Inferential statistics is a branch of statistics that makes the use of various analytical tools to draw inferences about the population data from sample data. Apart from inferential statistics, descriptive statistics forms another branch of statistics. Inferential statistics help to draw conclusions about the population while descriptive statistics summarizes the features of the data set.

There are two main types of inferential statistics - hypothesis testing and regression analysis. The samples chosen in inferential statistics need to be representative of the entire population. In this article, we will learn more about inferential statistics, its types, examples, and see the important formulas.

1. | What is Inferential Statistics? |

2. | Types of Inferential Statistics |

3. | Inferential Statistics Examples |

4. | Inferential Statistics vs Descriptive Statistics |

5. | FAQs on Inferential Statistics |

## What is Inferential Statistics?

Inferential statistics helps to develop a good understanding of the population data by analyzing the samples obtained from it. It helps in making generalizations about the population by using various analytical tests and tools. In order to pick out random samples that will represent the population accurately many sampling techniques are used. Some of the important methods are simple random sampling, stratified sampling, cluster sampling, and systematic sampling techniques.

### Inferential Statistics Definition

Inferential statistics can be defined as a field of statistics that uses analytical tools for drawing conclusions about a population by examining random samples. The goal of inferential statistics is to make generalizations about a population. In inferential statistics, a statistic is taken from the sample data (e.g., the sample mean) that used to make inferences about the population parameter (e.g., the population mean).

## Types of Inferential Statistics

Inferential statistics can be classified into hypothesis testing and regression analysis. Hypothesis testing also includes the use of confidence intervals to test the parameters of a population. Given below are the different types of inferential statistics.

### Hypothesis Testing

Hypothesis testing is a type of inferential statistics that is used to test assumptions and draw conclusions about the population from the available sample data. It involves setting up a null hypothesis and an alternative hypothesis followed by conducting a statistical test of significance. A conclusion is drawn based on the value of the test statistic, the critical value, and the confidence intervals. A hypothesis test can be left-tailed, right-tailed, and two-tailed. Given below are certain important hypothesis tests that are used in inferential statistics.

**Z Test:** A z test is used on data that follows a normal distribution and has a sample size greater than or equal to 30. It is used to test if the means of the sample and population are equal when the population variance is known. The right tailed hypothesis can be set up as follows:

Null Hypothesis: \(H_{0}\) : \(\mu = \mu_{0}\)

Alternate Hypothesis: \(H_{1}\) : \(\mu > \mu_{0}\)

Test Statistic: z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\). \(\overline{x}\) is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the population standard deviation and n is the sample size.

Decision Criteria: If the z statistic > z critical value then reject the null hypothesis.

**T Test:** A t test is used when the data follows a student t distribution and the sample size is lesser than 30. It is used to compare the sample and population mean when the population variance is unknown. The hypothesis test for inferential statistics is given as follows:

Null Hypothesis: \(H_{0}\) : \(\mu = \mu_{0}\)

Alternate Hypothesis: \(H_{1}\) : \(\mu > \mu_{0}\)

Test Statistics: t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\)

Decision Criteria: If the t statistic > t critical value then reject the null hypothesis.

**F Test:** An f test is used to check if there is a difference between the variances of two samples or populations. The right tailed f hypothesis test can be set up as follows:

Null Hypothesis: \(H_{0}\) : \(\sigma_{1}^{2} = \sigma_{2}^{2}\)

Alternate Hypothesis: \(H_{1}\) : \(\sigma_{1}^{2} > \sigma_{2}^{2}\)

Test Statistic: f = \(\frac{\sigma_{1}^{2}}{\sigma_{2}^{2}}\), where \(\sigma_{1}^{2}\) is the variance of the first population and \(\sigma_{2}^{2}\) is the variance of the second population.

Decision Criteria: If the f test statistic > f test critical value then reject the null hypothesis.

**Confidence Interval: **A confidence interval helps in estimating the parameters of a population. For example, a 95% confidence interval indicates that if a test is conducted 100 times with new samples under the same conditions then the estimate can be expected to lie within the given interval 95 times. Furthermore, a confidence interval is also useful in calculating the critical value in hypothesis testing.

Apart from these tests, other tests used in inferential statistics are the ANOVA test, Wilcoxon signed-rank test, Mann-Whitney U test, Kruskal-Wallis H test, etc.

