STATISTICAL ANALYSIS AND ITS APPLICATIONS

STATISTICS It refers to the body of technique or

methodology which has been developed for the collection ,presentation and analysis of quantitative data and for the use of such data in decision-making

OR The science of statistics is the method of judging

collection ,natural or social phenomenon from the results obtained from the analysis or enumeration or collection of estimates

Gupta,S.C., Kapoor,v.k., (2013) fundamentals of mathematical statistics. 11th Ed,Sultan Chand & Sons educational publishers

STATISTICAL METHODOLOGIES1) Descriptive Statistics:-summarizes data from a sample using indexes such as the mean or standard deviation2) Inferential Statistics:- Draws conclusion from data that are subject to random variations e.g observational errors,sampling variations

Measures of central tendency When a series of observations have been tabulated in the form of

frequency distribution it is felt necessary to convert a series of observation in a single

value, that describes the characteristics of that distribution,→ called Measure Of Central Tendency

All data or values are clustered round it These values enable comparisons to be made between one series

of observations and another Individual values may overlap, two distributions have different

central tendency E.g., average incubation period of measles is 10 days and that of

chicken pox is 15 days.

Types : Central tendency

Measures of Central tendency

Mean Mode Median

Arithmetic Geometric Harmonic Mean(AM) Mean(GM) Mean(HM)

Arithmetic mean:Sum of all observations divided by number of

observationsMean(x)=Sx/n; x is a variable taking different

observational values & n= no. of observationsExmp.ESR of 7 subjects are 8,7,9,10,7,7, & 6 mm for 1st hr.

Calculate mean ESR.- Mean(x)= (8+7+9+10+7+7+6)/7=54/7=7.7 mmPROPERTIES:Uniqueness:- Given set of data one and only one

arithmetic meanSimplicity:- easily understood and easy to compute

Median : when observations are arranged in ascending or

descending order of magnitude, the middle most value is known as Median.

Problem: From same example of ESR, observations are arranged first

in ascending order: 6,7,7,7,8,9,10. Median= {7+1}/2=8/2=4th observation I,e., 7 When n is Odd no., Median={n+1}2 th observation When n is Even no., Median={n/2th + (n/2+1)th}/2 th

observation Problem: suppose, there are 8 observations of ESR like

5,6,7,7,7,8,9,10 Median={8/2th +(8/2+1)th}/2={4th+5th obs}/2=(7+7)/2=7

Mode:The observation, which occurs most frequently in

seriesProblem: ESR of 7 subjects are 8,7,9,10,7,7, & 6 mm

for 1st hr. Calculate the Mode.- Mode is 7.

Calculation of weighted arithmetic mean:Following methods are utilized in case of large

no. of observationsFor Ungrouped Data:Suppose we have x₁, x₂, x₃,…nth observations

with corresponding frequencies f₁, f₂,f₃,…fn

Mean=

For grouped Date:Data are arrange in groups & frequency

distribution table are preparedMean value of each group is multiplied by

frequencySum of product value is divided by total no of

observationsMean such obtained is called “ weighted

mean”Mean(x)=

Geometric mean: Used when data contain a few extremely large or small

values It’s the nth root product of n observastions GM=ⁿ√(x₁.x₂.x₃….xn) Harmonic Mean: Reciprocal of the arithmetic mean of reciprocals of

observations arithmetic mean of reciprocals of observations=S(⅟x) HM=n/S⅟x got limited use A.M>GM>HM

Measures of dispersion• Measures of central tendency do not provide information

about spread or scatter values around them• Measures of dispersion helps us to find how individual

observations are dispersed or scattered around the mean of a large series of data

• Different measures of Dispersion are:i. Rangeii. Mean deviationiii. Standard deviationiv. Variancev. Coefficient of variation

Range:- Difference between highest & lowest value- Defines normal value of a biological characteristic• Problem: Systolic blood pressure (mm of Hg) of 10 medical

students as follows: 140/70, 120/88, 160/90, 140/80, 110/70, 90/60, 124/64, 100/62, 110/70 & 154/90

• Range of Systolic BP of medical students = highest value- lowest value=160-90=70mm of Hg

• Range of Diastolic BP= 90-60=30 mm of Hg

Mean deviation:- Average deviations of observations from mean value- Mean Deviation(S) =(x-x)/n, where x=observation, x=Mean

Gupta,s.c.,kapoor,v.k,. ( 2013 ) fundamentals of mathematical statistics .11th Ed,sultan chand & sons

Standard Deviation: Most frequently used measures of dispersion Square root of the arithmetic mean of the square of

deviations taken from the arithmetic mean. In simple term “ Root-Mean-Square-Deviation” s)

Where x= observation X=Mean

n=no. of observations

To estimate variability in population from values of a sample, degree of freedom is u in placed of no. of observations

Standard deviation is calculated by following stages: Calculate the mean Calculate the difference between each observation & mean Square the difference Sum the squared values Divide the sum of squares by the no. of observations(n) to get mean

square deviation or variances(s) Find the square root of variance to get “Root-Mean-Square-

Deviation” Use: sample size calculation of any study - Summarizes deviation of a large series of observation around mean

in a single value

Coefficient of Variation:- Used to denote the comparability of variances of two or

more different sets of observations- Coefficient of Variation=(sd/Mean) x 100- Coefficient of Variation indicates relative variability

Statistics and its application

Pharmaceutical statistics is the application of statistics to matters concerning the pharmaceutical industry. This can be from issues of design of experiments, to analysis of drug trials, to issues of commercialization of a medicine.

Evaluate the activity of a drug; e.g.; effect of caffeine on attention; compare the analgesic effect of a plant extract and NSAID

To explore whether the changes produced by the drug are due to the action of drug or by chance

To compare the action of two or more different drugs or different dosages of the same drug are studied using statistical methods.

To find an association between disease and risk factors such as Coronary artery disease and smoking

Gupta,S.c.,Kapoor,v.k,.(2013) fundamentals of mathematical statistics . 11th Ed,sultan & chand sons

Statistics and its application cont….

Public health, including epidemiology, health services research, nutrition, environmental health and healthcare policy & management.

Design and analysis of clinical trials in medicine Population genetics, and statistical genetics in order to link variation in genotype

with a variation in phenotype. In biomedical research, this work can assist in finding candidates for gene alleles that can cause or influence predisposition to disease in human genetics

Analysis of genomics data. Example: from microarray or proteomics experiments. Often concerning diseases or disease stages.

Systems biology for gene network inference or pathways analysis Demographic studies: Age, gender, height, weight, BMI Epidemiology: deficiency of iron in anemia, iodized salt and goiter, hygiene and

microbial disease