Sampling in Statistics
A sample statistic refers to quantity from the sample of the given population. A sample is a group of elements that are chosen from the population. The features which we use to describe the population are called the parameters and the properties of the sample data are known as statistics. Population and sample both are the important part of statistics. A sample statistic is a piece of information that we collect from a fraction of a population. Here we will study about sampling statistics methods, hypothesized mean, mean standard deviation and distribution of means.
What is a Sample in Statistics?
A sample statistic is a numerical descriptive measure of a sample data points. A statistic is generally derived from measurements of the individual data from the sample. The statistics are a characteristic of a sample data distribution such as mean, median, mode, standard deviation and proportions. A sample statistic can be used to measure any characteristic of the sample.
Hypothesized Mean
Hypothesis testing is an essential procedure in statistics. A hypothesis test used to evaluate two mutually exclusive statements about a population that determine which statement is best and also supported by the sample data.
The process of hypothesis testing involves setting up two competing hypotheses, first is null hypothesis and second one is alternate hypothesis.
The techniques for hypothesis testing depend on
(i) the type of outcome variable being analyzed (continuous, dichotomous, discrete)
(ii) the number of comparison groups in the investigation
(iii) whether the comparison groups are independent
Estimating the Mean
Following are the steps for estimating the mean :
Step 1. First we have to add a new column to the table writing down the midpoint (middle value) of each group.
Step 2. Multiply each midpoint value by the frequency of that group and then add the results in a new column.
Step 3. Add the values in the midpoint × frequency column.
Step 4. Finally, divide that value by the total frequency to get the estimate of the mean.
Sample Standard Deviation
The sample standard deviation formula is:
s = \[\sqrt{\frac{\sum (X - \bar{X})^{2}}{n-1}}\]
Sample standard deviation formula
where,
s = sample standard deviation
\[\sum\] = sum
\[\bar{X}\] = sample mean
n = number of scores in the sample.
Sampling Distribution
A sampling distribution is similar to a probability distribution of a statistic that we choose from random samples of a given population. It is also known as a finite-sample distribution, it represents the distribution of frequencies for how to spread apart various outcomes for a specific population.
The sampling distribution depends on multiple factors such as statistics, sample size, sampling process, and the overall population. It is used to help calculate statistics such as means, ranges, variances and standard deviations for the given sample.
Sample Mean
The sample mean refers to the average value found in a sample. A sample is just a small part of a whole data. For example, if we work for a polling company and want to know how much people pay for food a year, you aren’t going to want to poll over 300 million people. Instead of that, we take a fraction of that 300 million (perhaps a thousand people) that fraction is called a sample. In other words, mean refers to “average.” So in this example, the sample mean will be the average amount therefore those thousand people will have to pay for food a year.
The sample mean is useful when we have to estimate what the whole population is doing, without surveying everyone. Suppose sample mean for the food example was ＄2400 per year. The odds that we will get is a very similar figure if we surveyed all 300 million people. So the sample mean is a way to save a lot of time as well as money.
The Sample Mean Formula
The sample mean formula is: \[\bar{X}\] = \[\frac{\sum x_{i}}{n}\]
Here
\[\bar{X}\] just stands for the “sample mean”
\[\sum\] is summation notation
x\[_{i}\] “all of the x-values”
n is number of items in the sample mean
Mean and Standard Deviation
The mean refers to average or the most common value in a collection of numbers. There are multiple ways to calculate the mean. There are the two most popular methods i.e Arithmetic mean and geometric mean.
A standard deviation is the measurement of the distribution of a dataset which is related to its mean and it is calculated by the square root of the variance. It is calculated as the square root of variance by determining each data point's deviation which is relative to the mean. If the data points are further from the mean, then there is a chance of higher deviation within the data set. Therefore, the more spread out the data, the higher is the standard deviation.
The Formula for Standard Deviation is Given Below:
Standard deviation = s = \[\sqrt{\frac{\sum_{i=1}^{n} (X_{i} - \bar{X})^{2}}{(n-1)}}\]
Where
X\[_{i}\] = It is the of the i\[^{th}\] point in the data set
\[\bar{X}\] = It is the mean value of the data set
X = It is the number of data points in the data set
Probability Sample
Probability sampling is a sampling technique that is used by researchers to choose samples from a larger population using a method that is based on the theory of probability. For a participant to be considered as a probability sample, they must be selected using a random selection.
