An Introduction to 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.
FAQs on What is Sample Statistic?
1. What is a sample statistic in simple terms?
A sample statistic is a numerical measure or value, like an average or percentage, that is calculated from a subset (or sample) of a larger population. For instance, if you survey 100 students in a school of 1000 to find their average height, that average height is a sample statistic. It is used to estimate the corresponding value for the entire population.
2. How does a sample statistic differ from a population parameter?
The main difference lies in the group they describe. A population parameter is a value that describes a characteristic of the entire population (e.g., the average height of all students in a country). A sample statistic describes a characteristic of a smaller, manageable sample drawn from that population (e.g., the average height of 1000 students surveyed). We use statistics to make educated guesses about parameters.
3. What are the key formulas for calculating the sample mean and sample standard deviation?
The two most common sample statistic formulas as per the CBSE syllabus are:
- Sample Mean (x̄): This is the average of the sample data. The formula is:
x̄ = (Σxᵢ) / n
Where Σxᵢ is the sum of all data points and n is the number of data points in the sample. - Sample Standard Deviation (s): This measures the spread or dispersion of data around the sample mean. The formula is:
s = √[ Σ(xᵢ - x̄)² / (n-1) ]
Where (xᵢ - x̄) is the deviation of each data point from the sample mean.
4. What are some common types of sample statistics used in Maths?
Besides the mean and standard deviation, several other sample statistics are important in data analysis. The most common ones include:
- Sample Median: The middle value in an ordered dataset, which is less affected by outliers.
- Sample Variance (s²): The square of the sample standard deviation, representing the average squared deviation from the mean.
- Sample Proportion (p̂): The fraction of a sample that possesses a certain trait, often used in surveys and polls.
- Sample Mode: The value that appears most frequently in the sample data.
5. Why is the denominator (n-1) used for sample standard deviation instead of 'n'?
Using (n-1) in the denominator, known as Bessel's correction, provides an unbiased estimate of the true population standard deviation. When we use the sample mean (x̄) to calculate deviations, it's already an estimate of the true population mean (μ). This introduces a slight underestimation of the data's true variability. Dividing by a smaller number, (n-1), inflates the result slightly to correct for this bias, giving a more accurate picture of the population's spread.
6. What is the importance of a sample statistic in real-world research?
Sample statistics are fundamental to inferential statistics, which is the practice of drawing conclusions about a population based on sample data. It is often impossible or too expensive to collect data from an entire population. Therefore, researchers in fields like medicine, economics, and engineering rely on sample statistics to:
- Test hypotheses (e.g., Is a new drug effective?).
- Estimate population characteristics (e.g., What is the average income in a city?).
- Control quality in manufacturing (e.g., Does a batch of products meet the required standards?).
7. How does the size of a sample affect the reliability of a sample statistic?
The size of a sample has a direct impact on the reliability of a sample statistic. Generally, a larger sample size leads to a more reliable and precise estimate of the population parameter. This is because a larger sample is more likely to be representative of the population, reducing the effect of random fluctuations or outliers. The standard error, which measures the variability of a sample statistic, decreases as the sample size (n) increases, indicating greater precision.
