## A population parameter and its point estimate.

Population parameters refer to the statistical measures that are fixed and when used as variables, they make the population distribution descriptive hence descriptive statistics. A good example of population parameters is the mean and variance in a normal distribution. When these two variables are clearly stated it is possible to determine the type of distribution without much effort. There are distributions with single population parameters an example being the chi-square distribution.

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A population parameter can also be described in simpler terms as the statistical representation of a model or population. These can be classified as either unobservable or fixed. A population parameter is unobservable when the random variables are inferred in relation to the distribution at hand. It is fixed when the sampling techniques used are specific to the population, an example being the use of degrees of freedom in chi-square estimation.

The point estimate of a population parameter on the other hand is a statistical value used to describe that particular population. An example of this is the mean of sample data which is taken to be the point estimate of the mean for the population. The point estimate of a population can never be 100% precise but the level of accuracy is dependent on the characteristics of the distribution that is whether it is a normal distribution, t – distribution, chi-square distribution, and so on. The level of errors in the point estimate can be known from the shape of the distribution and this also applies to the credibility of the point estimate. In other words, the point estimate of a population parameter refers to the most reliable estimate available to that parameter.

A point estimate is especially useful when testing the hypothesis of the population parameters. This is what determines whether the hypothesis is to be accepted or rejected without having to rely on chance. When conducting hypothesis testing, the null value of the parameter is specified. This is the value that negates the theory behind the hypothesis. The point estimate of the sample population is determined and the null value assigned to it to decide whether the sample population holds true. From this, therefore, we can deduce that point estimate is a value used to describe the population parameter being used in statistical inference.

## A population and a corresponding sample mean

The population in statistics refers to the whole collection of objects being analyzed. This could be students in a school, vehicles in a certain street, trees in a forest, and patients in a hospital among others. A population consists of all the numbers of objects available irrespective of how many they are. Since it consists of a large number of objects, populations cannot be studied in their entirety since this could be time-consuming and very tiresome.

Some objects in a population have the same characteristics and so it is pointless to study all of them. This is why a sample is drawn from the population for statistical inference. A sample is a representation of the whole population normally drawn from the population under scrutiny.

This is what is studied in statistics and the results thereof are considered to be the characteristics of the whole population. In the study of death and birth rates in a certain country, the population is the total number of recorded births and deaths in a particular year. A sample can be selected at random or following a specific procedure such as two babies born each month. An assumption is made that the two babies represent all the babies born in that month.

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The mean of the population is the anticipated outcome in that when the unbounded number of observations is considered, it results in the average of the same number of observations. The population means is the real or true mean value since it represents the characteristics of each object under observation. The problem with the population means is that it is difficult to determine since in most cases, the population normally consists of a finite number of objects. The mean of a sample on the other hand is the value that gives the average of the selected objects. Since a sample consists of a finite number of objects selected from the population the sample mean is easier to arrive at compared to the population mean. However, this value is an estimate and usually contains a certain degree of errors.

## Range

Range in descriptive statistics is the value of the distance between the least observation and the greatest observation. In a normal distribution, this is calculated by subtracting the variable with the least number of observations from the variable with the greatest number of observations. The range is usually measured according to the units used in the population or sample. If the observation is in the currency, for example, the range will be in the currency as well.

This measure is only dependent on two observations – the smallest and the largest hence proving to be a poor statistical inference, except when the population is considerably large. In any population, the range is usually greater than the standard deviation with the only exception being the Bernoulli distribution where the range is equal to the standard deviation.

## Variance

Variance in statistics is the parameter that determines the level at which a given number of observations has diverted from the average. It is used in descriptive statistics since it describes the dispersion of the variable from the mean variable. Variance is mostly viewed as an approach used in differentiating different statistical distributions such as the t-distribution, z-distribution, chi-square distribution, and so on. In cases where samples are used, the variance is considered to be an estimate while the variance calculated from the entire population is the accurate variance.

## Standard deviation

Standard deviation refers to the measure of variation from the mean. When this value is low, it is an indication that the observations are clustered around the mean and when it is high, the observations seem to be dispersed away from the mean. The value of the standard deviation in a population or sample is determined by getting the square root of the variance. Both variance and standard deviation are used to determine the same statistical inference, but the difference is that standard deviation has units of measurement which are the same as that of the population in question while variance does not.

Besides measuring the variability from the mean, the standard deviation also measures the degree of confidence in the statistical inferences. Confidence in statistics refers to the level at which the results of statistical analysis are accurate. Standard deviation is mostly used in scientific experiments and the results which are considered statistically significant are the ones outside the values of the standard deviation. Besides this it is also used in measuring the viability of investments in finance. The viability of an investment is determined by computing the standard deviation of the rate of investment.