What is the difference between population parameters and sample statistics?

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 ComparisonStatisticParameterMeaningStatistic is a measure which describes a fraction of population.Parameter refers to a measure which describes population.Numerical valueVariable and KnownFixed and UnknownStatistical Notationx̄ = Sample Meanμ = Population Means = Sample Standard Deviationσ = Population Standard Deviationp̂ = Sample ProportionP = Population Proportionx = Data ElementsX = Data Elementsn = Size of sampleN = Size of Populationr = Correlation coefficientρ = 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)

Illustration

1. A researcher wants to know the average weight of females aged 22 years or older in India. The researcher obtains the average weight of 54 kg, from a random sample of 40 females.
   Solution: In the given situation, the statistics are the average weight of 54 kg, calculated from a simple random sample of 40 females, in India while the parameter is the mean weight of all females aged 22 years or older.
2. A researcher wants to estimate the average amount of water consumed by male teenagers in a day. From a simple random sample of 55 male teens the researcher obtains an average of 1.5 litres of water.
   Solution: In this question, the parameter is the average amount of water consumed by all male teenagers, in a day whereas the statistic is the average 1.5 litres of water consumed in a day by male teens, obtained from a simple random sample of 55 male teens.

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.

Parameters are numbers that describe the properties of entire populations. Statistics are numbers that describe the properties of samples.

For example, the average income for the United States is a population parameter. Conversely, the average income for a sample drawn from the U.S. is a sample statistic. Both values represent the mean income, but one is a parameter vs a statistic.

Remembering parameters vs statistics is easy! Both are summary values that describe a group, and there’s a handy mnemonic device for remembering which group each describes. Just focus on their first letters:

• Parameter = Population
• Statistic = Sample

A population is the entire group of people, objects, animals, transactions, etc., that you are studying. A sample is a portion of the population.

Types of Parameters and Statistics

Both parameters and statistics describe groups.

Parameters and statistics use numbers to summarize the properties of a population or sample. There is a range of possible attributes that you can evaluate, which gives rise to various types of parameters and statistics. For example, are you measuring the length of a part (continuous) or whether it passes or fails an inspection (categorical)?

When you measure a characteristic using a continuous scale, you can calculate various summary values for statistics and parameters, such as means, medians, standard deviations, and correlations.

When the characteristic is categorical, the parameter or statistic will often be a proportion, such as the proportion of people who agree with a particular law.

Related post: Discrete vs Continuous Data

Statistic vs Parameter Symbols

While parameters and statistics have the same types of summary values, statisticians denote them differently. Typically, we use Greek and upper-case Latin letters to signify parameters and lower-case Latin letters to represent statistics.

Summary ValueParameterStatisticMeanμ or Mux̄ or x-barStandard deviationσ or SigmasCorrelationρ or rhorProportionPp̂ or p-hat

Parameter vs Statistic Examples

In the examples below, notice how the same subject and summary value can be either a parameter or a statistic. The difference depends on whether the value summarizes a population or a sample.

ParameterStatisticMean weight of all German Shepherd dogs.Mean weight of a random sample of 200 German Shepherds.Median income of a county.Median income of a random sample of 50 from that county.Standard deviation of all transaction times in a particular bank.Standard deviation of a random sample of 500 transaction times at that bank.Proportion of all people who prefer Coke over Pepsi.Proportion of a random sample of 100 people who prefer Coke over Pepsi.

Identifying a Parameter vs Statistic

If you’re listening to the news, reading a report, or taking a statistics test, how do you tell whether a summary value is a parameter or a statistic?

Real-world studies almost always work with statistics because populations tend to be too large to measure completely. Remember, to find a parameter value exactly, you must be able to measure the entire population.

However, researchers define the populations for their studies and can specify a very narrowly defined one. For example, a researcher could define the population as a specific neighborhood, U.S Senators (n=100), or a particular sports team. It’s entirely possible to survey the entirety of those populations!

The trick is to determine whether the summary value applies to an entire population or a sample of a population. Carefully read the narrative and make the determination. Consider the following points:

• A description that specifies the use of a sample indicates that the summary value is a statistic.
• If the population is very large or impossible to measure completely, the summary value is a statistic.
• However, if the researchers define the population as a relatively small group that is reasonably accessible, the researchers could potentially measure the entire group. The summary value might be a parameter.

Researchers and Parameters vs Statistics

Researchers are usually more interested in understanding population parameters. After all, understanding the properties of a relatively small sample isn’t valuable by itself. For example, scientists don’t care about a new medicine’s mean effect on just a few people, which is a sample statistic. Instead, they want to understand its mean effect in the entire population, a parameter.

Unfortunately, measuring an entire population to calculate its parameter exactly is usually impossible because they’re too large. So, we’re stuck using samples and their statistics. Fortunately, with inferential statistics, analysts can use sample statistics to estimate population parameters, which helps science progress.

Using a sample statistic to estimate a population parameter is a process that starts by using a sampling method that tends to produce representative samples—a sample with similar attributes as the population. Scientists frequently use random sampling. Then analysts can use various statistical analyses that account for sampling error to estimate the population parameter. This process is known as statistical inference.

What is the difference between sample and parameter?

What is the difference between a sample statistic and a population parameter quizlet?

What is the difference between population and sample?

What is population and parameter in statistics?