors of InterpretationFailure to see the problem proper

perspective Investigator may have an inadequate

grasp of the problem in its broad sense and too close a focus in its immediate aspect.

Failure to appreciate the relevance of various elements

The investigator may fail to see the relevance of the various elements of the situation due to an inadequate grasp of the problem, too rigid a mind set or even a lack of imagination.