This is a summary including all the lectures of this course. It also includes SPSS output, formulas, explanations and examples. This is everything you need to know about PSRM 2.
Explanatory theories provide potential answers to ‘why’ questions.
Our observations are never perfect or complete and that is the problem with empirical observations.
- It is a problem because this requires an inferential step. We have to generalize from
observations to underlying (theoretical) process.
- We cannot prove the validity of this inferential step just by making more observations
because we can never make all the observations = the problem of induction.
Also what we observe and how we observe depends on our choice of theories. What you do is
already guided by a theory you already have. = problems of theory dependent/ observation.
The dilemma: the ‘why’ question is well supported by observations. However, there is the problem of
induction which means that verification is not justified.
- Solution: even though we can’t prove empirically that a theory is right, we can prove
empirically that a theory is wrong. = falsification. (Karl Popper).
Research questions → theories → methods → conclusions.
Methods are a set of rules or theories that help you make the step from theories to conclusions
through empirical methods.
The goal of theory-driven statistical research is to stringently test theories.
- We do that by taking the theories/hypothesis and turn them into precise models and
assumptions.
Quantitative = statistical research.
Lecture B
Central tendency = descriptive statistic for central value of a distribution of a variable.
- A variable is a characteristic of what we have measured for our cases. The variable can have
different values.
So the central tendency is not so much at the extremes. There are likely not to be many extreme
values of the variable. The values differ not so much around the middle. The central value indicates
where most of the values are concentrated. You can give this statistic a number in different ways:
- (arithmetic) mean (average) , median (the middle) , mode (most common).
Mean: the mean is the top of the distribution. The most likely value that you can find in your data. It
is a measure of the central tendency which will give you the most likely value.
Mean = (add all the cases of i together until you
have done this for the whole population) / size
of population.
N = size of population
n = size of sample. (just for distinguish).
.i= index. (sum off your individual populations).
This is the number you give your variable. E.g. you have 5 students and you give the students the
number 1 through 5. They have different variable values, but this is how you name the students.
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,e.g. mean:
Deviation score: how far are the values removed from the mean value in your population/sample. So
what is the difference between the observed value of the case and the mean. This is just subtraction.
e.g. deviation of student 1 is 8-5.4=2.6
e.g. deviation of student 2 is 4-5.4=-1.4 etc.
what is the sum of the deviation scores? The sum of deviation scores from mean is always 0!!
There is also a number of the sum of the deviation scores which you can calculate. But you do not do
this with adding it all up. So how do we quantify the deviation scores? We have to get rid of the
negative number by taking the square of the deviation scores.
- So e.g. student 1 is 2.62 = 6.76.
- E.g. student 2 is (-1.4)2 = 1.96 etc.
So the sum of the squared deviation values is 23.2.
Because the sum of the deviation values from the mean is always 0, that also implies that the sum of
squared deviation values from the mean is the smallest possible sum we can get. So only the actual
mean can get the smallest sum of squared deviation values. For any other number, the sum will be
larger!
Dispersion = descriptive statistics for variability of a variable. How do the individual variables vary
from one another. How much variation is there in the data?
- E.g. variance, standard deviation, range (difference between highest and lowest observed
variable) , interquartile range (25%-50%-25%).
Variance: the average squared distance to the mean. How much variability is there in the data?
Greek letters for population. Rome letters for sample size.
σ2 = population variance.
s2 =. Sample variance.
n – 1 has to do with the degree of freedom.
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,Standard deviation: square root of the variance. The average distance to the mean.
σ = standard deviation population
s = standard deviation sample.
- So the only difference with variance is that the square is gone. So it is the square root of the
variance. The variance is on the square variable, but your other variables are not squared. So
it is much easier to get rid of the square so you can measure on the same scale.
Inference: from sample to population.
Inference = taking a sample and taking this sample to talk about a larger population.
- Can we do this? When and how can we do this justifiable?
We can use the sample mean as a (point) estimate (statistic) to make inference about the population
parameter.
- The population parameter is usually not known. The sample mean is something you can
calculate.
The goal of inference is to say something about something that is usually not known by taking a
sample which can be measured.
How reliable is it to take a sample and say something about a bigger population? You can make a
judgement about this with mathematics.
- A fundamental concept is the sampling distribution!!
Sampling distribution is a sort of thought experiment: what would be the distribution of all possible
values and how can we distribute evenly in a sample.
- The distribution of all values for (mean)X we get by drawing all possible samples from the
population.
The mean of all the mean values of the samples is identical to the population mean.
- So if your population is 180 and you take samples of random 5 people than the mean of all
the samples is equal to the mean of all the individual 180.
How far removed are the sample mean removed from the population mean?
- We measure this with the standard error of the mean. It is a standard deviation but of a
sample mean to the population mean. Standard deviation is from the single valuable to the
mean.
if you get a value from the standard error than you can say something about this. The standard error
is related to the standard deviation so there is a formula with both in them to calculate the standard
error.
so the standard error = the standard deviation /
square root of the sample size.
The larger the sample becomes, the smaller the
standard error becomes.
The standard error says something about the sample distribution. The larger the sample is, the
smaller the standard error, indicating that the sample is more reliable.
The standard error is never larger than the standard deviation. At most it is equal, but usually not.
3
, Central limit theorem: it shows us that the sampling distribution will start to look more like the
normal distribution, the larger the sample size is.
- Even if original distribution of the variable is not normal
- Roughly if n > 30
This normal distribution is known and can be used to know how likely different values for the sample
mean are for any given µ.
So the formula: the population
mean +- ( 1.96*standard
error).
This can help us if we only have one sample of all the possible samples of the entire population
because now we know the sampling distribution. By knowing this we can say something about our
confidence in the sample mean. We can do this by looking at a confidence interval
So the population mean is now the
sample mean. So this means there
is a 95% chance of drawing a
sample with a mean when the
population mean falls in the range
of this interval.
Problem: if we only have a sample, we do not know what the population standard error is. However,
we know that the population standard error is connected with the standard deviation and sample
size. You can put it in the standard error formula and calculate the population standard error.
However, it is an estimate of the population standard error so there is uncertainty.
- This is the t-distribution instead of the normal distribution.
T distribution is a normal distribution but wider. The larger the sample, the more the t-distribution
become like normal distribution.
Now we have all the steps to go from a sample to a population.
We do this by testing the hypotheses by making inferences about the nature of the population based
on the sample.
- Null hypothesis (H0) and the alternative hypothesis (Ha) .
We cannot prove the alternative hypothesis, but we can test the null hypothesis and reject it or not
reject it. If we reject the null hypothesis, the alternative hypothesis gets more support.
- For testing a mean: H0: µ=value and then the Ha: µ≠value or < >
Problems with confidence interval:
- Hard to interpret
- You are tied to one significant level.
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