Inferential Statistics

Inferential statistics is a field in statistics that involves using information derived from a sample to make estimates about and test hypotheses regarding the population the sample represents. This form of statistics is used whenever studying the whole population is impractical, unaffordable, or time-consuming.

Hypothesis testing, confidence intervals, and regression analysis, all of which are techniques used in inferential statistics, are all driven by one question – whether the results in the sample apply to the population as well.

What Is Inferential Statistics?

Inferential statistics is a statistical tool that analyzes sample data and makes conclusions about a population with the associated uncertainty level. Population refers to a total group under consideration while sample refers to a subset of the population.

For instance, a population can include all registered voters in a nation, all patients suffering from a particular disease in a certain year, and all products manufactured in a particular production line in a certain month. Samples taken from such a population would include 2,000 random voters, 300 patients in five different hospitals, and 50 products from one production line.

Inferential statistics differs from descriptive statistics by producing a conclusion not only about a sample but also about the population, and that conclusion always comes with a certain degree of uncertainty.  This uncertainty is measured either by a confidence level or a p-value.

Descriptive vs. Inferential Statistics

Descriptive statistics summarizes data that has already been collected. Inferential statistics uses that data to make a claim about data that has not been collected.

Aspect	Descriptive Statistics	Inferential Statistics
Purpose	Summarizes and organizes known data	Estimates and tests claims about unknown population data
Scope	Limited to the sample or dataset at hand	Extends conclusions to the full population
Common tools	Mean, median, mode, standard deviation, range	Hypothesis testing, confidence intervals, regression analysis
Output	A number or chart describing the data	A conclusion with a stated margin of error or significance level
Example	“The average order value in this sample of 500 customers is $42.”	“The average order value across all customers is likely between $40 and $44, at 95% confidence.”

They are not methods that compete against each other. Research usually begins with the use of descriptive statistics to arrange the data, after which inferential statistics is used to generalize the results.

Why Inferential Statistics Matters

Inferential statistics helps make decisions about an entire population without gathering data on each individual member of the population. There are four industries that depend directly on inferential statistics.

• Healthcare – In a medical experiment, 400 patients are selected to test the effect of a certain drug in lowering blood pressure. It does not involve treating all the people with the disease; it involves sampling some individuals and then generalizing the result. The majority of experiments reported in clinical settings use a significance level of 0.05.
• Marketing – A retail website tries out two prices to find out the one that will generate maximum sales in the future through a technique called A/B testing.
• Manufacturing – A company inspects 100 items from the total production of 10,000 to find out if the proportion of defective products falls within the accepted margin.
• Opinion polls – A poll of 1,500 respondents determines the opinions of the rest of the population. The margin of error in national polls usually ranges from 2 to 4 percent.

The Core Building Blocks

Population and Sample

The population is made up of all of the individuals belonging to the group being studied. The sample is made up of those individuals selected for analysis. The validity of any inference is based on the extent to which the sample resembles the population.

Sampling Techniques

The four methods of sampling that are commonly employed in inferential statistics include:

1. Simple random sampling – every individual has an equal probability of being selected from the population.
2. Stratified sampling – the population is stratified into groups such as age or geographical location; then a sample is selected from each group. 
3. Cluster sampling – the population is split into clusters such as geographical location; then random samples of the whole clusters are selected. 
4. Systematic sampling – members of the population are selected at equal intervals.

A sampling method that is biased will lead to a conclusion that is also biased irrespective of how big the sample size is and how accurately the statistical test is done afterwards.

Central Limit Theorem

According to the Central Limit Theorem, the distribution of the sample means will take on a normal distribution as the sample size gets larger, irrespective of the population distribution from which the sample was drawn. This holds true once the sample size reaches about 30. 

Using the Central Limit Theorem, it is possible to employ methods that are associated with normal distributions, such as the Z test.

Estimation — Point Estimates, Confidence Intervals & Margin of Error

Point Estimate

The point estimator is a value that is computed based on the sample observations used to estimate a population parameter. The most commonly used point estimate for the population mean is the sample mean. The most commonly used point estimate for the population proportion is the sample proportion.

Confidence Interval

The confidence interval can be defined as the range of values obtained through sampling that contains the true population parameter with a certain degree of confidence. The formula for a confidence interval for the population mean is:

Here, x̄ is the sample mean, Z(α/2) is the Z value corresponding to the selected level of confidence (1.96 for 95% confidence), σ is the population standard deviation, and n is the sample size. 

A 95% confidence interval means that if you repeated the whole procedure 100 times, approximately 95 of them would include the true parameter value. 

Margin of Error

Margin of error is the additional amount that is added or subtracted from an estimate in order to come up with a confidence interval. The margin of error reduces with increasing sample size and increases with increasing variability.

Hypothesis Testing, Step by Step

Hypothesis testing is defined as a formal way of deciding whether sample data supports an assumption about a population. 

Let us suppose there is an online retailer which sells some goods at $49. Now, the retailer wants to check whether reducing the price to $44 would increase the average daily sales of units.

