The Building Blocks of Inferential Statistics, Explained

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The Building Blocks of Inferential Statistics

Research Methods in Psychology · Chapter 15: Inferential Statistics and Hypothesis TestingSection 15.1 · lesson 87 of 949 min · free with a free account

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In this lesson

From a sample in hand to claims about the world that produced it.

  • Separate populations from samples, parameters from statistics
  • Explain how a sampling distribution links the two
  • Define and interpret the standard error of the mean
  • Identify the assumptions behind classical inference

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Everything said in the video, section by section (1,343 words). Read along, or search it.

Inferential Statistics and Hypothesis Testing

The Building Blocks of Inferential Statistics

You measured fifty people. You want to say something about millions. That leap is the entire subject of inferential statistics. By the end of this section you will distinguish a population from a sample. And a parameter from a statistic, which is the same distinction wearing different clothes. You will explain the role of sampling distributions in connecting the two. You will define and interpret the standard error of the mean. You will see exactly how sample size affects the precision of an inference. And you will identify the assumptions that underpin classical inferential procedures.

The Transition

What Inferential Statistics Add

Descriptive statistics summarize the data a researcher has collected. Inferential statistics go further, and the extra step is a real one. They use that summary to draw conclusions about the larger population. The population the sample was drawn from. That transition is not trivial, and it is worth seeing why. The mean of a sample describes that sample exactly. There is no uncertainty about it. You computed it from the numbers you have. But it only approximates the corresponding value in the population. So the central question of inferential statistics has two halves. How much does the sample tell us about the population? And how confident can we be in that answer?

Four Words

Population, Sample, Parameter, Statistic

Four words carry the framework, and they pair up neatly. A population is the complete set of individuals, observations, or units. The complete set about which the researcher wants to draw conclusions. A sample is the subset of that population actually included in the study. Now the second pair, which are the numbers rather than the groups. A parameter is a value that characterizes the whole population. It is fixed, and it is usually unknown. That combination is the difficulty. A statistic is a value computed from the data in hand. Its job is to estimate the corresponding parameter. So populations have parameters, and samples have statistics.

Notation

Greek for Parameters, Roman for Statistics

The notation encodes that distinction, and once you know it, results sections open up. Greek letters are reserved for population parameters. Roman letters are used for sample statistics. The population mean is mu. The sample mean is capital M, or x bar. The population standard deviation is sigma. The sample standard deviation is lowercase s, or S D. Variance follows the same pattern, as sigma squared and s squared. A population proportion is pi, and a sample proportion is lowercase p. A population correlation is rho, and a sample correlation is lowercase r. That last pair matters, because rho and r look alike and mean different things. Reading the alphabet correctly is genuinely half of reading a results section.

The Common Slip

What You Have vs What You Wish You Had

Here is the slip that beginners make constantly. Blurring the distinction between population and sample. Treating the sample mean as if it simply were the population mean. But the whole point of inferential statistics is that they are not the same thing. Put it as plainly as possible. The sample mean is what you have. The population mean is what you wish you had. Inferential procedures exist to bridge that gap. But they can only do it if the gap is acknowledged in the first place. A researcher who forgets the gap has no use for any of the tools that follow.

The Bridge

The Sampling Distribution

So how do you bridge it? With an idea that starts as a thought experiment. Imagine drawing a single sample of twenty five participants from a population. Compute its mean, and write the result down. Now do it again, with a fresh sample of twenty five from the same population. The new sample mean will be similar to the first, but not identical. Random fluctuation in who happens to be selected guarantees some variation. Now repeat the process many times over. The collection of sample means becomes a distribution in its own right. That is the sampling distribution of the mean. Notice that it is a distribution of means, not a distribution of people.

Three Properties

Why the Sampling Distribution Is Central

The sampling distribution has three properties that make it the central tool. First, its mean equals the population mean. Across enough samples, the average sample mean converges on the parameter you care about. So sample means are not biased. They scatter around the right answer. Second, its standard deviation is smaller than the population's. That standard deviation has its own name, the standard error. It is smaller because averaging across observations smooths out individual variability. Third, and most remarkably, it is approximately normal. That comes from the central limit theorem, for samples of reasonable size. And it holds even when the underlying population is not normal at all. That third property is why so many procedures work as widely as they do.

