What Effect Does Increasing The Sample Size Have On A Distribution Of Sample Means?

Lecture 4
Sample Size

In this cyberlecture, I'd like to outline a few of the important concepts relating to sample size. By and large, larger samples are expert, and this is the case for a number of reasons. And then, I'one thousand going to try to prove this in several different means.

Bigger is Better
ane. The first reason to understand why a big sample size is beneficial is simple. Larger samples more closely approximate the population. Because the primary goal of inferential statistics is to generalize from a sample to a population, it is less of an inference if the sample size is large.

two. A second reason is kind of the opposite. Small samples are bad. Why? If we pick a small-scale sample, we run a greater take a chance of the small sample being unusual just by chance. Choosing 5 people to represent the entire U.S., fifty-fifty if they are chosen completely at random, volition often result if a sample that is very unrepresentative of the population. Imagine how easy it would be to, only past chance, select 5 Republicans and no Democrats for example.

Let's take this bespeak a little further. If there is an increased probability of one minor sample being unusual, that means that if we were to draw many small-scale samples as when a sampling distribution is created (see the second lecture ), unusual samples are more frequent. Consequently, there is greater sampling variability with pocket-sized samples. This figure is another way to illustrate this:

Note: this is a dramatization to illustrate the issue of sample sizes, the curves depicted hither are fictitious, in club to protect the innocent and may or may not represent real statistical sampling curves. A more realistic depiction tin can exist constitute on p. 163.

In the bend with the "small size samples," notice that in that location are fewer samples with means around the centre value, and more samples with means out at the extremes. Both the correct and left tails of the distribution are "fatter." In the curve with the "big size samples," find that there are more samples with means around the middle (and therefore closer to the population value), and fewer with sample means at the extremes. The differences in the curves represent differences in the standard departure of the sampling distribution--smaller samples tend to have larger standard errors and larger samples tend to have smaller standard errors.

three. This point near standard errors can be illustrated a unlike way. One statistical test is designed to run into if a single sample mean is different from a population mean. A version of this exam is the t-test for a unmarried mean. The purpose of this t-test is to meet if there is a significant difference between the sample mean and the population mean. The t-test formula looks like this:

The t-test formula (also plant on p. 161 of the Daniel text) has two main components. First, it takes into account how large the deviation between the sample and the population mean is by finding the difference between them (). When the sample hateful is far from the population mean, the difference volition exist big. Second, t-examination formula divides this quantity past the standard fault (symbolized by ). By dividing past the standard error, we are taking into account sampling variability. But if the difference between the sample and population ways is large relative to the amount of sampling variability will nosotros consider the difference to be "statistically pregnant". When sampling variability is high (i.east., the standard error is large), the difference between the sample hateful and the population mean may not seem and then big.

Concept	Mathematic Representation
distance of the sample hateful from the population hateful
representation of sampling variability
Ratio of the altitude from the population mean relative to the sampling variability	t

At present, back to sample size... As we saw in the figure with the curves in a higher place, the standard fault (which represents the amount of sampling variability) is larger when the sample size is pocket-sized and smaller when the sample size is big. So, when the sample size is small, it can be hard to see a difference between the sample mean and the population mean, because there is too much sampling variability messing things up. If the sample size is large, it is easier to see a difference between the sample mean and population mean because the sampling variability is non obscuring the difference. (Kinda nifty how nosotros go from an abstract concept to a formula, huh? I took years of math, but until I took a statistics grade, I didn't realize the numbers and symbols in formulas actually signified anything).

4. Another reason why bigger is better is that the value of the standard error is directly dependent on the sample size. This is really the same reason given in #2 above, only I'll show it a different way. To calculate the standard error, nosotros divide the standard deviation by the sample size (actually there is a square root in there).

In this equation, is the standard error, southward is the standard departure, and north is the sample size. If nosotros were to plug in unlike values for north (try some hypothetical numbers if you desire!), using just ane value for s, the standard fault would exist smaller for larger values of northward, and the standard error would be larger for smaller values of n.

5. There is a rule that someone came up with (someone who had vastly superior brain to the population average) that states that if sample sizes are large enough, a sampling distribution will be normally distributed (retrieve that a normal distribution has special characteristics; see p. 107 in the Daniel text; an approximately usually distributed curve is likewise depicted by the large sample size curve in the effigy above). This is chosen the central limit theorem. If we know that the sampling distribution is normally distributed, nosotros can make meliorate inferences about the population from the sample. The sampling distribution will be normal, given sufficient sample size, regardless of the shape of the population distribution.

6. Finally, that last reason I can think of right now why bigger is better is that larger sample sizes give us more power. Remember that in the previous lecture power was divers as the probability of retaining the alternative hypothesis when the alternative hypothesis is really true in the population. That is, if we tin can increase our chances of correctly choosing the alternative hypothesis in our sample, we accept more power. If the sample size is large, nosotros volition have a smaller standard fault, and as described in the #3 and #4, we are more likely to find significance with a lower standard errror.

Do I seem similar I am repeating myself? Probably. Office of the reason is that information technology is important to try to explain these concepts in several different ways, but information technology is also considering, in statistics, everything is interrelated

How Big Should My Sample Be?
This is a good question to inquire, and it is frequently asked. Unfortunately, at that place is not a really elementary answer. It depends on the type of statistical exam one is conducting. It also depends on how precise your measures are and how well designed your written report is. And so, it just depends. I ofttimes hear a general recommendation that there exist about 15 or more participants in each grouping when conducting a t-test or ANOVA. Don't worry, we'll return to this question later.

What Effect Does Increasing The Sample Size Have On A Distribution Of Sample Means?,

Source: http://web.pdx.edu/~newsomj/pa551/lecture4.htm

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