### Regression Analysis

Regression analysis is used to quantify how one variable will change with respect to another variable. There are many types of regressions available such as simple linear, multiple linear, nominal, logistic, and ordinal regression. The most commonly used regression in inferential statistics is linear regression. Linear regression checks the effect of a unit change of the independent variable in the dependent variable. Some important formulas used in inferential statistics for regression analysis are as follows:

Regression Coefficients:

The straight line equation is given as y = \(\alpha\) + \(\beta x\), where \(\alpha\) and \(\beta\) are regression coefficients.

\(\beta = \frac{\sum_{1}^{n}\left ( x_{i}-\overline{x} \right )\left ( y_{i}-\overline{y} \right )}{\sum_{1}^{n}\left ( x_{i}-\overline{x} \right )^{2}}\)

\(\beta = r_{xy}\frac{\sigma_{y}}{\sigma_{x}}\)

\(\alpha = \overline{y}-\beta \overline{x}\)

Here, \(\overline{x}\) is the mean, and \(\sigma_{x}\) is the standard deviation of the first data set. Similarly, \(\overline{y}\) is the mean, and \(\sigma_{y}\) is the standard deviation of the second data set.

## Inferential Statistics Examples

Inferential statistics is very useful and cost-effective as it can make inferences about the population without collecting the complete data. Some inferential statistics examples are given below:

- Suppose the mean marks of 100 students in a particular country are known. Using this sample information the mean marks of students in the country can be approximated using inferential statistics.
- Suppose a coach wants to find out how many average cartwheels sophomores at his college can do without stopping. A sample of a few students will be asked to perform cartwheels and the average will be calculated. Inferential statistics will use this data to make a conclusion regarding how many cartwheel sophomores can perform on average.

## Inferential Statistics vs Descriptive Statistics

Descriptive and inferential statistics are used to describe data and make generalizations about the population from samples. The table given below lists the differences between inferential statistics and descriptive statistics.

Inferential Statistics | Descriptive Statistics |
---|---|

Inferential statistics are used to make conclusions about the population by using analytical tools on the sample data. | Descriptive statistics are used to quantify the characteristics of the data. |

Hypothesis testing and regression analysis are the analytical tools used. | Measures of central tendency and measures of dispersion are the important tools used. |

It is used to make inferences about an unknown population | It is used to describe the characteristics of a known sample or population. |

Measures of inferential statistics are t-test, z test, linear regression, etc. | Measures of descriptive statistics are variance, range, mean, median, etc. |

**Related Articles:**

- Probability and Statistics
- Data Handling
- Summary Statistics

**Important Notes on Inferential Statistics**

- Inferential statistics makes use of analytical tools to draw statistical conclusions regarding the population data from a sample.
- Hypothesis testing and regression analysis are the types of inferential statistics.
- Sampling techniques are used in inferential statistics to determine representative samples of the entire population.
- Z test, t-test, linear regression are the analytical tools used in inferential statistics.

## FAQs on Inferential Statistics

### What is the Meaning of Inferential Statistics?

Inferential statistics is a field of statistics that uses several analytical tools to draw inferences and make generalizations about population data from sample data.

### What are the Types of Inferential Statistics?

There are two main types of inferential statistics that use different methods to draw conclusions about the population data. These are regression analysis and hypothesis testing.

### What are the Different Sampling Methods Used in Inferential Statistics?

It is necessary to choose the correct sample from the population so as to represent it accurately. Some important sampling strategies used in inferential statistics are simple random sampling, stratified sampling, cluster sampling, and systematic sampling.

### What are the Different Types of Hypothesis Tests In Inferential Statistics?

The most frequently used hypothesis tests in inferential statistics are parametric tests such as z test, f test, ANOVA test, t test as well as certain non-parametric tests such as Wilcoxon signed-rank test.

### What is Inferential Statistics Used For?

Inferential statistics is used for comparing the parameters of two or more samples and makes generalizations about the larger population based on these samples.

### Is Z Score a Part of Inferential Statistics?

Yes, z score is a fundamental part of inferential statistics as it determines whether a sample is representative of its population or not. Furthermore, it is also indirectly used in the z test.

### What is the Difference Between Descriptive and Inferential Statistics?

Descriptive statistics is used to describe the features of some known dataset whereas inferential statistics analyzes a sample in order to draw conclusions regarding the population.

