Statistics: First Steps
Andrew Martin
PS 372
University of Kentucky
Variance
Variance is a measure of dispersion of data
points about the mean for interval- and ratio-level
data.
Variance is...
Standard Deviation
Standard deviation is a measure of dispersion of
data points about the mean for interval- and ratio-
le...
Normal Distribution

The bulk of observations lie in the center,
where there is a single peak.

In a normal distribution...
Normal Distribution

Mathematical theory allows us to know what
percentage of observations lie within one
(68%), two (95%...
Frequency Distribution
What about categorical variables?
Example
Calculate the ID and IQV for a former PS 372
class grades using the following frequencies or
proportions:
Grade Fr...
Index of Diversity
ID = 1 – (p2
a
+ p2
b
+ p2
c
+p2
d
+p2
e
)
ID = 1 - (.122
+ .212
+ .122
+ .212
+ .342
)
ID = 1 - (.0144...
Index of Qualitative Variation
1 – (p2
a
+ p2
b
+ p2
c
+p2
d
+p2
e
)
1 - (1/K)
Index of Qualitative Variation
.7674
(1 – 1/5)
.9592
Data Matrix
A data matrix is an array of rows and columns
that stores the values of a set of variables for all
the cases i...
Data Matrix from JRM
Properties of Good Graphs
Should answer several of the following questions:
(JRM 384)
1. Where does the center of the dist...
Properties of Good Graphs
5. Do observations tend to pile up at one end of
the measurement scale, with relatively few
obse...
Statistical Concepts
Let's quickly review some concepts.
Population
A population refers to any well-defined set of
objects such as people, countries, states,
organizations, and so...
Population

A sample is a subset of the population.

Samples are drawn in some known manner and
each case is chosen inde...
Populations

Parameters are numerical features of a
population.

A sample statistic is an estimator that
corresponds to ...
Two Kinds of Inference
Hypothesis Testing
Point and interval estimation
Hypothesis Testing
Many claims can be translated into specific
statements about a population that can be
confirmed or disc...
Point and Interval Estimation
The goal here is to estimate unknown population
parameters from samples and to surround thos...
Hypothesis Testing
Start with a specific verbal claim or proposition.
Ex: The chances of getting heads or tails when
flipp...
Hypothesis Testing
Hypothesis Testing
Next, the researcher constructs a null hypothesis.
A null hypothesis is a statement that a
population p...
Hypothesis Testing
Following up on the coin example, the null
hypothesis would equal .5.
Stated more formally: H0
: P = .5...
Hypothesis Testing

Next, specify an alternative hypothesis.

An alternative hypothesis is a statement
about the value o...
Hypothesis Testing
Suppose you believe the coin is unfair, but have
no intuition about whether it is too prone to come
up ...
Hypothesis Testing
Perhaps you believe the coin is more likely to
come up heads than tails. You would formulate
the follow...
Hypothesis Testing

After specifying the null and alternative
hypothesis, identify the sample estimator that
corresponds ...
Hypothesis Testing

Next, determine how the sample statistic is
distributed in repeated random samples. That
is, specify ...
Hypothesis Testing

Make a decision rule based on some criterion
of probability or likelihood.

In social sciences, a re...
Hypothesis Testing

In light of the decision rule, define a critical
region. The critical region consists of those
outcom...
Hypothesis Testing

Collect a random sample and calculate the
sample estimator.

