Statistics

Basics of estimation  In many real-life problems, the population parameter(s) is (are) unknown and  someone is interested to obtain the value(s) of parameter(s). But, if the whole  population is too large to study or the units of the population are destructive in  nature or there is a limited resources and manpower available then it is not  practically convenient to examine each and every unit of the population to find the value(s) of parameter(s). In such situations, one can draw sample from the  population under study and utilize sample observations to estimate the  parameter(s).  Every one of us makes estimate(s) in our day to day life. For example, a house  wife estimates the monthly expenditure on the basis of particular needs, a sweet  shopkeeper estimates the sale of sweets on a day, etc. So the technique of  finding an estimator to produce an estimate of the unknown parameter on the  basis of a sample is called estimati...

Random variables are always generated with a particular pattern of probability attached to them. Thus, based on the pattern of probability for the different values of random variable, we can distinguish them. Once we know these probability distributions and their properties, and if any random variable fits in a probability distribution, so we will see what is probability mass function and probability density function A probability mass function (PMF)also called a frequency function  gives you probabilities for   discrete random variables. “Random variables” are variables from experiments like dice rolls, choosing a number out of a hat, or getting a high score on a test. The “discrete” part means that there’s a set number of outcomes. For example, you can only roll a 1, 2, 3, 4, 5, or 6 on a die. for example pmf for binomial distribution... Probability Mass Function Probability mass function is a function of defining a discrete probability dist...

  Rao-Blackwell Theorem  Calyampudi Radhakrishna Rao was born on 10 September 1920, in a  small town in south India called Huvvinna Hadagalli, then in the integrated  Madras Province of British India but now in the state of Karnataka.  C. R. Rao has received numerous  awards for his precious work in statistics, including the Guy Medal  in Silver of the Royal Statistical Society, London (1965), the Megnadh  Saha Medal of the Indian National Science Academy (1969), the Jagdish  Chandra Bose Gold Medal and cash award (1979), the Silver Plate bearing  the monogram of the Andhra Pradesh Academy of Sciences (1984), the S.  S. Wilks Medal of the American Statistical Association (1989), and the  Mahalanobis Birth Centenary Gold Medal awarded by the Indian Science  Congress (1996). The government of India honored C. R. Rao in 1968 by  awarding him the title of Padma Bhushan, a high civilian award. Let X and Y be ra...

  Laws of probability and conditional probability As in last post we have seen sample space and events in probability,today we will see some rules of probability  and conditional probability  To see laws of probability we will require some knowledge about independence and some terminologies regarding events. 1)Exhaustive events :The total number of possible outcomes of random experiment is known as the exhaustive events. Example:If we roll a die,there are 6 exhaustive cases since any number can appear uppermost 2)Favorable events :The outcome which make necessary the happening of an event in a trial are called favorable events. Example:let us consider two dies are rolled,then the number of favorable events getting sum 3 is 2,that is (1,2),(2,1) 3)Mutually exclusive events(Disjoint events) :The events are said to be mutually exclusive or disjoint if they cannot both occur at same time. Example:Outcome of single coin toss (either head or tail) 4)Equally likely events :...

  Sample space and Event Sample space:  A collection of all possible outcomes of an experiment is called sample space.It is denoted by  Ω. Example: A coin is tossed   Ω ={H,T} Where H=head, T=tail.   Event: The collection of all outcomes favorable to phenomenon is called an event. It is denoted by capital letters A,B,C.. Example: Event A is containing at least one head,if two coins are tossed.   Ω ={HH,HT,TH,TT} A={HH,HT,TH} (1) Elementary event: An event containing only one element is called as elementary or simple event. Example: Event A is getting both heads ,if two coins are tossed.   Ω ={HH,HT,TH,TT} A={HH} (2) Compound event: An event which contains more then one sample point of sample space is called a compound event. Example: Event A is containing both head and tail,if two coins are tossed. Ω ={HH,HT,TH,TT} A={HT,TH} (3)Sure event or certain event: An event containing all the points of sample space is called sure event. Example:  In an exp...

Kurtosis is a statistical measure that defines how heavily the tails of a distribution differ from the tails of a normal distribution. In other words, kurtosis identifies whether the tails of a given distribution contain extreme values. The above formula are given by prof.karl Pearson which are known as 'covexity of frequency curve' or simply kurtosis. These formula will give us idea about flatness or peakedness.It is measured by above  coefficients.If value of gamma2 is more  than zero then the curve is leptokurtic,if value equals to zero then it is mesokurtic or normal curve and if value of gamma 2 is less than zero than curve said to be platykurtic. 1)Mesokurtic Data that follows a mesokurtic distribution shows an excess kurtosis of zero or close to zero. This means that if the data follows a normal distribution, it follows a mesokurtic distribution. 2. Leptokurtic Leptokurtic indicates a positive excess kurtosis. The leptokurtic distribution shows h...

