6.2: Mean and Standard Deviation of Sampling Distributions

In the last section, we saw how we can build a sampling distribution: take many samples from the same population, calculate a statistic (like the mean) for each sample, and plot those values.

Now we focus more carefully on two key questions:

• What is the center of that sampling distribution?
• How much do the sample statistics typically vary? That is, how wide is the distribution?

To answer these, we revisit two ideas from earlier chapters: mean and standard deviation. But this time, we’re not looking at raw data, we’re looking at a distribution of statistics from many samples. And that leads us to two new terms: unbiased estimators and standard error.

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Review: Mean and Standard Deviation

• Mean: The average of a set of values; center of a distribution
• Standard Deviation (SD): A measure of how spread out values are around the mean on average

In a sampling distribution, the values we’re working with are not individual data points, they’re sample statistics, like sample means from different samples. So when we talk about “the mean of all the sample means” or their spread, we’re talking about the sampling distribution of the statistic.

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Definition: Mean of the Sampling Distribution

The mean of the sampling distribution is the mean of all of the sample statistics from all possible samples.

For sample means, the mean of the sampling distribution is equal to the population mean:

\[ \mu_{\bar{x}} = \mu \]

For sample proportions, the mean of the sampling distribution is equal to the population proportion:

\[ \mu_{\hat{p}} = p \]

The sample mean and sample proportion are both called unbiased estimators since the mean of all the sample means matches the population mean and the mean of all the sample proportions matches the population proportion. The statistics "target" the population parameter. This means that on average the sample mean (and proportion) is correct, even though individual samples may vary. This is NOT true of other sample statistics like the range or median. The mean of all sample ranges may NOT match the population range.

This concept, known as unbiasedness, is important: it means that even though sample statistics fluctuate, the process doesn’t consistently overestimate or underestimate the population value. Remember, not all statistics are unbiased!

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Definition: Standard Error (SE)

The standard error is the standard deviation of a sampling distribution. It measures how much the sample statistic varies from sample to sample. We often abbreviate standard error as SE. However, it may also be written like \( \sigma_{\bar{x}} \) for sample means, \( \sigma_{\hat{p}} \) for sample proportions, or \( \sigma_{s} \) for sample standard deviations.

As an example, for sample means the standard error is:

\[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \]

where \( n\) is the sample size and \(\sigma\) is the population standard deviation.

Note the \(n\) in the denominator of the fraction. As sample size increases, the standard error shrinks. Larger samples give more stable sample means. This is generally true for all sampling distributions, not just sample means, but this particular formula \(\frac{\sigma}{\sqrt{n}}\) is specific to sample means. You can see an example of this plotted below.

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Interpreting the Horizontal Axis

When we draw or simulate a sampling distribution (like we did in Section 6.1), the x-axis now shows sample statistics, not individual data points.

The center of the distribution is \( \mu_{\bar{x}} \) or \(\mu_{\hat{p}}\) or whichever statistic you're using.
The spread of the distribution is \( \sigma_{\bar{x}} \), the standard error.

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Nightly Sleep Amounts

Suppose that the average college student gets 6.8 hours of sleep per night with a population standard deviation is 1.1 hours. If we were studying student sleep habits, we would not know these actual population values! Instead, we would take a sample to determine an estimate.

We take a random sample of 25 students and compute this sample mean. What would the center and spread of the sampling distribution of sample means look like?

Mean: \( \mu_{\bar{x}} = 6.8 \) hours. The average sample mean is just the pre-existing population mean.

Standard Error: \( \sigma_{\bar{x}} = \frac{1.1}{\sqrt{25}} = \frac{1.1}{5} = 0.22 \text{ hours} \). We have to calculate the standard error by dividing by the sample size. This is because as the sample size goes up, the likelihood of having a sample mean further from the population mean decreases.

Conclusion: Most of the sample means (not individual sleep values!) would fall within 0.22 hours of 6.8 on average. Using our rule of thumb, we can say that with perfectly random sampling, there is a 68% of getting a sample within one standard deviation, or between 6.58 and 7.02 hours.

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A Larger Sample

Let's suppose we change our minds and double our sample size for our sleep study. We now have \( n = 50 \) and can recompute the standard error.

\( \sigma_{\bar{x}} = \frac{1.1}{\sqrt{50}} = \frac{1.1}{7.07} = 0.156\)

Note that with a doubling of sample size, the standard error does not get twice as small. This is still helpful however, as our 68% area would now span between 6.64 and 6.96, which is closer to the true mean.

 

The plotted histograms show the theoretical results from sampling 25 or 50 students repeatedly and tracking what the average amount of sleep for the sample is. To generate this image, we used a computer to randomly sample 20,000 times. This means for the image on the left, there are 20,000 samples, each with 25 students for a total of 500,000 data points! In reality, we would take one sample, of 25 students, and have no idea where it lies on the distribution. Note that it is more likely to like close to the real mean, and with a larger sample that probability increases.

 

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Connecting to Chapter 5

Back in Chapter 5, we plotted raw data and observed shapes and spread. That was the distribution of the data.

Here in Chapter 6, we are plotting statistic values from many samples. It’s a shift in thinking:

• Before: Each point = one person’s data value
• Now: Each point = one sample's mean (or another statistic)

There are two major points to consider however:

1. In reality, we don't know the population values and therefore cannot calculate our sample distribution. Instead, we have a sample that came from that distribution.
2. We don't take multiple samples to build the distribution. It's a tool that is used to assess a single sample that comes from the distribution.

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