The most critical requirement of probability sampling is that everyone in the population is known and they have equal chance of getting selected. Suppose, if we have a population of 100 people, and every person would have odds of 1 in 100 for getting selected. In this case probability sampling gives us the best chance to create a sample that is mainly representative of the population.
It uses statistical theory while selecting a small group of people (or sample) from an existing large population and then predicts all their responses that will match with the overall population.
Errors in Sampling
Sampling error often occurs when the sample we use in the study is not representative of the whole population. It often occurs, that’s why, researchers always calculate a margin of error during final results as a statistical practice. The margin error is the amount of error that is allowed for miscalculation while representing the difference between the sample and the actual population. We can control and eliminate these sampling by creating a sample design, having a large enough sample to reflect the entire population, or using an online sample or survey audience to collect responses.
FAQs on Sampling
1. What is sampling in the context of statistics?
In statistics, sampling is the process of selecting a subset of individuals or items from a larger group, known as the population. The goal is to use the data collected from this subset, or sample, to make educated guesses or inferences about the characteristics of the entire population, without having to study everyone in it.
2. What is the fundamental difference between a sample and a population?
The main difference lies in their scope. A population includes every single member of a group being studied (e.g., all students in a school). A sample is a smaller, manageable portion of that population that is selected for actual study (e.g., 100 students randomly chosen from that school). We analyse the sample to understand the population.
3. What are the main types of sampling methods explained in the CBSE syllabus?
Sampling methods are broadly divided into two main categories:
- Probability Sampling: Every member of the population has a known, non-zero chance of being selected. This method relies on randomisation. Examples include Simple Random Sampling, Stratified Sampling, and Systematic Sampling.
- Non-Probability Sampling: Selection is not based on random chance. The researcher uses their judgment or convenience to choose the sample. Examples include Convenience Sampling and Quota Sampling.
4. How do you calculate the sample mean and sample standard deviation?
To analyse a sample, two key formulas are used:
Sample Mean (x̄): This is the average of the sample data. It is calculated by summing all the data points (Σxᵢ) and dividing by the number of data points (n). The formula is: x̄ = (Σxᵢ) / n.
Sample Standard Deviation (s): This measures the spread or dispersion of data points around the sample mean. The formula is: s = √[ Σ(xᵢ - x̄)² / (n-1) ], where 'n-1' represents the degrees of freedom.
5. What is a sampling distribution and why is it important in statistics?
A sampling distribution is a theoretical probability distribution of a statistic (like the mean or proportion) obtained from a large number of random samples of a specific size from a population. It's important because it allows us to understand how much a sample statistic is likely to vary from one sample to another, which is crucial for hypothesis testing and estimating population parameters with a certain level of confidence.
6. Why is 'n-1' used in the formula for sample standard deviation instead of just 'n'?
Using 'n-1' in the denominator, known as Bessel's correction, provides an unbiased estimate of the population standard deviation. A sample's variance is typically smaller than the true population variance. Dividing by the smaller number 'n-1' instead of 'n' slightly increases the result, correcting this tendency and giving a more accurate reflection of the spread within the entire population from which the sample was drawn.
7. In what real-world scenarios is probability sampling preferred over non-probability sampling?
Probability sampling is preferred in scenarios where the results need to be generalised to an entire population with high accuracy and minimal bias. For example:
- National Surveys: Government census or national opinion polls must accurately reflect the entire country's demographics and views.
- Scientific Research: Clinical trials or psychological studies require random samples to ensure the findings are statistically valid and applicable to a wider group.
- Quality Control: A factory might randomly sample products from an assembly line to estimate the defect rate of the entire production batch.
8. What is the difference between sampling error and non-sampling error?
Sampling error is the natural variation that occurs simply because a sample is not a perfect representation of the entire population. It is unavoidable but can be reduced by increasing the sample size. In contrast, non-sampling error arises from mistakes in the data collection process itself, such as poorly worded survey questions, measurement mistakes, or data entry typos. These errors can occur even if a whole population is studied and are not fixed by a larger sample.
