Parameters... measured characteristics of the population, usually unknown and/or unknowable yet

the thing we are most interested in knowing.

Statistics... measured characteristics of the sample, which we typically use to estimate population

parameters that we cannot measure directly. Statistics are the basis for all of statistical inference,

since to infer is to draw conclusions about a population from a sample.

Parameters... measured characteristics of the population, usually unknown and/or unknowable yet

the thing we are most interested in knowing.

Statistics... measured characteristics of the sample, which we typically use to estimate population

parameters that we cannot measure directly. Statistics are the basis for all of statistical inference,

since to infer is to draw conclusions about a population from a sample.

Exactly as Gevorg says, we calculate statistics from the sample to make inferences about the parameters of the distribution.

With respect to the idea of a sufficient statistic, informally a sufficient statistic is a complete summary of the information in a sample about some parameter of the distribution.    

From your example, if we assume that the variance of the Gaussian is known, then the sample mean is a sufficient statistic. If you know the sample mean, you have all the information in the sample about the population mean. If you don't know the variance, then the sample mean and the sample variance are sufficient.

Thank you Gevorg,

But in the attached paper, for Bernouli Distribusion, P=Teta^T*(1-Teta)^(n-T), the author say that Teta is the parameter of the distribusion and T can be sufficient statistic if Teta do not changed.

What is your idea?

In your example, you have a series of independent random variables, X_1 -- X_n, each is Bernoulli with parameter \theta. That is, each is 0 or 1., Now T = \sum_i X_i.

Now the conditional distribution of {X_i , i=1 ... n} given T=t and \theta does not depend on \theta, therefore T is sufficient for \theta.  

So, in this particular case the statistics and parameters can be considered as synonyms.

Mahdieh,

Let me explain them to you in a simple way:

Parameters:  unknown values in the population that we can know only if we know the values in the entire population. like means, rates , proportions, variances, correlation coefficients,... etc.

Statistics: functions that are computed from the sample data such as sums of observations, sums of their squares..

Estimators: statistics that can be used for estimating a specific unknown parameter based on  the sample data. So, any estimator is a statistic but not all statistics are estimators.

Sufficient statistics: the statistics that if we know their values then no need to know each single data point. They are sufficient to enable us to estimate the unknown parameters.

Example: Suppose that the phenomenon of interest is the prevalence rate of a specific disease in a population. Then the prevalence rate is the parameter and that it is unknown and can be only known if you have information on each single person in the population. Suppose that you want to estimate the prevalence rate of that disease in that population based on a sample of 1000 observations. If you compute the number of diseased persons in the sample then that is a statistic and it is also a sufficient statistic since you do not need to know any further information from the sample other than that sufficient statistic to estimate the unknown parameter (the prevalence rate). The number of diseased persons (the sufficient statistic) divided by 1000  (the sample size) is then the estimator of that parameter and it is also a statistic.

I hope that helps you.

Regards,,

More simple/basic information as regards population, sample, parameters and statistics:

parameters - are used to represent population parameters, usually/always whose information is unknown. Population is a collection of  all elements defined by certain characteristic e.g all male students at a particular university. Mathematical notation employs the Greek alphabet characters/letters e.g µ (Mu) for the mean, σ2 for the variance, so you have the mean (µ) weight or height of all the males at the particular university with a variance, σ2

statistic - is the estimator of the particular parameter. Its value is calculated from the data collected based on a sample which is a part of the particular population. Since a population is large, you may not be able to make measurements of all the population elements. Hence  you take a sample of the elements from the population. From the measurements you calculate the sample statistics e.g sample mean denoted by x̄ to estimate the population mean, µ, and the sample variance denoted by s2 to estimated the population variance, σ2.

Note also that the word statistics is also used to mean the data/the figures/the numbers/measurements/observations/information. And the word population is also used broadly in the subject of statistics - it does not only have the meaning associated with the number of people in a country i.e population.

Mahdieh, I also have problems understanding the Wikipedia sentence:

'a statistic is sufficient with respect to a statistical model and its associated unknown parameter if "no other statistic that can be calculated from the same sample provides any additional information as to the value of the parameter".[1] In particular, a statistic is sufficient for a family of probability distributions if the sample from which it is calculated gives no additional information than does the statistic, as to which of those probability distributions is that of the population from which the sample was taken.' because:

1. It suposes an apriori parametric distribution that givse frequences to each dataset value.

2. It seems to derive the whole parameter from the a priori distribution family while using another premise from Laplace when estimating U as the average of your N=100 values, and variance as a function of that U. This asumes equal frequences for each value and contradicts the parametric asumption.

3. It seems a tautological procedure based in a dominating premise and mixed with Laplace criterion that provide U and variance, and then it is thrown away saying that data set contains no more information, and declares the statistical method as "suficient", so the rest is explained only by the parametric equation and estimates entities called "estimators" that may explain the apriori model, but do not give a realistic model, or statistic for the dataset.

In my opinion it is a concept very hard to explain clearly and even harder to understand.

Thanks, emilio