What are Inferential Statistics?
Inferential statistics use samples to draw conclusions about populations. Typically, it is impractical to measure every population member. Instead, we collect a random sample from a small portion of the population, measure them, and use their data to estimate population properties. Using correct inferential statistics procedures, you can use samples to draw reasonable conclusions about whole populations.
For example, you can’t measure the heights of all adult women, but you can measure a random sample and use that information to infer the mean heights of all women.
You gain tremendous benefits by working with samples because it is usually impossible to measure the entire population. However, there are tradeoffs. Most samples are tiny compared to the whole population. Consequently, when you estimate a population’s characteristics using a sample, the estimates are unlikely to equal the actual population value exactly.
For instance, your sample height mean is unlikely to equal the population height mean exactly. Inferential statistics uses various analytical tools to account for this error in the results.
If you’re only describing the characteristics of your sample and not generalizing beyond it, you can use simpler descriptive statistics.
At a broad level, inferential statistics requires the following three steps:
- Define the population we are studying.
- Draw a representative sample from that population.
- Use analyses that incorporate the sampling error.
The first step is identifying the population you want to study. That’s relatively simple. Let’s dig into the latter two steps!
Learn more about Descriptive vs. Inferential Statistics.
Inferential Statistics and Representative Samples
Inferential statistics involves generalizing the sample results to the population. To make that generalization, you must be confident that the sample accurately represents the population. This requirement affects how you should obtain that sample. If you use an incorrect method, the sample might not represent the population, leading you to erroneous conclusions. Learn more about Representative Samples.
In inferential statistics, simple random sampling is the most well-known method to obtain an unbiased sample. Using this method, researchers have an equal probability of selecting all items in the population. This approach helps ensure that the sample reflects the population. Additionally, all relevant subpopulations tend to be incorporated into the sample and represented accurately on average. Simple random sampling minimizes bias and simplifies data analysis.
However, you can use other probability sampling methods to obtain representative samples. Probability sampling selects a sample where each population unit has a known, non-zero chance of being randomly chosen.
Inferential Statistics and Point Estimates
It’s crucial to understand the difference between sample statistics and population parameters in inferential statistics. They both use numbers to summarize the properties of a population or sample, such as means, medians, standard deviations, proportions, and correlations.
- Parameters summarize population properties.
- Statistics summarize sample properties.
Researchers are usually more interested in understanding population parameters because the properties of a relatively small sample aren’t valuable to science. For example, scientists don’t care about a new medicine’s mean effect on a small group, a sample statistic. Instead, they want to generalize the results and understand its impact on a whole population, a parameter.
Unfortunately, parameters are usually unknowable because you can’t measure the entire population. However, random sampling produces statistics, such as the mean, that do not tend to be too high or too low. Hence, using inferential statistics, analysts can use sample statistics to estimate population parameters. This process helps science progress even without being able to measure an entire population.
For example, imagine our study of adult women in the U.S. finds that the random sample has a mean height of 63.5 inches (1.61 m). We can use that sample mean as the estimate of the population mean. That value is our best parameter estimate. Statisticians refer to it as the point estimate.
In other words, you’re using the sample to infer the population characteristics. Hence, inferential statistics.
Learn more about Populations vs. Samples: Uses & Examples.
Types of Parameters and Statistics
While parameters and statistics use the same kinds of summary values, statisticians represent them differently. Usually, we use Greek and upper-case Latin letters to denote parameters and lower-case Latin letters for statistics.
| Summary Value | Parameter | Statistic |
| Mean | μ or Mu | x̄ or x-bar |
| Standard deviation | σ or Sigma | s |
| Correlation | ρ or rho | r |
| Proportion | P | p̂ or p-hat |
Learn more about Parameters vs. Statistics: Examples & Differences.
Inferential Statistics Analysis
In inferential statistics, it’s virtually guaranteed that sample statistics from a study will be at least somewhat wrong compared to their corresponding population parameters. The difference between a sample statistic and a population parameter is sampling error. However, because the parameters are unknown, the degree of error is not apparent.
For example, our sample’s mean height (63.5 inches) won’t exactly equal the population mean, but we don’t know how wrong it might be.
Some error is unavoidable in inferential statistics because samples are typically miniscule compared to the population. Samples introduce fuzziness because there’s a margin of error around all point estimates of population parameters. We need to account for that error when drawing conclusions by using analyses designed for inferential statistics.
Fortunately, inferential statistics can estimate and incorporate that uncertainty into the results. For this purpose, statisticians have developed two broad types of procedures: confidence intervals and hypothesis tests.
Confidence Intervals
Confidence intervals (CIs) are a critical tool in inferential statistics. Instead of producing a point estimate, CIs use sample data to present a range of values that incorporates a margin of error around the point estimate. This range of values is where the unknown parameter is likely to fall.
For our height study example, we found an average height of 63.5 inches. We know our estimate is incorrect to some degree, but by how much? The confidence interval gives us an idea.
For instance, a confidence interval of [63 64] indicates that the population height likely falls within that range. We can be confident that the actual population height is within 0.5 inches of our sample average.
Learn about Interval Notation, which is how we write confidence intervals.
Hypothesis Tests
Hypothesis tests are a form of inferential statistics that assesses mutually exclusive hypotheses about population parameters and determines which one the sample data support.
For example, a hypothesis test can determine whether a medication is effective in a population. Imagine a study that finds the treatment group outcome is better than the control group. However, that difference between sample statistics could be sampling error rather than an actual effect. A hypothesis test helps us separate sampling error from true population effects by assessing whether the outcome parameter is better in the treatment group.
Hypothesis tests occur everywhere in statistics that you see p-values, from comparing means and proportions to assessing regression coefficients. When your p-value is less than the significance level, the results are statistically significant. You can conclude the effect you observe in a sample also exists in the population.
In inferential statistics, confidence intervals and hypothesis tests have many details to consider. The information above is just the tip of the iceberg. To learn more, consider reading the following:
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