1. PA330
FEB 28, 2000

2. General information 2nd half of semester - switching gears to statistics
Goals - develop ability to:
Calculate statistics commonly used in public administration
Choose appropriate statistical tool for various problems/research questions
Interpret statistical applications used in other research (academic and applied)

3. How to accomplish these goals
Read before class - will enhance your understanding of lectures
Meier/Brudney is a great “cookbook”. Use it!
Keep a formula sheet, list of notations
Do repeated problems -the more you practice, the more comfortable and quicker you become
class exercises, homework, lab homework
don’t rush through homework

4. TYPES OF STATISTICS
Descriptive statistics - to describe a characteristic. The aggregated measurement of a single variable for a number of cases
Inferential statistics - uses a sample of cases to infer characteristics to a population. In other words, use data from cases studied to conclude something about an entire population that includes cases not studied. Inferential is built upon descriptive statistics

5. Univariate vs multivariate statistics
Univariate statistics - tell us about the distribution of the values of one variable
Multivariate statistics - measure the joint distribution of two or more variables and assess the relationships between and among the variables

6. Prelude to data analysis
Data coding
Array - present the measurements associated with each case
ALWAYS “eyeball” your data before analyzing
Overhead - Low Income Uninsured Children

7. Visual Presentation Tables Graphs bar graphs circle graphs line graphs
A:/Dagedata#5.wb3
Page 1 - data
Page 2 - charts

8. Bar graph (histogram) quantitative differences but not portion of the whole shows both #s and magnitude

9. Circle (pie) graph parts of a whole: relationships, percentages

10. Line graph Used to display trends, change over time

11. Statistical Tools for Describing the World - Distributions
P Intuitive Definition
< A bunch of numbers that measure a characteristic for a group of cases.
< May be represented by a set of numbers, a graph or picture, or even a mathematical equation.

12. Frequency Distribution
A frequency distribution is a graph or chart that shows the number of observations of a given value or class interval.

13. Frequency distribution
Frequency distribution - lists the values of categories of each variable and the number of cases with each of the values
Categories must be exhaustive and mutually exclusive (each case will fit into one and only one category)
Can be displayed both numerically and graphically
O’Sullivan, Table 11.5

14. The Frequency Histogram
To create a frequency histogram
Determine the class interval width
Determine the number of intervals desired
Tally number of observations in each range
Create bar chart from class totals

15. Frequency polygon
Same as a frequency histogram except the midpoints of the class intervals are used
Points are connected with a line graph
A large number of classes will make the distribution a smooth curve if there is a large sample size.

16. Stem and leaf plot Preserves data and creates histogram

17. Frequency Distributions Shape
Modality
The number of peaks in the curve
Skewness
An asymmetry in a distribution where values are shifted to one extreme or the other.
Kurtosis
The degree of peakedness in the curve

18. Frequency Distributions Modality
Unimodal
Bimodal
Multi-modal

19. Frequency Distributions Skewness
Right Skew (Positive Skew)
Left Skew (Negative Skew)

20. Frequency Distributions Kurtosis
Platykurtic
Leptokurtic
Mesokurtic

21. Characteristics of a distribution
When summarizing cases or the distribution of their values, we usually want to know two things:
how similar are the individual values to each other (measures of central tendency)
how different are the values from one another (measures of dispersion)

22. Measures of Central Tendency
Measures which provide some indication of the typical value or the 'middle' of the distribution

23. Measures of Central Tendency Listed in order of least to most useful
mode - most frequent category
median - the middle value
mean - arithmetic average

24. Mode The category or value that most commonly occurs among all cases
In a frequency distribution, is the value with the highest frequency
Can be determined for all levels of measurement
Used primarily for nominal data

25. Mode
The peak (or tallest) value of a frequency distribution histogram is also referred to as the mode.
The mode is the category or value, not the number of cases containing that value

26. Median The value or category that is the center of the distribution
Requires ordinal or interval/ratio level of measurement (can be determined only if the cases can be ordered)
Best measure for skewed distributions
One half of cases have a value less than the median and one half have a value more than the median

27. Median Place cases in order, then select middle value.
If even number of cases, average the two middle values

28. Mean Arithmetic average
Most commonly known measure of central tendency
Is the “balance point” or center of the distribution
Most appropriate for symmetric distributions
Influenced by extreme values

29. Mean
Created by summing values of each case and dividing by number of cases

30. Calculate mean, median, mode

31. Mean = 5
Median = 5
Mode = 6

32. Grouped data - median
If data is grouped (as in a frequency distribution) the exact value of the median can’t be found. It can be estimated by finding the class interval of the middle case and taking the mid-point of the interval.
Overhead - O’S, Table 11.11

33. Find midpoint of each class interval.
Grouped data - mean
Again, will not be exact.
Find midpoint of each class interval.
Multiply the frequency for that interval times the midpoint.
Sum and divide by number of cases.
O’Sullivan, table 11.12

34. How to select appropriate measure of central tendency
Level of measurement
median requires ordinal
mean requires interval
Shape of the distribution
unimodal vs bimodal
skewedness

35. If unimodal and symmetrical (or almost), mean is preferred measure
mean, median, mode will be the same (or almost)
If extreme values, mean is distorted and median should be used
Overhead, O’Sullivan, pg 341

36. Mathematical notation Important mathematical notation the student needs to know.å
PSummation
< The 3 is a symbolic representation of the process of adding up a specified series or collection of numbers.
< For instance, the sum of all Xi from {I=1} to n means: beginning with the first number in your data set, add together all n numbers. X i i = 1

37. Mean The sum of all of the cases (numbers) in a set, divided by the number of cases in the set
Sample mean
Population mean

38. Measures of dispersion
Two distributions can have very similar means but different overall values.
These measures tell us how much individual values vary from one another.
Small values on the measure of dispersion imply more uniformity; larger values imply more diversity.

39. Measures of Dispersion Range
Highest value minus the lowest value
Uses only two pieces of information, so is strongly influenced by these two values.
$58,000 - $12,000 = $46,000 (mean=$35,000)
$36,000 - $34,000 = $2,000 (mean=$35,000)

40. Measures of Dispersion standard deviation and variance
Standard deviation - measures the average distance of values in a distribution from the mean of the distribution
Variance - the square of the standard deviation
Important statistics used in many other statistical measures and tests

41. Measures of Dispersion variance
The mean of the squared deviations

42. Or more simply

43. Standard deviation
Square the deviations to remove minus signs
Take the square root to return to the original scale

44. Standard deviation
Again, more simply

45. Common notation if standard deviation of a sample, rather than a population

46. Calculating the Standard Deviation
The easiest way to calculate the standard deviation is to use a computer.
Can be done in SPSS and spreadsheets such as Excel and Quattro Pro

47. Interpretation of standard deviation
Standard deviation can only be interpreted in conjunction with mean and range