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Probability and Samples: The Distribution of Sample Means - Coggle Diagram
Probability and Samples: The Distribution of Sample Means
Samples, Populations, and the Distribution of Sample Means
The Distribution of Sample Means
Two separate samples will be different even though they're taken from the same population
Distribution of sample means: collection of sample means for all the possible random samples of a particular size (n) that can be obtained from a population
Sampling distribution: distribution of statistics obtained by selecting all the possible samples of a specific size from a population
often called the sampling distribution of M
Characteristics of the Distribution of Sample Means
Sample means should pile up around the population mean. They're not perfect, but they're representative of the population, therefore they should be relatively close to the population mean
Pile of sample means should tend to form a normal shaped distribution. Should have means close to the population mean (u)
In general, the larger the sample size, the closer the sample means should be to the population mean. Thus, sample means obtained with a large sample size should cluster relatively close to the population mean; the means obtained from small samples should be more widely scattered
Shape, Central Tendency, and Variability for the Distribution of Sample Means
The Central Limit Theorem
Provides a precise description of the distribution that would be obtained if you selected every possible sample, calculated every sample mean, and constructed the distribution of the sample mean
Describes the distribution of sample means for any population, no matter the same, mean, or SD
The distribution of sample means "approaches" a normal distribution very rapidly...by the time the sample size reaches n=30, the distribution is almost perfectly normal
Central Limit Theorem: For any population with mean (u) and standard deviation (o), the distribution of sample means for sample size n will have a mean and SD of.. SD/square root of n (number) and will approach a normal distribution as n approaches infinity
The Shape of the Distribution of Sample Means
The population from which the samples are selected is a normal distribution
The number of scores (n) in each sample is relatively large, around 30 or more
When n>30, the distribution is almost normal regardless of the shape of the original population
The Standard Error of M
The standard error of M provides a measure of how much distance is expected on average between a sample mean (M) and the population mean (u)
The standard errors describes the distribution of sample means. It provides a measure of how much difference is expected from one sample to another
If SE is small, they'll be close together and have similar values. If they're large they'll be scattered over a wide range and there will be big differences from one sample to another
SE measures how well an individual sample mean represents the entire distribution. Specifically, providing a measure of how much distance is reasonable to expect between sample mean and the overall mean for the distribution of sample means
Law of large numbers: as the sample size increases, the error between the sample mean and the population mean should decrease
a. As sample size (n) increases, the size of the standard error decreases. (Larger samples are more accurate.)
b. When the sample consists of a single score (n=1), the SE is the same as the standard deviation
Defining the Standard Error in Terms of Variance
Three Different Distributions
First, we have the original population of scores- contains thousands or millions of individual people, and has its own shape, mean, and standard deviation
Next, we have a sample that's selected from the population. The sample consists of a small set of scores for people who have been selected to represent the entire population
The third distribution is the distribution of sample means. This is a theoretical distribution consisting of the sample means obtained from all the possible random samples of specific sizes
z-Scores and Probability for Sample Means
Because the distribution of sample means presents the entire set of all possible samples means, we can use proportions of this distribution to determine the probability of obtaining a sample with a specific mean
A z-Score for Sample Means
z-score for a sample mean can be defined as a signed number that identifies the location of the sample mean in the distribution of sample means so that
the sign tells whether the sample mean is located above (+) or below (-) the mean for the distribution
the number tells the distance between the same mean and population mean in terms of the number of standard errors
More about Standard Error
We are now using the distribution of sample means instead of a distribution of scores
We are now using the standard error instead of the standard deviation
Whenever you're working with a sample mean, you must use the standard error
Sampling and Standard Error
Sampling Error
A sample is usually not a perfect representation of the population.
The difference between a sample statistic (e.g., sample mean) and the population parameter (e.g., population mean) is called sampling error.
About 50% of sample means underestimate the population mean, and 50% overestimate it.
Some amount of sampling error is expected in every sample
Standard Error
Most sample means are close to the population mean, but some are farther away.
The error for a sample is the distance between the sample mean and the population mean.
Standard error measures the typical (average) size of this sampling error.
A smaller standard error means sample means tend to be closer to the population mean and are more accurate.