## FAQs

### What is the formula for inferential statistics? ›

**s X = s/√n**. This formula shows how it is that the accuracy of the estimate provided by a sample increases as the sample size increases.

**What is the formula for statistics? ›**

Mean | x ¯ = ∑ x n |
---|---|

Median | If n is odd, then M = ( n + 1 2 ) t h term If n is even, then M = ( n 2 ) t h t e r m + ( n 2 + 1 ) t h t e r m 2 |

Mode | The value which occurs most frequently |

Variance | σ 2 = ∑ ( x − x ¯ ) 2 n |

Standard Deviation | S = σ = ∑ ( x − x ¯ ) 2 n |

**Why do we calculate inferential statistics? ›**

Inferential statistics **allow you to test a hypothesis or assess whether your data is generalisable to the broader population**.

**What are the 4 methods in statistics? ›**

It all comes down to using the right methods for statistical analysis, which is how we process and collect samples of data to uncover patterns and trends. For this analysis, there are five to choose from: **mean, standard deviation, regression, hypothesis testing, and sample size determination**.

**What are the 4 basic elements of statistics? ›**

**Sample size, variables required, numerical summary tools, and conclusions** are the four elements of a descriptive statistics problem.

**What are the 5 types of statistics? ›**

...

**Descriptive Statistics**

- Measure of frequency.
- Measure of dispersion.
- Measure of central tendency.
- Measure of position.

**What are 3 examples of statistics? ›**

**8 Examples of How Statistics is Used in Real Life**

- Example 1: Weather Forecasting.
- Example 2: Sales Tracking.
- Example 3: Health Insurance.
- Example 4: Traffic.
- Example 5: Investing.
- Example 6: Medical Studies.
- Example 7: Manufacturing.
- Example 8: Urban Planning.

**What is an inferential test examples? ›**

inferential test

any statistical procedure used to evaluate hypotheses about differences between sample and population distributions. Examples include **the chi-square test, the F test, and the t test**. Inferential tests more commonly are known as significance tests (see significance testing).

**What is an inferential sample? ›**

Inferential statistics helps study a sample of data and make conclusions about its population. A sample is **a smaller data set drawn from a larger data set called the population**. If the sample does not represent the population, one cannot make accurate estimations related to the latter.

**What are the 3 components of statistics? ›**

There are three real branches of statistics: **data collection, descriptive statistics and inferential statistics**.

### What is inferential statistics and types? ›

Inferential statistics **helps to suggest explanations for a situation or phenomenon**. It allows you to draw conclusions based on extrapolations, and is in that way fundamentally different from descriptive statistics that merely summarize the data that has actually been measured.

**What is the formula for probability? ›**

In general, the probability is the ratio of the number of favorable outcomes to the total outcomes in that sample space. It is expressed as, Probability of an event **P(E) = (Number of favorable outcomes) ÷ (Sample space)**.

**What is the formula for population in statistics? ›**

Population Mean **(μ) = ∑X / N**

And, N is the count of data in X.

**How to find standard deviation? ›**

**Step 1: Find the mean.** **Step 2: For each data point, find the square of its distance to the mean.** **Step 3: Sum the values from Step 2.** **Step 4: Divide by the number of data points**.

**How is probability used in inferential statistics? ›**

Answer and Explanation: In inferential statistics, the probability (or p value) is used **to accept or reject a null hypothesis on the basis of which conclusions about the sample and population relationship can be made**. It provides information on how often the result may occur by chance alone.

**Is chi-square an inferential statistic? ›**

**The chi-square test of independence is an inferential statistical test**, meaning that it allows you to draw conclusions about a population based on a sample. Specifically, it allows you to conclude whether two variables are related in the population.

**What are the 5 common statistical tools? ›**

- Mean: The statistical analysis methods utilized mean, which is all the more normally alluded to as the average. ...
- Standard deviation: ...
- Regression: ...
- Hypothesis testing: ...
- Sample Size Determination:

**What are the 5 types of methods? ›**

A popular and helpful categorization separate qualitative methods into five groups: **ethnography, narrative, phenomenological, grounded theory, and case study**. John Creswell outlines these five methods in Qualitative Inquiry and Research Design.

**What are the 5 basic methods of statistical analysis? ›**

The five basic methods are **mean, standard deviation, regression, hypothesis testing, and sample size determination**.

