1. Univariate Statistics
Analysis of a single variable
Two general varieties:
Descriptive Statistics: Describe Variables (where data are any collection of observations, sample/population)
Inferential Statistics: Make inferences about the population based on characteristics of sample data

2. List of Variable Values
Raw
Curved
Grade
100
103 A B 80 83

3. Frequency Distribution
A summary of the observations for a variable
Includes a list of the values of the variable and the frequency of observations for each value

4. Example – Interval/Ratio
Freq. distribution of midterm grades

5. Example – Interval/Ratio

6. Example – Interval/Ratio
Freq. / Total

7. Example – Interval/Ratio
Freq. / Total*100

8. Example - Nominal
Freq. distribution of active hate group organizations in 1999

9. Example - Nominal

10. Summarizing Data in Graphs
Pie charts, Bar charts: appropriate for nominal variables and ordinal variables (small number of categories)

11. Example – Bar Chart

12. Summarizing Data in Graphs
Histograms: appropriate for all interval/ratio variables with a large number of possible values; data are collapsed into intervals, and axis labels represent interval boundaries or interval midpoints

13. Histogram of County Unemployment Rates in Fla

14. Measures of Central Tendency
Mean _
Y =  Yi / N
Appropriate for interval/ratio variables ONLY

15. Measures of Central Tendency
Median: Defined as the value of the variable in the “middle” of the distribution.
Odd# of obs: median=5
Even# of obs:
median=(5+9)/2 = 7
Appropriate for ordinal, interval and ratio

16. Measures of Central Tendency
Mode: Defined as the value that occurs most often.
Mode=2
Appropriate for all levels of measurement

17. Measures of Dispersion
1. Range |Ymax - Ymin|
Weakness?
2. Percentiles - For variable Y, the pth percentile represents the value of Y below which p% of the observations fall.
50th percentile = median
IQR = |Y75pct - Y25pct|

18. Measures of Dispersion (cont’d)
More complex measures: Based on “mean deviations” _ Yi – Y _
Average Mean Deviation(?): S (Yi – Y) / N
Mean Absolute Deviation: S |Yi – Y| / N
could use as measure of variation
Mean Squared Deviation: S(Yi – Y)2 / N

19. Variance (sample) Standard Deviation _ s2Y= S (Yi - Y)2 / (N-1)
Numerator = “Sum of Squares”
Denominator = “degrees of freedom”

20. The Normal Distribution
Symmetric
Bell-shaped
Mean=Median=Mode

21. The Normal Distribution

22. Deviations from the normal distribution
Bimodal distributions
Skewed distributions
Left skew vs. right skew
Mean is pulled in direction of skew

23. Histogram of County Unemployment Rates in Fla

24. Descriptive Statistics for County Unemployment Rates in Fla
. sum unemp, detail
unemp
Percentiles Smallest 1% 5%
10% Obs
25% Sum of Wgt
50% Mean
Largest Std. Dev
75%
90% Variance
95% Skewness
99% Kurtosis

25. Sampling Distribution (sample means)
Population
Draw Random Sample of Size N
Calculate sample mean
Repeat until all possible random samples are exhausted
The resulting collecting of sample means is the sampling distribution of sample means

26. Sampling Distribution of Sample Means
A frequency distribution of all possible sample means for a given sample size (N)
The mean of the sampling distribution will be equal to the population mean.

27. Sampling Distribution of Sample Means
When N is reasonably large (>30), the sampling distribution will be normally distributed
The standard error of the sampling distribution can be reliably estimated as (where sY = sample standard deviation for Y and N= sample size).
sY /√N

28. Standard Error
How the sample means vary from sample to sample (i.e. within the sampling distribution) is expressed statistically by the value of the standard deviation (i.e. standard error) of the sampling distribution.
(Standard deviation = the “average” distance of each observation from the mean)

29. Using the Standard Error to Calculate a 95% Confidence Interval
Calculate the mean of Y
Calculate the standard deviation of Y
Calculate the standard error of Y
Calculate a 95% confidence interval for the population mean of Y: _
95% CI = Y ± 1.96*(standard error)

30. Example
Hillary Clinton Feeling Thermometer (NES 2004)

31. Example Hillary Clinton Feeling Thermometer (NES 2004)
Mean = , s.d. = , N = 1212

32. Example Hillary Clinton Feeling Thermometer (NES 2004)
Mean = , s.d. = , N = 1212
Standard Error = / √1212 = 2.539

33. Example Hillary Clinton Feeling Thermometer (NES 2004)
Mean = , s.d. = , N = 1212
Standard Error = / √1212 = 2.539
95% CI = ± 1.96 * 2.539
= ,