Chapter 7: Probability and Samples: The Distribution of Sample Means
so far, we’ve been looking at probability in terms of a single sample score; however, more often than not, research studies involve larger samples, in which case we use the sample mean (rather than a single score) to answer qs about the population.
we can take the mean of a sample, compute its z-score, and use that to help explain our sample—close to 0 = representative sample, > ±2.00 = extreme sample, etc. lets us describe a specific sample in terms of all other possible samples. also usually lets us find the probability of obtaining a specific sample regardless of how many scores the sample contains.
sampling error: the natural discrepancy, or amount of error, between a sample statistic and its corresponding population parameter.
generally, we can obtain thousands of different samples from a single population; this means it’s crucial to know how your specific sample measures up to others as well as the general population.
distribution of sample means: the collection of sample means for all the possible random samples of a particular size (n) that can be obtained from a population. often called the sampling distribution of M.
any probability question [about a sample] requires complete info about the population from which the sample is being selected!!
note that in the past, we’ve only talked about distributions in terms of scores; now, however, we’re talking about a distribution of statistics (sample means). we often refer to this as a sampling distribution (a distribution of statistics obtained by selecting all the possible samples of a specific size from a population). in other words, the distribution of a sample is a kind of sampling distribution!
to construct the distribution of sample means, you first select a random sample of specific size n from the population, calculate the sample mean, and place the sample mean in a frequency distribution. then you select another random sample with the same number of scores and do the same. you keep doing this until you have a complete distribution of every single possible sample in the population. then your distribution will show the distribution of sample means.
in general, this distribution has the following characteristics:
sample means should pile up around the population mean. samples aren’t expected to be perfect but only to be representative of the population, so most of the sample means should be relatively close to the population mean. in other words, the distribution of sample means should be normal-shaped!
the pile of sample means should tend to form a normal-shaped distribution—it should be relatively rare to find a sample substantially different from μ.
in general, the larger the sample size, the closer the sample means should be to the population mean. (logically, larger samples are more representative of the population than smaller samples, so larger sample sizes have less variability/spread and smaller ones have more variability/spread.)
we can use the distribution to answer questions like, what are the chances of pulling a sample from the population that has a mean of over #?
in situations with large populations and samples, the number of possible samples is huge, and it is virtually impossible to actually obtain every possible random sample. that said, it’s possible to determine exactly what the distribution of sample means looks like without taking hundreds or thousands of samples, according to central limit theorem: for any population with a mean μ and standard deviation σ, the distribution of sample means for sample size n will have a mean of μ and a standard deviation of σ/√n and will approach a normal distribution as n approaches infinity.
this theorem describes the distribution of sample means for any population regardless of shape, mean, or standard deviation.
the DOSM “approaches” a normal distribution very rapidly—by the time n = 30, the distribution is almost perfectly normal. this means we don’t need an enormous sample of, say, 1000 people in order to get a normal distribution.
the DOSM is almost perfectly normal so long as either (A) the population from which the samples are selected is a normal distribution, or (B) the number of scores, n, in each sample is relatively large (≥ 30). in other words, when n>30, the distribution is almost normal no matter how wack the original population was.
expected value of M: the mean of the distribution of sample means, which is equal to the mean of the population scores, μ. in other words, the sample mean is expected to be near its population mean, so that when all possible sample means are obtained, the average value is identical to μ.
recalling the definition of an (un)biased statistic (ch4; the sample does (not) give an accurate representation of the total population), we can see that the sample mean is an unbiased statistic—on average, the sample stat produces a value exactly equal to the population parameter.
note: sometimes the expected value of M is expressed as μM; however, we won’t be using that in this textbook.
standard error of M (σM): the standard deviation of the distribution of sample means; provides a precise measure of how much distance is expected on average between a sample mean M and the population mean μ. measure of variability.
describes the distribution of sample means—provides a measure of how much difference is expected from one sample to another. when standard error is small, all sample means are close together and have similar values.
