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Probability & Samples: the distribution of sample means - Coggle…
Probability & Samples: the distribution of sample means
Sampling Error
: discrepancy that naturally occurs between a selection statistic and its corresponding population parameter
Distributions of Sample Means for any Population & any Sample Size
Central Limit Theorem:
a mathematical theorem that specifies the characteristics of the distribution of sample means
the standard deviation divided by the square root of the sample size (n)
1.) describes distribution of sample means for any population, no matter what shape, mean, or standard deviation
2.) distribution of sample means "approaches" a normal distribution very rapidly
Theorem describes:
Shape
1.) Population from which the samples are selected is a normal distribution
2.) Number of scores (n) in each sample is relatively large - around 30 or more
Central tendency
Mean of the distribution of a sample means the expected values of M
Variability
Standard Error of M:
standard deviation of the distribution of the sample
Standard error provides a measure of how much distance is expected on average between a sample mean ans a population mean
1.) describes the distribution of sample means
error = small = sample means are close together
error = large = sample means are scattered
provides a measure of how much difference is expected from one sample to another
2.) measures how well an individual sample mean represents the entire distribution
provides measure of how much distance is reasonable to expect between sample mean and overall mean
Magnitude of standard error is determined by:
1.) size of sample
as the sample size increases, the error between the sample mean & the population mean decreases
Large of large numbers
: larger the sample size (n), the more probable it is that the sample mean will be close to the population mean
2.) the standard deviation of the population from which the sample is selected
when n = 1, standard error is identical to the standard deivation
as the sample size (n) increases, the size of the standard error decreases (larger samples are more accurate)
when the sample consists of a single score (n=1), the standard error is the same as the standard deviation
Three Different Distributions
1.) Original population of scores
2.) sample selected from the population
3.) Sample means
Distribution of Sample Means
: collection of sample means for all possible random samples of a particular size
Sampling Distribution:
a distribution of statistics obtained by selecting all the possible samples of a specific size from a population
(sampling distribution of M)
Characteristics
1.) Sample means should pile up around the population mean
2.) Pile of sample means should tend to form a normal-shaped distribution
3.) The larger the sample size, the closer the sample mean should be to the population mean
Probability & the Distribution of Sample Means
Primary use of distribution of sample means is to find the probability associated with any specific sample
Z-Score for Sample Means
1.) Sign tells if location is above (+) or below (-) the mean
2.) Number tells distance between location and mean in terms of the number of standard deviation
z= (mean of original population - mean of sample means) / standard deviation of sample means
More about Standard Error
1.) Now using distribution of sample means instead of distribution of scores
2.) Now using the standard error instead of standard deviation
Distribution of sample means provides a method to organize different sample means into a single picture
Sampling Error
Typically won't provide a perfectly accurate representation of its population
there will be some discrepancy between the mean for a sample and the mean for the population from which the sample was obtained
Standard Error
A way to measure the average or standard distance between the sample mean and the population mean