Parameters and Statistics

Statistics and parameters are two fundamental concepts in statistical theory. Although they may sound equal, there is a sharp difference between the two. One is used to represent the population, and the other is used to represent the sample. Now we will focus on the sample and population:

Population: A population refers to the whole data. It is the dataset that the statisticians use to derive conclusions or gain insights about the data.

Sample: Sample refers to the small dataset. It is considered to be a subset of population. Since population can be huge and may be difficult to examine, Statisticians usually consider a subset or sample, perform Statistical analysis and derive conclusions about the Population.

It is to be noted that the sample to be selected should be random in nature. If the subgroup or sample is not randomly selected, it may produce biased results.

A parameter is a numerical quantity that focuses on the whole population. It represents the characteristics of the whole population. It is to be noted that parameters are usually unknown and that they are used to summarize the whole population. To derive these values, we make use of statistical values, which will be discussed later. Common examples of parameters include:

• Population Mean: We use population mean when we want to find the average of the population. Population mean is denoted by μ.

• Population standard deviation: It is denoted by σ. It is used to find the dispersion of data in the population.

• Population proportion: We use p to denote population proportion. It is a parameter that is used to calculate percentage of the data points that follow some specific pattern.

Statistics is a numerical quantity that is derived from a sample. Since sample vary in nature, the Statistics are also variable in nature. Since statistical values refer to the sample, using these values, we can derive the parameter values.

• Sample mean (x̄): is used to find the average of the sample.

• Sample standard deviation: It is denoted by s. This is used to find the standard deviation of the sample data.

• Sample proportion (p̂): Often read as p-hat, this is used to find the fraction of the data points that possess some characteristic.

Sample is a subset of population. Basically we make use of statistical values to estimate parameters. There are different techniques. Let us go through each of them:

Sampling

Sampling is a fundamental concept in which we select random samples from the population. We use those sample to draw statistical inference about the population. It is to be noted that Random Sampling should be used so that our samples are diverse in nature. If samples are not randomly selected, then it will generate biased results. Different Sampling Techniques include:

• Simple random sampling
• Systematic sampling
• Stratified sampling
• Cluster sampling
• Probabilistic Sampling

Law of Large Numbers

Law of Large Numbers is a mathematical theorem which states that as the sample size increases, the sample mean becomes closer to the population mean. For example: Suppose we have a huge dataset which comprises of 50 numbers. Now if we select two or three numbers as our samples, then the average will not be even close to the average of the number. However if consider sample size 20-20, then the sample mean will come closer to the population mean.

Central Limit Theorem

Before going through the central limit theorem, we must be familiar with Normal Distribution. Normal Distribution or Gaussian distribution is a bell shaped curve which maintains its symmetry about the mean.

This theorem states that as the sample size increases the sample means follow a normal distribution or the Gaussian Distribution.

Let us follow some steps to estimate the population parameter using Statistical values:

• Derive a sample from the Population. The sample should be random in nature.

• Calculate the values of sample statistics.

• Use the above results to estimate population parameters.

Now there are two techniques to estimate population parameters. They are as follows:

Point Estimate

A point estimate is a numerical quantity or a statistical value of a sample that is used to estimate population parameter. It is used to estimate the population parameter. Although it provides best estimates but the drawback is for large population size it can generate erroneous results.

Interval Estimate

As the name suggests, interval estimate focuses on the intervals. Using the statistical values we estimate the intervals in which most likely the population parameter will lie. The most commonly used technique is Confidence Interval. Confidence Interval is a probabilistic value that provides estimation that a population parameter might fall between the set of values that has been provided. It is to be noted that Interval Estimate works better for large population as compared to Point Estimate.

Types of parameters and statistics in tabular form:

Parameters describe a population, while statistics describe a sample drawn from that population. Parameters are usually unknown and need to be estimated, while statistics are calculated from observed sample data.

Q1. Suppose we want to find the average height of the students from all over the country. We have sample of size 50. The average height of the sample is 160 cm. The sample standard deviation is 4. Construct a 95% confidence interval for population mean

Solution:

Sample mean=160 cm

Number of samples=50

Sample Standard Deviation = 4

The formula for the 95% confidence interval for the population mean is

[Tex]\bar{x} \pm Z \left( \frac{\sigma}{\sqrt{n}} \right) \\=160\pm1.96\times(\frac{4}{\sqrt50}) \\=(158.89,161.11)[/Tex]

So the average height of the population ranges between 158.89 cm and 161.11 cm

Q2. Calculate sample variance of the data 40,20,30,50,20

Solution:

To calculate the sample variance first we need to find sample mean

[Tex]\bar{x}=\frac{(40+20+30+50+20)}{5} \\=32[/Tex]

Now we will calculate the sample variance by finding the difference between the datapoint and the sample mean, then perform the sum of squared differences and then divide by n-1 where n is the number of samples

[Tex](40-32)^2=64 \\(20-32)^2=144 \\(30-32)^2=4 \\(50-32)^2=324 \\(20-32)^2=144[/Tex]

Let sum of differences be denoted by m

Sum of differences = 64+144+4+324+144=680

Sample variance = [Tex]\frac{m}{n-1} \\=\frac{680}{5-1} \\=170[/Tex]

Q3. Now calculate the standard deviation of the above data

Solution:

By definition we know that standard deviation is the square root of the variance.

So [Tex]s=\sqrt{170} \\=13.0384 \\\approx13.04[/Tex]

Q4. Calculate sample mean of the values 63,65,62,67.

Solution:

The sample size is 4

The sample values are 63,65,62,67

The sample mean is

[Tex]\bar{x}=\frac{(63+65+62+67)}{4} \\=\frac{257}{4} \\=64.25[/Tex]

Q1. Suppose we want to find the average weight of the students from all over the country. We have sample of size 100. The average weight of the sample is 60 kg. The sample standard deviation is 7. Construct a 90% confidence interval for population mean.(Refer to Z score table)

Q2. Calculate sample proportion of the random sample of size 200 where 80 samples follow certain trend. Also construct a 95% confidence interval for the true population proportion.

Q3. Find the sample mean of the dataset 1,2,3,4,5,6,7,18,19

Q4. Calculate sample standard deviation of the dataset 10,20,15,20,25.

Related Articles:

1. Population vs Sample in Statistic
2. Introduction of Statistics and its Types
3. Statistical Inference
4. Variance and Standard Deviation

Why do we use Statistics ?

It is used to estimate the population parameters. For large size population, we use the statistical values to draw conclusion from the sample which gives an overview of the population.

Explain the law of large numbers

The Law of Large Numbers states that as the sample size increases the statistical mean becomes almost equal to population mean.

Give examples of statistics and parameters

The examples of Statistics are Sample Mean, Sample Variance, Sample Standard Deviation etc. The examples of Parameters include Population Mean, Population Variance, Population Standard Deviation.

How is hypothesis testing useful in estimating population parameters?

Hypothesis Testing is a powerful statistical inference technique that is used to estimate population parameters using statistical values and probabilistic curves to solve the scenarios.

What is sampling and mention some types of sampling.

Sampling is the process of selecting samples from a population. It is varying in nature. Some common Sampling Methods are:

• Simple random sampling.
• Systematic sampling.
• Stratified sampling.
• Cluster sampling
• Probabilistic Sampling