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PROBABILITIES AND SAMPLE DISTRIBUTION - Coggle Diagram
PROBABILITIES AND SAMPLE DISTRIBUTION
Standard deviation
in probability and sample distribution examines
how much individuals in a sample set vary from the sample mean.
Two things to know about the population standard deviation:
Whenever you are working with a sample mean, you must use the standard error.
The standard error decreases in relation to the square root of the sample size.
The
Law of large numbers
explains that large samples are more representative of the population from which they are selected, thus making it possible to predict the characteristics of a sample with more accuracy. The larger the sample, the higher the probability that its mean will be close to the population mean.
Standard deviation and sample sizes
:
The smallest population sample is n = 1, which makes the SD for distribution of sample means identical to the standard deviation for the distribution of scores.
As 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 σM = σ .
Formula for standard error in a population variance:
σM = σ divided by square root of N
Distribution of sample means
(
M
) is the collection of sample means for all the possible random samples of a particular size (
n
) that can be obtained from a population.
The primary use is to find the probability of selecting a sample with a specific mean.
3 characteristics
of the distribution of sample means:
Distributions should pile up around the population mean.
That pile tends to form a normal-shaped distribution, with means close to ų.
The sample means obtained with a large sample size should cluster relatively close to the population mean. The larger the sample size, the closer the sample means should be to the population mean, ų.
A
sample
is a small portion of the population we can draw conclusions about, but it does not provide a perfectly accurate reflection of the population.
-
Random sampling
requires sampling with replacement
-The mean of a sample is an
unbiased statistic
; generally, the sample statistic produces a value that is exactly equal to the corresponding population parameter.
Sampling distribution
is a distribution of statistics obtained by selecting all the possible samples of a specific size from a population.
Distributions, all of which are interrelated
A
sample from a population
consists of a small set of scores for people who have been selected to represent the entire population. The sample has its own mean and standard deviation.
Distribution of sample means
is a theoretical distribution consisting of the sample means obtained from all the possible random samples of a specific size.
Original population of scores
has its own shape, mean, and standard deviation.
The Central Limit Theorem identifies
3 basic characteristics
of any distribution:
Shape
,
central tendency
, and
variability
.
The
Central Limit Theorem
is a mathematical proposition that helps us to determine exactly what the distribution of sample means looks like without taking hundreds or thousands of samples. It's a cornerstone for much of inferential statistics.
This theorem represents a precise description of the distribution as if you had obtained every possible sample, calculated every sample mean, and constructed the distribution of the sample mean.
This theorem is based on 2 facts:
It describes the distribution of sample means for any population, no matter what shape, mean, or standard deviation.
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 tendency
mean, median, and mode
The expected value of M
The mean of the distribution of sample means is
always
identical to the population mean, μ; The average value of M is equal to μ. μM - the mean of the distribution of sample means, which is also μ.
The
shape
tends to be a normal distribution, and almost perfectly normal if:
-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.
Variability
Differences among sample results when taken from the population
The standard error of M
is the standard deviation of the distribution of sample means (
σ M
). It provides a measure of how much distance to expect between a sample mean (M) and the population mean μ. In other words, how much to expect between a sample mean and the population mean.
Formula for standard error of M
: σ M = SD divided by square root of n; OR σ M = square root of σ divided by n, then find the square root of that result.
The standard error of M has 2 purposes:
To describe the distribution of sample means. When the standard error is small, all the sample means are close together and have similar values.
To measure how well an individual sample mean represents the entire distribution. This shows how much distance is reasonable to expect between a sample mean and the overall mean for the distribution of sample means.
z-Scores and probability for sample means
The
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:
We know if the sample mean is located above (+) or below (−) the mean for the distribution (which is the population mean, μ ).
We know the distance between the sample mean and μ in terms of the number of standard errors.
Formula for calculating the sample means in both directions: M = μ + z (σM)
Formula for a sample mean: z = M - μ divided by σM
Sampling error vs.
Standard error
A
sampling error
shows how much error there is between a sample statistic and the corresponding population parameter. We need this because a sample typically will not provide a perfectly accurate representation of its population.
A
standard error
shows how much error there is from the population mean when representing a population. Some samples with extremes produce means that are out in the tails of the distribution, relatively far from the population mean–and therefore don’t accurately measure the population.
The standard error for a sample mean is often reported in scientific papers instead of the standard deviation.