Calculate the observed test statistic. ...
Statistics 091208004734-phpapp01 (1)
Statistics 091208004734-phpapp01 (1)
Statistics 091208004734-phpapp01 (1)
Statistics 091208004734-phpapp01 (1)
Statistics 091208004734-phpapp01 (1)
Statistics 091208004734-phpapp01 (1)
Statistics 091208004734-phpapp01 (1)
Statistics 091208004734-phpapp01 (1)
Statistics 091208004734-phpapp01 (1)
Statistics 091208004734-phpapp01 (1)
Statistics 091208004734-phpapp01 (1)
Statistics 091208004734-phpapp01 (1)
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  1. 1. Statistics: First Steps Andrew Martin PS 372 University of Kentucky
  2. 2. Variance Variance is a measure of dispersion of data points about the mean for interval- and ratio-level data. Variance is a fundamental concept that social scientists seek to explain in the dependent variable.
  3. 3. Standard Deviation Standard deviation is a measure of dispersion of data points about the mean for interval- and ratio- level data. Like the mean, standard deviation is sensitive to extreme values. Standard deviation is calculated as the square root of the variance.
  4. 4. Normal Distribution  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 below it.  The mean, median and mode have the same statistical values.  Fewer and fewer observations fall in the tails.  The spread of the distribution is symmetric.
  5. 5. Normal Distribution  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.
  6. 6. Frequency Distribution
  7. 7. What about categorical variables?
  8. 8. Example Calculate the ID and IQV for a former PS 372 class grades using the following frequencies or proportions: Grade Freq. Prop. A 4 (.12) B 7 (.21) C 4 (.12) D 7 (.21) E 12 (.34)
  9. 9. Index of Diversity ID = 1 – (p2 a + p2 b + p2 c +p2 d +p2 e ) ID = 1 - (.122 + .212 + .122 + .212 + .342 ) ID = 1 - (.0144 + .0441 + .0144 + .0441 + .1156) ID = 1 - (.2326) ID = .7674
  10. 10. Index of Qualitative Variation 1 – (p2 a + p2 b + p2 c +p2 d +p2 e ) 1 - (1/K)
  11. 11. Index of Qualitative Variation .7674 (1 – 1/5) .9592
  12. 12. Data Matrix A data matrix is an array of rows and columns that stores the values of a set of variables for all the cases in a data set. This is frequently referred to as a dataset.
  13. 13. Data Matrix from JRM
  14. 14. Properties of Good Graphs Should answer several of the following questions: (JRM 384) 1. Where does the center of the distribution lie? 2. How spread out or bunched up are the observations? 3. Does it have a single peak or more than one? 4. Approximately what proportion of observations in in the ends of the distributions?
  15. 15. 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 another variable?
  16. 16. Statistical Concepts Let's quickly review some concepts.
  17. 17. Population 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.
  18. 18. Population  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.
  19. 19. Populations  Parameters are numerical features of a population.  A sample statistic is an estimator that corresponds to a population parameter of interest and is used to estimate the population value.  Y is the sample mean, (μ) is the population mean.  ^ is a “hat”, caret or circumflex
  20. 20. Two Kinds of Inference Hypothesis Testing Point and interval estimation
  21. 21. Hypothesis Testing Many claims can be translated into specific statements about a population that can be confirmed or disconfirmed with the aid of probability theory. Ex: There is no ideological difference between the voting patterns between the voting patterns of Republican and Democrat justices on the U.S. Supreme Court.
  22. 22. 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 precision.
  23. 23. Hypothesis Testing 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 same.
  24. 24. Hypothesis Testing
  25. 25. Hypothesis Testing Next, the researcher constructs a null hypothesis. A null hypothesis is a statement that a population parameter equals a specific value.
  26. 26. Hypothesis Testing 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. H0 is typically used to denote a null hypothesis.
  27. 27. Hypothesis Testing  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 null hypothesis.
  28. 28. Hypothesis Testing 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: HA : P ≠ .5
  29. 29. Hypothesis Testing Perhaps you believe the coin is more likely to come up heads than tails. You would formulate the following alternative hypothesis: HA : 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 direction: HA : P < .5
  30. 30. Hypothesis Testing  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.
  31. 31. Hypothesis Testing  Next, determine how the sample statistic is distributed in repeated random samples. That is, specify the sampling distribution of the estimator.  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)?
  32. 32. Hypothesis Testing  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 common..  Make the decision rule before collecting data.
  33. 33. Hypothesis Testing  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 occur.  So there are areas of “rejection” (critical areas) and nonrejection.
  34. 34. Hypothesis Testing  Collect a random sample and calculate the sample estimator.  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 critical values.  Examine the observed test statistic to see if it falls in the critical region.  Make practical or theoretical interpretation of the findings.
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