What is Sample Statistic?

A sample statistic is arbitrary data computed from sample data. A sample is a group of elements chosen from a particular statistical population. The characteristics that represent the population are called parameters and the properties of the sample data are called statistics. Both population and sample are important parts of statistics.

Inferential statistics provide a way to generalize the characteristics of a population using sample statistics. Suppose it is reported that "45% of preschool children in India watch TV more than 5 hours a day". This conclusion is not reached when examining the television viewing habits of all preschool children in India. In contrast, this conclusion should have been reached by estimating the population parameters using the sampling space. It is generally impractical to investigate all the elements of a population to find population parameters. In most cases, population parameters are generalized using representative, undistorted sample statistics.

Here, we will learn about sample statistics and some related important concepts such as sample mean, sample standard deviation, population parameter, etc.

Sample Mean

Now let us understand what we mean by sample statistics or sampling distribution. Assume that we want to determine the meanof a certain population. In real practice, we would essentially consider just one sample from the given population. However, in general, we consider the sample following the next sample, all the samples will be of the same size, say $n$, and then we compute the sample mean $\bar{x}$ each time.

The sample mean $x$ is an arbitrary variable, and it varies from sample to sample in a way that is quite unpredictable certainly. We write $\bar{x}$ when the sample means are thought of as an arbitrary variable, and write x for the values that it takes. The random variable $\bar{x}$ has a mean, it is denoted by $\mu \bar{x}$ and a standard deviation denoted by $\sigma \bar{x}$.

Thus, the sample mean is usually defined according to sample statistics in the following way. Assume that we have random samples of the count nthat are extracted from a population with mean and standard deviation be . Then sample mean will be the product of the mean population and average of the sample, i.e., $\mu \bar{x}$ and standard deviation: $\sigma \bar{x}$. Both the quantities must satisfy the following condition:

$\Rightarrow \mu_{\bar{x}} = \mu$ and $\sigma_{\bar{x}} = \dfrac{\sigma}{\sqrt{n}}$

At the same time, the sample mean formula is written differently in mathematics; estimating the sample mean is not different from estimating the population mean. The sample mean of the population can be easily derived by adding all the data points and dividing the sum of all data points by the total number of values in the given data set. Therefore, the sample mean is:

$\Rightarrow \bar{x}=\dfrac{\sum_{i=1}^{n} x_{i}}{n}$

Where,

$\sum_{i=1}^{n} x_{i}$ - The total sum of all the data points

$n$ - The total number of data points

Sample Standard Deviation

Sample statistics are numerical descriptive statistics of a sample. Statistics are generally derived from individual measurements in a sample. Statistics are the characteristics of the sample data distribution, such as mean, median, mode, standard deviation, and ratio. The sample statistic can be a measure of any sample characteristic.

The sample standard deviation of the population is usually determined using the formula $\sqrt{\dfrac{\left(x_{i}-\bar{x}\right)^{2}}{n}}$ to estimate the sample standard deviation of a tiny sample.

To get a fair estimate of the sample standard deviation, then in the numerator must be replaced by $(n-1)$. Therefore, the sample Standard Deviation formula or mean standard deviation is given by

$\Rightarrow S=\sqrt{\dfrac{\left(x_{i}-\bar{x}\right)^{2}}{n-1}}$

Population Parameter in Sample Statistics

Now, let us understand the Population parameter in statistics. One indicator found from the analysis of sample data is sample statistics. An important aspect of inference statistics is an attempt to estimate population parameters using sample statistics. The mean of the unbiased samples collected in a random way can be used as an estimator of the mean of the population represented by the sample and the population with a nearly normal distribution.

In particular, if the sample mean $\bar{x}$ is used as an estimate of the population mean, then $\bar{x}$ is said to be a point estimate of the population mean $\mu$. The sample mean $\bar{x}$ is the best estimate of the population mean over the sample median and sample mode. The reason is that the mean of several samples from the same population varies less than the mean and method of different samples.

Example:

1. The class teacher's sample set has seven examination scores: 78, 89, 93, 95, 88, 78, 95. Estimate the sample mean and sample standard deviation of the given set of data points.

Sol: Given, The total number of data points $= n =7$

The sum of all the data points $=\sum_{i=1}^{7} x_{i}=616$

We have a set of data points that are taken from various examinations. Let us tabulate the required information to calculate the sample mean and standard deviation.

We know that the sample mean is calculated using the mathematical formula:

$\Rightarrow \bar{x}=\dfrac{\sum_{i=1}^{n} x_{i}}{n} \ldots ...(1)$

Where,

$\sum_{i=1}^{n} x_{i}$ - The total sum of all the data points

$n$ - The total number of data points

Substituting all the values in the equation (1) and simplifying:

$\Rightarrow \bar{x}=\dfrac{\sum_{i=1}^{n} x_{i}}{n}$

$\Rightarrow \bar{x}=\dfrac{616}{7}$

$\Rightarrow \bar{x}=88$

Therefore, the sample mean for a given set of data is 88.

Now, the sample standard deviation can be found using the following formula:

$\Rightarrow S=\sqrt{\dfrac{\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}}{n-1}}$

Where,

$\bar{x}$ - The sample mean of the data points

Substituting all the required data in the above expression and simplifying:

$\Rightarrow S=\sqrt{\dfrac{\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}}{n-1}}$

$\Rightarrow S=\sqrt{\dfrac{(616-88)^{2}}{6}}$

$\Rightarrow S=215.5$

Therefore, the sample standard deviation and mean sample of the given data points are 215.5 and 88 respectively.

Conclusion:

A sample statistic is when your statistical information comes from a set of data or a set of a population.

A sample is simply a part of the population. Assume your population is every Indian, and you want to know how much the average person earns. Because time and money prevent you from knocking on every door in India, you decide to ask 1,000 random people. This is your sample of a thousand people.

You'll get some kind of statistic once you have your sample. A statistic is simply a piece of data—in this case, average earnings.