The Number

The Standard Error of the Mean

The second property deserves its own treatment, because it becomes a number you compute. The standard error of the mean, abbreviated S E or S E M. It is the standard deviation of the sampling distribution. What it quantifies is how much sample means typically deviate from the population mean. Which is exactly what you need to know. It tells you how precisely a single sample mean estimates its parameter. And the formula is straightforward. The standard error equals s, divided by the square root of n. Where s is the sample standard deviation, and n is the sample size.

Reading the Formula

Two Features Worth Attention

Two features of that formula deserve attention. First, the standard error is smaller than the standard deviation of the data. Smaller by a factor of the square root of n. In plain terms, the mean of a sample is more stable than any individual observation in it. And that stability is the basic reason research collects multiple participants. Rather than relying on one, however carefully chosen. Second, the standard error shrinks as sample size grows. But only as the square root, which is slower than most people expect. Doubling the sample size reduces the standard error by a factor of about one point four one. Not by half. To halve the standard error, you have to quadruple the sample size.

Diminishing Returns

Where the Extra Participants Pay Off

That square root relationship has practical consequences for study design. Compare two ways of enlarging a study, each a fourfold increase. Moving from twenty five participants to one hundred is a large gain in precision. Moving from one hundred to four hundred is a much smaller gain per participant. Small samples produce wide sampling distributions, and imprecise estimates. Differences between groups may be impossible to detect against the noise. Large samples produce narrow sampling distributions, and tighter estimates. But the gain per additional participant diminishes as you go. So most empirical research aims for a middle ground. Samples large enough to detect effects of theoretical interest with adequate power. And small enough to actually be feasible.

Easy to Confuse

Standard Deviation Is Not Standard Error

Two quantities in this section are constantly confused, so separate them clearly. The standard deviation describes the spread of individual observations within a sample. The standard error describes the spread of sample means around the population mean. One is about people. The other is about samples. They are easy to confuse in writing, and impossible to interchange in analysis. A report that gives a standard deviation where it should give a standard error is misleading. It communicates something quite different from what the author intended. And because the standard error is always the smaller number, error bars look tighter. So reading published papers carefully means checking which quantity is reported.

Assumptions

Guardrails, Not Walls

Most of the procedures in this chapter rest on a set of shared assumptions. Independent observations, meaning each data point is independent of every other. Violate that, and your apparent precision is inflated, making Type one errors more likely. Random sampling, meaning the sample is drawn at random from the population. Violate that, and generalization to the broader population is limited. An approximately normal sampling distribution of the mean. Large samples often satisfy that automatically, via the central limit theorem. Adequate sample size, so that theorem can actually apply. And for some tests, equal variances across the comparison groups. Now here is the right attitude toward all of that. Assumptions are guardrails, not walls. Independence is fundamental, while modest non-normality with large samples is usually fine.

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FAQ

Questions students ask us

About The Building Blocks of Inferential Statistics

What is the difference between a parameter and a statistic?
A parameter describes a whole population and is usually unknown, while a statistic is computed from sample data to estimate it. Parameters use Greek letters and statistics use Roman ones: the population mean is mu and the sample mean is M or x-bar, the population standard deviation is sigma and the sample value is s, and the population correlation is rho while the sample correlation is r.
What is a sampling distribution of the mean?
A sampling distribution of the mean is the distribution of means from many samples of the same size drawn from one population. Its mean equals the population mean, its standard deviation, called the standard error, is smaller than the population standard deviation, and by the central limit theorem it is approximately normal for reasonably large samples even when the population is not. It describes sample means, not individual people.
How do you calculate the standard error of the mean?
The standard error of the mean equals the sample standard deviation divided by the square root of the sample size, s divided by the square root of n. For example, with s = 10 and n = 25, the standard error is 10 / 5 = 2. Because of the square root, doubling the sample reduces the standard error only by a factor of about 1.41, and halving it takes four times as many participants.
What is the difference between standard deviation and standard error?
The standard deviation describes how spread out individual observations are within a sample, while the standard error describes how much sample means vary around the population mean. One is about people, the other about samples. For any sample larger than one, the standard error is the smaller number, so error bars showing standard errors look tighter. Readers should check which one a paper reports, because the two are not interchangeable.

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Research Methods in Psychology (Psychology 111) carries a NCCRS credit recommendation for 3 semester credits. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.
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