Chi sqare test

Populations, Samples, Parameters, and Statistics

The field of inferential statistics enables you to make educated guesses about the numerical characteristics of large groups. The logic of sampling gives you a way to test conclusions about such groups using only a small portion of its members.

A population is a group of phenomena that have something in common. The term often refers to a group of people, as in the following examples:

• All registered voters in Crawford County
  
• All members of the International Machinists Union
  
• All Americans who played golf at least once in the past year

But populations can refer to things as well as people:

• All widgets produced last Tuesday by the Acme Widget Company
  
• All daily maximum temperatures in July for major U.S. cities
  
• All basal ganglia cells from a particular rhesus monkey

Often, researchers want to know things about populations but do not have data for every person or thing in the population. If a company's customer service division wanted to learn whether its customers were satisfied, it would not be practical (or perhaps even possible) to contact every individual who purchased a product. Instead, the company might select a sample of the population. A sample is a smaller group of members of a population selected to represent the population. In order to use statistics to learn things about the population, the sample must be random. A random sample is one in which every member of a population has an equal chance of being selected. The most commonly used sample is a simple random sample. It requires that every possible sample of the selected size has an equal chance of being used.

A parameter is a characteristic of a population. A statistic is a characteristic of a sample. Inferential statistics enables you to make an educated guess about a population parameter based on a statistic computed from a sample randomly drawn from that population (see Figure 1).

Figure 1.Illustration of the relationship between samples and populations.

 

For example, say you want to know the mean income of the subscribers to a particular magazine—a parameter of a population. You draw a random sample of 100 subscribers and determine that their mean income is $27,500 (a statistic). You conclude that the population mean income μ is likely to be close to $27,500 as well. This example is one of statistical inference.

Different symbols are used to denote statistics and parameters, as Table 1 shows.

A population is a group of phenomena that have something in common. The term often refers to a group of people, as in the following examples:

• All registered voters in Crawford County
  
• All members of the International Machinists Union
  
• All Americans who played golf at least once in the past year

But populations can refer to things as well as people:

• All widgets produced last Tuesday by the Acme Widget Company
  
• All daily maximum temperatures in July for major U.S. cities
  
• All basal ganglia cells from a particular rhesus monkey

Often, researchers want to know things about populations but do not have data for every person or thing in the population. If a company's customer service division wanted to learn whether its customers were satisfied, it would not be practical (or perhaps even possible) to contact every individual who purchased a product. Instead, the company might select a sample of the population. A sample is a smaller group of members of a population selected to represent the population. In order to use statistics to learn things about the population, the sample must be random. A random sample is one in which every member of a population has an equal chance of being selected. The most commonly used sample is a simple random sample. It requires that every possible sample of the selected size has an equal chance of being used.

A parameter is a characteristic of a population. A statistic is a characteristic of a sample. Inferential statistics enables you to make an educated guess about a population parameter based on a statistic computed from a sample randomly drawn from that population (see Figure 1).

Figure 1.Illustration of the relationship between samples and populations.

 

For example, say you want to know the mean income of the subscribers to a particular magazine—a parameter of a population. You draw a random sample of 100 subscribers and determine that their mean income is $27,500 (a statistic). You conclude that the population mean income μ is likely to be close to $27,500 as well. This example is one of statistical inference.

Different symbols are used to denote statistics and parameters, as Table 1 shows.

Difference Between Statistic and Parameter

In statistics vocabulary, we often deal with the terms parameter and statistic, which play a vital role in the determination of the sample size. Parameter implies a summary description of the characteristics of the target population. On the other extreme, the statistic is a summary value of a small group of population i.e. sample.

The parameter is drawn from the measurements of units in the population. As against this, the statistic is drawn from the measurement of the elements of the sample.

While studying statistics it is important to the concept and difference between parameter and statistic, as these are commonly misconstrued.

Content: Statistic Vs Parameter

Comparison Chart

BASIS FOR COMPARISON	STATISTICS	PARAMETER
Meaning	Statistic is a measure which describes a fraction of population.	Parameter refers to a measure which describes population.
Numerical value	Variable and Known	Fixed and Unknown
Statistical Notation	x̄ = Sample Mean	μ = Population Mean
	s = Sample Standard Deviation	σ = Population Standard Deviation
	p̂ = Sample Proportion	P = Population Proportion
	x = Data Elements	X = Data Elements
	n = Size of sample	N = Size of Population
	r = Correlation coefficient	ρ = Correlation coefficient

Definition of Statistic

A statistic is defined as a numerical value, which is obtained from a sample of data. It is a descriptive statistical measure and function of sample observation. A sample is described as a fraction of the population, which represents the entire population in all its characteristics. The common use of statistic is to estimate a particular population parameter.

From the given population, it is possible to draw multiple samples, and the result (statistic) obtained from different samples will vary, which depends on the samples.

Definition of Parameter

A fixed characteristic of population based on all the elements of the population is termed as the parameter. Here population refers to an aggregate of all units under consideration, which share common characteristics. It is a numerical value that remains unchanged, as every member of the population is surveyed to know the parameter. It indicates true value, which is obtained after the census is conducted.

Key Differences Between Statistic and Parameter

The difference between statistic and parameter can be drawn clearly on the following grounds:

1. A statistic is a characteristic of a small part of the population, i.e. sample. The parameter is a fixed measure which describes the target population.
2. The statistic is a variable and known number which depend on the sample of the population while the parameter is a fixed and unknown numerical value.
3. Statistical notations are different for population parameters and sample statistics, which are given as under:
• In population parameter, µ (Greek letter mu) represents mean, P denotes population proportion, standard deviation is labeled as σ (Greek letter sigma), variance is represented by σ2, population size is indicated by N, Standard error of mean is represented by σx̄, standard error of proportion is labeled as σp, standardized variate (z) is represented by (X-µ)/σ, Coefficient of variation is denoted by σ/µ.
• In sample statistics, x̄ (x-bar) represents mean, p̂ (p-hat) denotes sample proportion, standard deviation is labeled as s, variance is represented by s2, n denotes sample size, Standard error of mean is represented by sx̄, standard error of proportion is labeled as sp, standardized variate (z) is represented by (x-x̄)/s, Coefficient of variation is denoted by s/(x̄)

4. Conclusion

   To sum up the discussion, it is important to note that when the result obtained from the population, the numerical value is known as the parameter. While, if the result is obtained from the sample, the numerical value is called statistic.