sampling distribution explained
Draw one sample and average it and you get one number. Draw another sample and you get another number. The statistic computed from a sample, here the sample mean, is itself a random variable that shakes every time you draw. That shaking has a rule. The distribution of the values the sample mean can take is called the sampling distribution, and its width is called the standard error. If you saw in normal-clt that adding many values makes a bell, here you see how far that mean spreads from sample to sample, and how the spread shrinks when you grow the sample. The core formula is one line. The standard error is the standard deviation divided by a square root. That one line explains why a poll can ask only a modest number, why an experiment is repeated, and where the ± beside an estimate comes from. The width of the shaking is the answer.
Press the button and a new sample is drawn. Even when you draw samples of the same size from the same population, their mean x̄ differs a little every time. One draw, one point, another draw, another point. Stack these sample means into a histogram and it starts jagged and then settles into a mound with a raised middle. This growing distribution is the sampling distribution, the distribution that the statistic called the sample mean draws for itself. What matters is that the population mean we are measuring is fixed, while the tool that estimates it, the sample mean, shakes. Knowing the shape and the width of that shaking is the whole of this lesson. The tool shakes.
Now see the two distributions side by side. The wide one is how the raw values scatter one by one, and its width is the population standard deviation. The narrow one rising above it is how the means of samples of size n scatter, gathered much more tightly to the middle. In the averaging, large values and small values cancel, so the mean shakes much less than a single value. This narrow width is the standard error. It has a name close to the standard deviation and is a completely different quantity. The standard deviation says how different the individuals are. The standard error says how untrustworthy the mean of those individuals is. Grow the size and the wide one stays while only the narrow one grows narrower. Averaging reduces the shaking.
Now pin that width with a formula. The standard error is the standard deviation divided by a square root. The point is that the denominator holds the square root, not the count itself. Grow the size and the standard error falls along the curve, steeply at first and then more gently. You have to grow the size 4 times for the standard error to halve, and to cut the error to 1/10 you have to gather 100 times the sample. Because of this square-root wall, raising precision costs more and more. Past a point it is better to improve the measurement itself than to gather still more. More data is better, but the gain grows only at the speed of √n, and knowing that is practical sense. The gain grows slowly.
So far the population was bell-shaped, which was generous. What if the distribution is openly skewed? This exponential distribution crowds near 0 and drags a long tail to the right. It is not symmetric at all. When the size is 1, the distribution of the sample mean is that skewed original as it stands. Grow the size and something surprising happens. Even though the source is this crooked, the distribution of the sample mean approaches a left-right symmetric bell and fits the normal curve. This is the sampling-distribution face of the central limit theorem. If normal-clt showed that adding values makes a bell, here the distribution of the mean, that sum divided by the count, converges to N(μ, σ/√n) regardless of the original shape. So we can treat the sample mean as normal even when we do not know the population's shape. The shapes differ and the mean is still a bell.
Last, overlap directly the effect of sample size on the sampling distribution. The wide one is the distribution of the mean of a small sample, n=4, spread out. The narrow one is the distribution of the mean of a large sample that you grow. As you raise the size the narrow one rises taller and tighter around the true mean. The centers of the two distributions overlap at the same place, and only the width shrinks at the speed of a square root. What this means in practice is plain. A large sample lands our estimate closer to the truth, and more often. The estimate is more precise. Polls, clinical trials, and quality checks grow the sample size for the reason in this picture. As lesson 3 showed, the gain grows only as a square root, so how narrow is enough is always weighed against cost. Precision and cost are read together.