1. The null and alternative hypotheses are as follows. Null Hypothesis (H0): There is no impact of the price change on the average number of units sold per day. Alternative Hypothesis (H1): The price change increases the average number of units sold per day. 
2. Choose the significance level (α). The retailer assumes α = 0.05, meaning the risk of rejecting the null hypothesis and making an error is 5%.
3. Collect the sample data. Daily units sold are gathered for 30 days at $49 and 30 days at $44.
4. Find the value of the test statistic. A two-sample t-test compares the average number of units sold per day by two samples while considering their sizes and dispersions.
5. Compare the p-value with the significance level. If the p-value calculated after that procedure turns out to be smaller than 0.05, then the null hypothesis will be rejected.
6. Conclusion.The p-value is less than 0.05; therefore, the null hypothesis will be rejected. 

A Full Worked Example, Start to Finish

The retailer from the previous section collects the following data over 30 days per group:

• $49 price group: mean = 82 units sold per day, standard deviation = 10, n = 30
• $44 price group: mean = 89 units sold per day, standard deviation = 11, n = 30

Step 1 — Calculate the pooled standard deviation:

Step 2 — Calculate the standard error:

Step 3 — Calculate the t-statistic:

Step 4 — Compare to the critical value. At α = 0.05, two-tailed, with 58 degrees of freedom, the critical t-value is approximately 2.00. Since 2.58 exceeds 2.00, the retailer rejects the null hypothesis.

Step 5 — Calculate the p-value. A t-statistic of 2.58 with 58 degrees of freedom produces a two-tailed p-value of approximately 0.012, below the 0.05 threshold.

Step 6 — Calculate the 95% confidence interval for the difference:

7 ± (2.00 × 2.71) = 7 ± 5.42 = 1.58 to 12.42 additional units sold per day

Conclusion: The retailer rejects the null hypothesis. The $5 price reduction is associated with an estimated increase of 1.6 to 12.4 units sold per day, with a point estimate of 7 additional units per day, at 95% confidence.

Choosing the Right Statistical Test

Choosing the right test depends on the number of groups being compared, the type of data collected, and whether or not the data is consistent with the assumptions for a parametric test. 

• Comparison of the means of two independent groups with known population standard deviation and sample size above 30 → Z-test
• Comparison of the means of two independent groups with unknown population standard deviation and sample size below 30 → t-test
• Comparison of the variance of two samples → F-test
• Comparison of the means of three or more groups → ANOVA
• Association between two categorical variables → Chi Square Test
• Comparing groups where data is not normally distributed → Non-parametric test (Mann-Whitney U, Wilcoxon Signed-Rank or Kruskal-Wallis)
• Predicting the continuous variable with respect to one or more variables → Regression analysis (simple or multiple linear regression)
• Predicting a binary variable → Logistic regression

Parametric Tests at a Glance

Test	When to Use	Key Assumption	Statistic
Z-test	n ≥ 30, population standard deviation known	Data approximately normal	Z = (x̄ − μ) / (σ / √n)
T-test	n < 30, population standard deviation unknown	Data approximately normal	t = (x̄ − μ) / (s / √n)
F-test	Comparing variances of two samples	Both samples approximately normal	F = s1² / s2²
ANOVA	Comparing means of three or more groups	Group variances approximately equal	F = between-group variance ÷ within-group variance

Non-Parametric Tests at a Glance

Test	When to Use	Key Characteristic
Chi-Square test	Testing association between two categorical variables	Does not assume a normal distribution
Mann-Whitney U test	Comparing two independent groups with non-normal data	Ranks values instead of using raw means
Wilcoxon Signed-Rank test	Comparing two related (paired) samples with non-normal data	Ranks the differences between paired values
Kruskal-Wallis test	Comparing three or more independent groups with non-normal data	Extends the Mann-Whitney U test to multiple groups

Common Mistakes in Inferential Statistics

• Taking the p-value as the probability of the null hypothesis being true. A p-value simply determines the probability of getting such an outcome based on the truth of the null hypothesis.
• Drawing population inference from a sample that is not randomly selected. A sample that is not randomly selected cannot be considered a representative of the population regardless of how large the sample is.
• Mistaking statistical significance for practical significance. The finding may lead to a p-value that is lower than 0.05 but the finding may be insignificant to making any decision.
• Performing different hypotheses on the same data set without adjusting the significance level. The chances of obtaining a Type I error become higher when several hypothesis tests are conducted using the same data.
• Not considering the assumptions of the particular test. Using a T-test without checking the assumption of the normality of the data yields meaningless conclusions.

When Not to Use Inferential Statistics

• When the sample is not a random selection from the population.
• When the sample size is too small to satisfy the assumptions of the test being used.
• When the dataset already constitutes the whole population and therefore descriptive statistics are adequate alone.
• When the dataset does not satisfy the basic assumptions of the test being used without using any non-parametric test.

Tools for Running Inferential Statistics

• R — an open-source statistical programming language with built-in functions for hypothesis testing and regression analysis.
• Python — supports inferential statistics through libraries including SciPy, StatsModels, and Pandas. Running scipy.stats.ttest_ind() on two samples returns both the t-statistic and the p-value directly.
• SPSS — a menu-driven statistical software package used widely in social science research.
• Excel — supports basic inferential functions, including T-tests and confidence intervals, through the Data Analysis ToolPak.

Frequently Asked Questions