Introduction to Statistics

Statistics is mainly used to study quantitave research. It is used to organise and analyse data, to analyse percentage scores. Statistics enables us to serve two purposes namely description and pridiction. Descriptive statistics is used to study the distribution of data, pridiction is based on general forcasts.

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Statistics Variables

A variable is an attribute that varies when different entities are observed. Variables in statistics can be classified broadly into two types, Qualitative and Quantitative variables. Qualitative variables are also called categorical variables while quantitative variables are commonly known as numerical variables.

Types of Variables in Statistics

  • Numerical variable

     Example : Age, height and weight of a person.
                         Outcomes in the roll of a die 1,2,3,4,5 and 6

  • Categorical Variable

   Example : Hair color, nationality , status , position and the role of a person.
                         Outcomes in a toss of a coin head and tail

Variables in Statistics

Statistics Quantitative variables can be further classified as discrete and continuous variables. While discrete variable assume countable values like 1,2,3…. continuous variable take in all values in an interval.

Discrete Variable:  A numerical variable is said to be discrete if it assumes only countable or infinitely countable numbers.

Continuous Variable: Continuous variables take in all the values in an interval formed by two given numbers.  Continuous variables are measured and not counted.

Random Variable: Random variable is a numerical variable whose value is determined by chance and it is used as a function to assign numerical values to the outcomes of a sample space.
While the discrete random variable assigns countable numbers to sample space elements, the continuous random variables assigns intervals on the number line to the outcomes.

Using the mapping ability of random variables even the non numerical outcomes of a sample space gets translated into numerical values and can be analytically treated. For example the two outcomes the head and tail of the simple experiment of tossing a coin can be translated into numerical values of 0 and 1 using the one-to-one mapping of random variable.

Discrete Variables

Suppose three coins are thrown simultaneously.  The number of heads turned is noted.  The possible outcomes are three head turning, two heads turning, one head turning and no head in the three tosses.  The sample space can be written as a range of random variable function ‘X’ consisting of discrete variables as follows:
X = {0, 1, 2, 3}
Both the probabilities of turning a head and tail in a single throw =  .  Hence the probabilities of the compound events represented by the discrete variables 0,1,2 and 3 are as follows.
P(x=0) =  x  x  =
P(x=1) =3 x  x  x  =
P(x=2) =3 x  x  x  =
P(x=3) =  x  x  =
The discrete variables can be tabled along with the respective probabilities in a manner similar to a frequency distribution.

 X  0   1 
 2 
 3 
 P(X)         

This mapping of discrete random variables to their probabilities is called a probability function or probability distribution if the following conditions are met.

  • P(x)  0  for all values of x in the range space
  • The sum of all such probabilities = 1.  i.e   = 1.

Binomial and Poisson distributions are examples for discrete probability distributions.  Binomial distributions are related to Bernoulli trials of two outcome probability experiment.  The two outcomes in a single trial are defined as success and failure.  The probabilities of these two complimentary events are used in calculating the probability function of the random variable. The expected value of a probability distribution is commonly known as the mean of the distribution.  The parameters of a binomial distribution are ‘n’ and ‘p’ where n is the number of trials and p the probability of success in a single trial. The mean and standard deviation of the probability distribution are given by the formulas
mean = np  and standard deviation =   where q = 1-p, the probability of failure in a single trial.

Continuous Variable

Suppose the maximum rainfall recorded in your locality is 15cm for the month of April.  So the amount of rainfall in the month of April can be a continuous variable taking all values in the interval (0,15).  Similarly the heights of student in a class range from 122 cm to 155 cm then the height of a random student in the class is the continuous variable assuming all the values in the interval (122,155). When we say that the continuous variable assumes all values in the interval, it does not refer to a single value in the interval.  The whole interval is taken as one unit rather as a range of values. When we say the interval (0,15) represents the rainfall in the month of April, it means the rainfall could be anything between 0 and 15 cm including integer and decimal or fractional values. Hence the probabilities can be measured only for the intervals and not for any particular value.  For example if x is a continuous random variable, you can compute the probabilities for P(x a) and P(a
Probability density function can be defined for a continuous random variable x, just as a probability distribution function is defined for a discrete variable.

A function f(x) is said to be the probability density function of a continuous random variable if the following conditions are met

f(x)  0 for all x
 =1

Normal probability distribution with its bell shaped curve is a commonly known continuous probability distribution.  Normal probability distribution is a symmetric distribution and the area under the curve for  an interval represent the probability of the interval being the value of the variable.  An area table for standardized normal distribution is used to find the required probabilities.  The mean of the standard normal distribution is 0 and the standard deviation is 1.For this purpose the given variables are transformed into standardized normal variable, which is popularly known as z-score. The formula used for transforming a normal variable ‘x’ is as follows:

z=   where  is the mean and  the standard deviation of the distribution under study.

Statistical Inference

The probability theory and the random sampling allow statistical analysis to draw conclusion about the population tendencies. The systematic of statistical inference in the context of parametric model consists of the following stages.
  • Sampling done using random methods.
  • Determining the sample error expected.
  • Estimating the population parameter using the sample statistic.
  • Analyzing the estimates and drawing conclusion within the allowance of confidence levels.

In classical approach,hypothesis testing is a decision making process applied to check the validity of claims about a population.  'z-tests' , 't-tests' and χ2 tests are few types to hypothesis testing we study in the beginning study of inferential statistics.

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Help with Statistics tutoring is concerned with collection, analysis and explanation of data. Statistics help from Tutornext deals with learning the basic concepts which include finding mean, mode, median and variance from the collected data. Our online statistics tutors do their best to make the students get the best out of the online session. Not only tutoring, but also homework help can be availed whenever needed.

Choose your Statistics topic below:

Chapter 1: Introduction to Statistics

Chapter 2: Mean, Median and Mode

Chapter 3: Stem and Leaf Plots

Chapter 4:Bar Graphs

Chapter 5: Pie Charts

Chapter 6: Normal Distribution

Chapter 7:Data Handling

Chapter 8: Survey Sampling

Chapter 9: Regression and Regression Analysis