**What is the 2 types of statistics? ›**

**Descriptive and Inferential Statistics**

The two major areas of statistics are known as descriptive statistics, which describes the properties of sample and population data, and inferential statistics, which uses those properties to test hypotheses and draw conclusions.

### What is statistical question Give 5 examples? ›

Examples of Statistical Questions

**What time did the students in this class get up this morning?** How many votes did the winning candidate for the Presidents of the Student Body receive in each of the past 20 years? What were the high temperatures in all of the Latin American capitals today?

**What is descriptive statistic examples? ›**

If you want a good example of descriptive statistics, look no further than **a student's grade point average (GPA)**. A GPA gathers the data points created through a large selection of grades, classes, and exams then average them together and presents a general idea of the student's mean academic performance.

**What is statistics definitions and examples? ›**

A statistic is **a number that represents a property of the sample**. For example, if we consider one math class to be a sample of the population of all math classes, then the average number of points earned by students in that one math class at the end of the term is an example of a statistic.

**What are the 8 descriptive statistics? ›**

Descriptive statistics are broken down into measures of central tendency and measures of variability (spread). Measures of central tendency include the **mean, median, and mode, while measures of variability include standard deviation, variance, minimum and maximum variables, kurtosis, and skewness**.

**What is an inferential method? ›**

Inferential statistical analysis is **the method that will be used to draw the conclusions**. It allows users to infer or conclude trends about a larger population based on the samples that are analyzed. Basically, it takes data from a sample and then makes conclusions about a larger population or group.

**What is basic inferential statistics? ›**

The goal of inferential statistics is to discover some property or general pattern about a large group by studying a smaller group of people in the hopes that the results will generalize to the larger group.

**What is sample size in inferential statistics? ›**

Sample size, **the number of participants in a study**, is a key factor in Type II errors. Typically, the higher your sample size, the lower your chance of a Type II error. In other words, the more people you have from a population, the more generalizable (able to be applied to the general population) your results are.

**What is the 4 step process in statistics? ›**

Consider statistics as a problem-solving process and examine its four components: **asking questions, collecting appropriate data, analyzing the data, and interpreting the results**.

**How to calculate the mean? ›**

It's obtained by simply **dividing the sum of all values in a data set by the number of values**. The calculation can be done from raw data or for data aggregated in a frequency table.

**What is inferential statistics in math? ›**

Inferential statistics:

Inferential statistics **aim to test hypotheses and explore relationships between variables, and can be used to make predictions about the population**. Inferential statistics are used to draw conclusions and inferences; that is, to make valid generalisations from samples.

### Is chi square test inferential statistics? ›

**The chi-square test of independence is an inferential statistical test**, meaning that it allows you to draw conclusions about a population based on a sample. Specifically, it allows you to conclude whether two variables are related in the population.

**What is the logic of inferential statistics? ›**

The goal in classic inferential statistics is to prove the null hypothesis wrong. The logic says that **if the two groups aren't the same, then they must be different**. A low p-value indicates a low probability that the null hypothesis is correct (thus, providing evidence for the alternative hypothesis).

**Is an ANOVA test inferential? ›**

Another fundamental set of **inferential statistics falls under the general linear model and includes analysis of variance (ANOVA)**, correlation and regression. To determine whether group means are different, use the t test or the ANOVA.

**Is standard deviation inferential statistics? ›**

False. Descriptive statistics gives the characteristics of the sample and the population. For example, **the mean and standard deviation are examples of descriptive statistics**. On the other hand, inferential statistics refers to making conclusions about the population using the descriptive statistics.

**What are the methods used in inferential statistics? ›**

The most common methodologies in inferential statistics are **hypothesis tests, confidence intervals, and regression analysis**. Interestingly, these inferential methods can produce similar summary values as descriptive statistics, such as the mean and standard deviation.

**What is p value in inferential statistics? ›**

The P value is defined as **the probability under the assumption of no effect or no difference (null hypothesis), of obtaining a result equal to or more extreme than what was actually observed**. The P stands for probability and measures how likely it is that any observed difference between groups is due to chance.

**What is P value in statistics? ›**

What exactly is a p value? The p value, or probability value, **tells you how likely it is that your data could have occurred under the null hypothesis**. It does this by calculating the likelihood of your test statistic, which is the number calculated by a statistical test using your data.