Mode : Mode is an important measure of central tendency.Ya Lun Chou statement about the mode is ,"Mode is that values of series which appears most frequently than any other." In simple  words,"The mode is a value which has highest frequency." The mode may not be unique .that is there are several values or classes with the same highest frequency. I) Mode for discrete data:      Mode of sample is the observation with highest frequency. Example: A sample is 4,8,3,8,3,9,3,7,11,3,12,11 Here number 3 has maximum frequency(4). Hence given sample has mode 3. II)Mode for grouped data:      Mode of grouped data is the class centre of the modal class,while the modal class is a class which has highest frequency. Example Here 10-20 class has maximum frequency (15). Hence 10-20 is modal class. Here L=10,f1=15,fo=5,f2=8,h=20-10= 10 Mode=10+[(15-5)/(30-5-8)]×10 Mode=10+(10/17)×10 Mode=10+5.88 Mode=15.88 For example,In clothing Store,they operate their business to incl...

  Today    we will see some important measures of central tendency called Quartile,Deciles and Percentiles... 1)Quartile Three variate  value of the variables which divides the series(data) into four equal parts are called Quartiles. There  are three quartiles namely Q1,Q2 and Q3.Q1 is known as lower or first quartile,Q2 is  the second quartile (median) and last one Q3 is known as upper or third quartile.25% values are less then Q1 and 25% values are larger than Q3 and the rest 50% values lie between Q1 and Q3.And Q2 devides data into two equal parts and due to this this is also known as median.The quartiles can help us to find some insights of given data set.These widely used in data analytics and economics.By these values we can about shape of distribution. 2)Deciles The variate value of the variables which divides the series(data) into ten equal parts are known as Deciles.They are denoted by D1,D2....so on.fifth decile divides data into two parts so it i...

  As we described the importance and use of skewness in last post,Now  we are going to see the different formula's derived famous mathematicians and statistician.These measures are easy to calculate and have commenting capacity.We will see two methods elucidated by karl pearson and prof. Bowley. I) Pearson's coefficient of Skewness     Karl Pearson developed two methods for finding skewness in sample,by using mode and second one is by median 1) Pearson's coefficient of skewness using mode- Interpretation of calculation done by using formula If Sk<0 - negatively skewed If Sk=0 - symmetric If Sk>0 - positively skewed. Let us consider following example  A sample with mean 15 ,mode 14 and standard deviation 2.. Sk=(15-14)/2=1/2=0.5 Sk>0 . Hence the curve of given sample is positively skewed  2) Pearson's coefficient of skewness using median: Example: A sample with mean 20 ,median 22 and standard deviation 3. Sk=3×((20-22)/3)=-6/3=-2 Here Sk...

  Median for continuous data As we said in  previous post,We will come with post to find median for continuous data.we seen median for discrete data series. Now,median for grouped data.let us take the following data. 1) When classes are continuous Where , L=lower limit of median class f=frequency of median class N=total frequency c.f=cumulative frequency of previous class to median class i=class interval Here N=39  N/2=39/2=19.5 Hence cumulative frequency just greater than 19.5 is 24. Therefore 20-30 is median class. Here L=20,N=39,c.f=12,i=10-0=0,f= 12. Median=20+(((39/2)-12)/12)×10) Median=26.5 II)When classes are inclusive First we will convert classes into continuous type.... For that we will find: (11-10)/2=1/2=0.5           Add 0.5 to upper limit and and           Subtract 0.5 from lower limit. New table: Here N=39            N/2=39/2=19.5 Hence cumulative frequency j...

  What is Skewness? Skewness is asymmetry (lack of symmetry) in a statistical distribution, in which the curve appears distorted or skewed either to the left or to the right. Skewness can be quantified to define the extent to which a distribution differs from a normal distribution. In a normal distribution, the graph appears as a classical, symmetrical "bell-shaped curve." The mean, or average, and the mode, or maximum point on the curve, are equal. In a perfect normal distribution, the tails on either side of the curve are exact mirror images of each other. When a distribution is skewed to the left, the tail on the curve's left-hand side is longer than the tail on the right-hand side, and the mean is less than the mode. This situation is also called negative skewness. When a distribution is skewed to the right, the tail on the curve's right-hand side is longer than the tail on the left-hand side, and the mean is greater than the mode. This situation is also cal...

  variance and standard deviation In  statistics we come across some words which are used in so many situations and one of them are variance and standard deviation... Variance  The variance can be defined as the average of squared differences from mean value.In rough terms it is measure of how far set of data(numbers) are spread out from their mean (average) value.Variance never be negative.By variance we can draw some useful insights for given data.For comparison purpose variance can be useful in many cases. variance is very useful in lot of statistical measures,test,analysis and many more... Standard deviation Standard deviation measures how spread out the values in a data set are around the mean. More precisely, it is a measure of the average distance between the values of the data in the set and the mean. If the data values are all similar, then the standard deviation will be low (closer to zero). If the data values are highly variable, then the standard variation is ...