# Inferential Statistics – A Guide With Examples

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Inferential statistics is a branch of statistics that uses sampled data to make predictions or draw conclusions about a larger population or dataset. Using inferential statistics, you attempt to draw conclusions beyond the immediate facts. For instance, we use inferential statistics to infer what the population may believe from sample data. Alternatively, we utilize inferential statistics to determine whether a difference between groups seen in this study is either reliable or the result of random chance.

## Inferential Statistics – In a Nutshell

• Inferential statistics help conclude an entyre population by looking at only a population sample.
• Inferential statistics analyse a sample to conclude the population, whereas descriptive statistics describe the features of some known dataset.
• In inferential statistics, there are ways to test and validate our results from an experiment that involves hypothesis testing.
• Hypothesis tests are divided into three categories:
• Regression tests (Simple linear, Multiple linear)
• Comparison tests (T-test, ANOVA)
• Correlation tests (Chi-square, Pearson’s) that cheque different variables and parameters.

## Definition: Inferential statistics

Inferential statistics is a discipline that collects and analyses data based on a probabilistic approach. It helps us make conclusions and references about a population from a sample and their application to a larger population.

There are many types of inferential statistics, and each is appropriate for a research design and sample characteristics. It is used to compare two models to find which one is more statistically significant compared to the other.

Example

The following statements are clear examples of inferential statistics:

• Based on a survey, the mean weekly hours spent on gaming consoles by teenagers in the United Kingdom is 9.00 hours.
• In 2025, city b’s population will be 2.5 million.

## Descriptive statistics vs. inferential statistics

Descriptive statistics organise, summarize, and display the characteristics of a data set using bar graphs, histograms, or pie charts. They involve the measures of central tendency: Mean, Median, and Mode, measures of dispersion as the tools, and measures of variability: Range, variance, and standard deviation.

Inferential statistics allow us to test a hypothesis and assess whether the data is generalizable to the broader population. Sample data is also used to make inferences and draw conclusions about the people, and the results are in the form of probability.

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## Inferential statistics: Hypothesis testing

Hypothesis testing is a tool for making statistical analyses using inferential statistics. The aim is to compare populations between variables using samples.

It involves the following steps:

### 1. Determine the null and alternative Hypotheses

The null hypothesis (Ho) states the value of the population is assumed to be true. The alternative hypothesis (H1) contradicts the null hypothesis. It’s the informed guess of all contingencies not covered by the null hypothesis.

### 2. Selecting significance level

The criterion upon which we decide whether the claim is being tested is true or is determined.

### 3. Determine the rejection region

These consists of the values of the test statistic for which the null hypothesis is rejected.

Comparison of the samples and making two decisions based on the significance level. These include:

• Rejecting the null hypothesis: The sample average is associated with a low probability of occurrence when the null hypothesis is true, if the probability of obtaining a sample is less than 5%.
• Failing to reject the null hypothesis: The sample average is associated with a high probability of occurrence when the null hypothesis is true if the probability of obtaining a sample mean is greater than 5%.

Hypotheses are tested using inferential statistical tests that can be parametric (ANOVA, T-test), which is based on assumptions about the population distribution from which the sample is taken, or non-parametric (spearman’s correlation) not based on an assumption.

### 4. Comparison test

This inferential statistics test assesses whether there are differences in means, medians, or rankings of scores of two or more groups.

 Comparison test Parametric What’s being compared? Samples t-test ✔ Means 2 samples ANOVA ✔ Means 3+ samples Mood’s median ✘ Medians 2+ samples Wilcoxon signed-rank ✘ Distributions 2 samples Wilcoxon rank-sum (Mann-Whitney U) ✘ Sums of rankings 2 samples Kruskal-Wallis H ✘ Mean rankings 3+ samples

### 5. Correlation test

These inferential statistics tests determine the extent to which two variables are associated.

 Correlation test Parametric? Variables Pearson’s r ✔ Interval/ratio variables Spearman’s r ✘ Ordinal/interval/ratio variables Chi square test of independence ✘ Nominal/ordinal variables

### 6. Regression test

These inferential statistics tests demonstrate whether changes in predictor variables cause changes in an outcome variable.

 Regression test Predictor Outcome Simple linear regression 1 interval/ratio variable 1 interval/ratio variable Multiple linear regression 2+ interval/ratio variable(s) 1 interval/ratio variable Logistic regression 1+ any variable(s) 1 binary variable Nominal regression 1+ any variable(s) 1 nominal variable Ordinal regression 1+ any variable(s) 1 ordinal variable

## Inferential statistics example

The t-test value can be calculated with the following formula:

Example

After new sales training is given to employees, the mean sale goes up to £50 (a sample of 25 employees examined) with a standard deviation of £12. Before the training, the average sale was £100. cheque if the training helped at α = 0.05.

Solution: The t-test in inferential statistics solves this problem with the formula:

x = 150, μ = 100, s= 12, n = 25

H0: μ=100

H1: μ=100

= 20.83

The degree of freedom is given by 25 – 1 = 24. Using the t table at α = 0.05, the critical value is T(0.05, 24) = 1.71 . As 20.83 > 1.71 thus, H0 is rejected. The conclusion is that the training helped in increasing the average sales.

## FAQs

#### What is a sampling error?

It’s the difference between a population parameter and a sample statistic.

#### What is the difference between a population parameter and sample statistics?

Sample statistics involve change, as it depends upon sample values chosen randomly, hence becoming constant, while the population parameter is a descriptive measure for an entyre population.

#### What are the different sampling methods used in inferential statistics?

Random sampling, stratified sampling, clustre sampling, and systemic sampling are the most efficient methods used in inferential statistics.

#### What are the limitations of inferential statistics?

One limitation is that data provided is not fully measured. Therefore, you cannot be sure that the values or statistics you calculate are correct.

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