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Let \(X_1,X_2,\ldots, X_n\) be a random sample of size \(n\) from a distribution (population) with mean \(\mu\) and variance \(\sigma^2\). What is the mean, that is, the expected value, of the sample mean \(\bar{X}\)?
- Independent Samples vs. Dependent Samples
- Pros and Cons of Independent and Dependent Samples
- Types of Statistical Analyses For Independent and Dependent Groups
- Example of Dependent Groups and Their Extra Statistical Power
- Understanding The Different Results
Hypothesis testsand statistical modeling that compare groups have assumptions about the nature of those groups. Choosing the correct test or model depends on knowing which type of groups your experiment has. Additionally, when designing your study, selecting the best type can help you tailor the design to meet your needs.
When thinking about comparing groups, you frequently picture independent groups. For instance, when you imagine comparing a treatment group to a control group, you’re probably assuming these groups contain different subjects. However, by understanding the pros and cons of independent and dependent samples, you can design a study to meet your needs ...
After choosing the type of samples and conducting the experiment, you need to use the correct statistical analysis. The table displays pairs of related analyses for independent and dependent samples. Several notes about the table. While analyses for dependent groups typically focus on individual changes, McNemar’s test is an exception. That test co...
I’m closing with an example that illustrates the extra statistical powerthat dependent samples can provide. Imagine two studies that, by an amazing coincidence, obtain the same measurements exactly. The only difference is that one has independent groups, while the other has dependent groups. It should go without saying, but I’ll say it anyway—you w...
The analyses make different assumptions about the nature of the samples. For the 2-sample t-test, the two groups contain entirely different individuals. While the treatment group has a higher mean IQ score than the control group, we don’t know each subject’s starting score because there was no pretest. Perhaps the treatment group started with highe...
Mar 26, 2023 · The sample mean is a random variable; as such it is written \(\bar{X}\), and \(\bar{x}\) stands for individual values it takes. As a random variable the sample mean has a probability distribution, a mean \(μ_{\bar{X}}\), and a standard deviation \(σ_{\bar{X}}\).
Apr 23, 2022 · Suppose that \bs {x} = (x_1, x_2, \ldots, x_n) is a sample of size n from a real-valued variable. The sample mean is simply the arithmetic average of the sample values: m = \frac {1} {n} \sum_ {i=1}^n x_i. If we want to emphasize the dependence of the mean on the data, we write m (\bs {x}) instead of just m.
The sample mean ( sample average) or empirical mean ( empirical average ), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables .
Center. The mean difference is the difference between the population means: μ x ¯ 1 − x ¯ 2 = μ 1 − μ 2. Variability. The standard deviation of the difference is: σ x ¯ 1 − x ¯ 2 = σ 1 2 n 1 + σ 2 2 n 2. (where n 1 and n 2 are the sizes of each sample). This standard deviation formula is exactly correct as long as we have:
1 comment. ( 42 votes) Upvote. Downvote. Flag. jacob.930321. 11 years ago. Is there any difference if I take 1 "sample" with 100 "instances", or I take 100 "samples" with 1 "instance"? (By sample I mean the S_1 and S_2 and so on. With instances I mean the numbers, [1,1,3,6] and [3,4,3,1] and so on.)
- 11 min
- Sal Khan
