Statistics: First Steps
University of Kentucky
Variance is a measure of dispersion of data
points about the mean for interval- and ratio-level
Variance is a fundamental concept that social
scientists seek to explain in the dependent
Standard deviation is a measure of dispersion of
data points about the mean for interval- and ratio-
Like the mean, standard deviation is sensitive to
Standard deviation is calculated as the square
root of the variance.
The bulk of observations lie in the center,
where there is a single peak.
In a normal distribution half (50 percent) of the
observations lie above the mean and half lie
The mean, median and mode have the same
Fewer and fewer observations fall in the tails.
The spread of the distribution is symmetric.
Mathematical theory allows us to know what
percentage of observations lie within one
(68%), two (95%) or three (98%) standard
deviations of the mean.
If data are not perfectly normally distributed, the
percentages will only be approximations.
Many naturally occurring variables do have
nearly normal distributions.
Some can be transformed using logarithms.
Properties of Good Graphs
Should answer several of the following questions:
1. Where does the center of the distribution lie?
2. How spread out or bunched up are the
3. Does it have a single peak or more than one?
4. Approximately what proportion of observations
in in the ends of the distributions?
Properties of Good Graphs
5. Do observations tend to pile up at one end of
the measurement scale, with relatively few
observations at the other end?
6. Are there values that, compared with most,
seem very large or very small?
7. How does one distribution compare to another
in terms of shape, spread, and central tendency?
8. Do values of one variable seem related to
Let's quickly review some concepts.
A population refers to any well-defined set of
objects such as people, countries, states,
organizations, and so on. The term doesn't simply
mean the population of the United States or some
other geographical area.
A sample is a subset of the population.
Samples are drawn in some known manner and
each case is chosen independently of the other.
From here on out, when the book uses the term
sample, random sample or simple random
sample, it's making reference to the same
concept, which is a sample chosen at random.
Parameters are numerical features of a
A sample statistic is an estimator that
corresponds to a population parameter of
interest and is used to estimate the population
Y is the sample mean, (μ) is the population
^ is a “hat”, caret or circumflex
Two Kinds of Inference
Point and interval estimation
Many claims can be translated into specific
statements about a population that can be
confirmed or disconfirmed with the aid of
Ex: There is no ideological difference between the
voting patterns between the voting patterns of
Republican and Democrat justices on the U.S.
Point and Interval Estimation
The goal here is to estimate unknown population
parameters from samples and to surround those
estimates with confidence intervals. Confidence
intervals suggest the estimates reliability or
Start with a specific verbal claim or proposition.
Ex: The chances of getting heads or tails when
flipping the coin is are roughly the same.
Ex: The chances of the United States electing a
Republican or Democrat president are roughly the
Next, the researcher constructs a null hypothesis.
A null hypothesis is a statement that a
population parameter equals a specific value.
Following up on the coin example, the null
hypothesis would equal .5.
Stated more formally: H0
: P = .5
Where P stands for the probability that the coin
will be heads when tossed.
is typically used to denote a null hypothesis.
Next, specify an alternative hypothesis.
An alternative hypothesis is a statement
about the value or values of a population
parameter. It is proposed as an alternative to
the null hypothesis.
An alternative hypothesis can merely state that
the population does not equal the null
hypothesis, or is greater than or less than the
Suppose you believe the coin is unfair, but have
no intuition about whether it is too prone to come
up heads or tails.
Stated formally, the alternative hypothesis is:
: P ≠ .5
Perhaps you believe the coin is more likely to
come up heads than tails. You would formulate
the following alternative hypothesis:
: P > .5
Conversely, if you believe the coin is less likely to
come up heads than tails, you would formulate
the alternative hypothesis in the opposite
: P < .5
After specifying the null and alternative
hypothesis, identify the sample estimator that
corresponds to the parameter in question.
The sample must come from the data, which in
this case is generated by flipping a coin.
Next, determine how the sample statistic is
distributed in repeated random samples. That
is, specify the sampling distribution of the
For example, what are the chances of getting
10 heads in 10 flips (p = 1.)? What about 9
heads in 10 flips (p = .9)? 8 flips (p = .8)?
Make a decision rule based on some criterion
of probability or likelihood.
In social sciences, a result that occurs with a
probability of .05 (that is, 1 chance in 20) is
considered unusual and consequently is
grounds for rejecting a null hypothesis.
Other common thresholds (.01, .001) are also
Make the decision rule before collecting data.
In light of the decision rule, define a critical
region. The critical region consists of those
outcomes so unlikely to occur that one has
cause to reject the null hypothesis should they
So there are areas of “rejection” (critical areas)
Collect a random sample and calculate the
Calculate the observed test statistic. A test
statistic converts the sample result into a
number that can be compared with the critical
values specified by your decision rule and
Examine the observed test statistic to see if it
falls in the critical region.
Make practical or theoretical interpretation of