Null Distribution vs Sampling Distribution

The null distribution is the sampling distribution of a statistic in a world where $H_0$ holds: centered at the value $H_0$ names, with a spread computed under that same assumption. It is not the distribution of the population and not the distribution of the values in your sample. It describes one summary number across every sample of size $n$ you might have drawn.

With $H_0: \mu = 100$, $\sigma$ (sigma) known to be 15 and $n = 25$, the null distribution of $\bar{x}$ (x-bar) is normal, centered at 100, with standard error $15/\sqrt{25} = 3$. An observed $\bar{x} = 106.2$ sits $\frac{106.2 - 100}{3} = 2.07$ standard errors above the center, and the area beyond it, 0.0194, is the one-sided p-value. Change the hypothesized mean and the whole curve slides sideways, while the sample mean you actually observed leaves it exactly where it is.

That last point has a consequence students meet without recognizing it. Testing $H_0: p = 0.50$ with $n = 100$, the standard deviation is $\sqrt{0.50(0.50)/100} = 0.05$, built from the hypothesized 0.50 even when the sample gave $\hat{p}$ (p-hat) of 0.60. A confidence interval has no hypothesis to borrow from, so it uses $\hat{p}$ instead and gets $\sqrt{0.60(0.40)/100} = 0.049$. Same data, two denominators, which is one reason an interval and a test can disagree.

A simulated null distribution, the kind a randomization test builds, is a finite pile of reshuffled outcomes rather than a curve, so its p-value moves in steps of 1 divided by the number of trials and shifts a little when you rerun it. A theory-based curve earns its shape only when the conditions hold; without them the p-value is an area under the wrong curve, correct arithmetic and all.

The area this distribution hands back is AP Statistics topic 3.6, p-Values.

Three different distributions get called "the distribution" and only one of them is this one. The population distribution holds a value for every individual. The distribution of a sample holds the values in the one sample you actually collected. A sampling distribution is neither: its individuals are entire samples of a fixed size $n$, and the number recorded for each is a statistic computed from it, such as $\bar{x}$ (x-bar) or $\hat{p}$ (p-hat).

Small cases can be written out in full. Take the population 2, 4, 6, 8, with $\mu = 5$ (mu) and $\sigma = 2.2361$ (sigma), and draw samples of size 2 with replacement. There are exactly 16 such samples, and their means come out as 2 once, 3 twice, 4 three times, 5 four times, 6 three times, 7 twice, and 8 once. That list is the sampling distribution of $\bar{x}$. Its mean is 5, matching $\mu$, and its standard deviation is 1.5811, which is $\sigma/\sqrt{2}$. The population is flat; the sampling distribution is already triangular at $n = 2$.

The misreading is blunt: "the sampling distribution is the distribution of my sample." A histogram of the 40 numbers you collected is a picture of the sample. As $n$ grows it comes to look more like the population, skew and all, and it does not narrow. The sampling distribution is the thing that narrows, and $\sigma/\sqrt{n}$ describes it rather than your data.

Outside enumerable toy cases you never actually build one. Theory stands in for it, the central limit theorem for means and the binomial for counts, or a simulation approximates it: 10,000 simulated sample means draw a close picture of the sampling distribution without being it.

Topic 2.12 introduces sampling distributions. Every confidence interval and every p-value later in the course is a statement read off one.

One sampling distribution gets reported on the news every month. The US unemployment rate is not a census, it is an estimate from a household survey of about sixty thousand homes, which is why it is revised and why it carries a margin of error at all: the unemployment rate.