standard error measures how well an individual sample represents the entire distribution—provides a measure of how much distance is reasonable to expect between a sample mean and the overall mean for the distribution of sample means. however, since the overall mean is equal to μ, the standard error also provides a measure of how much distance to expect between a sample mean M and a population mean μ.
remember: samples aren’t expected to perfectly reflect a population, only to be representative of the population mean; thus there is typically some error between samples and populations, which is measured by the standard error.
specifies precisely how well a sample mean estimates its population mean/exactly how much error to expect!! found with σ/√(n), meaning it’s determined by the size of the sample and the standard deviation of the population from which it was selected.
law of large numbers: states that the larger the sample size n, the more probable it is that the sample mean will be close to the population mean.
when a sample consists of 1 score, each sample is a single score, so the distribution of sample means is identical to the original distribution and its standard deviation is also identical.
you can think of standard deviation as the ‘starting point’ for standard error—when n=1, σM=σ; as n→∞, σM→0. this is that same equation, σ/√(n), which fulfills our requirements.
also note that due to it being the square root of n, accuracy increases substantially up to n=30, but increasing past there doesn’t do much.
remembering that standard deviation is just √(σ2), we can substitute variance into the above equation to find √(σ2/n).
we are now dealing with 3 different but interrelated distributions! we have (1) the original population of scores, (2) a sample selected from the population, and (3) the distribution of sample means!
note that the scores for the sample were taken from the original population and that the mean for the sample is one of the values contained in the distribution of sample means. thus, the 3 distributions are all connected, but they’re distinct.
we can use the DOSM to determine the probability of obtaining a sample with a specific mean. eg, to determine the probability of drawing a sample of 16 students with an SAT score higher than 525, we first picture the distribution based on the central limit theorem (distribution is normal because SAT distribution is normal; distribution has mean of 500 because μ=500; and for n=16, the distribution is σ/√(16), or σ/4). then we find a z-score for M=525 in the distribution, which is equal to 1 standard deviation from the mean of the DOSM. thus to find the proportion with scores over 525, we just find the percentage in the tail of z=1.00 (15.87%). in conclusion, we have a 15.87% chance of pulling a sample of 16 students whose mean SAT score is greater than 525.
we find the z-score of a DOSM with the same math as we use elsewhere, but we use different notation for the formula: .
we can also use the DOSM to make quantitative predictions about the kinds of samples that should be obtained from any population. eg, using SAT scores again, we can find the exact range of values expected for the sample mean 80% of the time [using samples of n=25]. we find that standard error=100/√(25)=20. from there, we want to find the range that makes up the middle 80% of the distribution. we can use our unit normal table to find what z-score corresponds to ±40% from the mean, which we find to be 1.28, meaning our lower boundary is -1.28 and our upper boundary is +1.28. we then calculate which specific means correspond to those z-scores, calculating that 1.28(20)=25.6 points, so mean±25.6 is 474.4 and 525.6. this means that 80% of samples of 25 will have means that fall between 474.4 and 525.6.
essentially the only thing in this chapter that is “new” is that we need to use the standard error whenever working with the sample mean. otherwise, the math and shit is pretty much just the same.
in most research situations, the population mean is unknown and the population is too large to study as a whole; thus researchers select a sample to help obtain info about the unknown population. similarly ,the sample mean provides info about the value of the unknown population mean—the sample mean isn’t expected to be perfect but representative, and the standard error tells you exactly how much error, on average, that will be.
note: since standard error is often more important than sample standard deviation, it’s often the one included in scientific literature. SE and SEM (“standard error of the mean”) are the usual abbreviations. could be expressed in a table or graph. on a bar graph, the height of the bar represents the mean, and an I-shaped bracket at the top of each bar depicts the standard error. we use the same bracket on line graph, which otherwise looks the same as any other line graph. the bracket extends 1 standard error above and below the mean!
when looking at studies, there’s always a margin of error when going from sample to population. we use the sample statistics so we can infer things about the population, but we’re never 100% certain. we can get really close to 100% sure, though, if we have a big sample and a small standard error for our sample.