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

List of Variable Values Raw Curved Grade 100 103 A 98.46154 101.4615 95.38462 98.38462 92.69231 95.69231 90.76923 93.76923 89.23077 92.23077 88.46154 91.46154 88.07692 91.07692 86.15385 89.15385 B 85.38462 88.38462 84.61538 87.61538 84.23077 87.23077 83.07692 86.07692 80 83

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

Example – Interval/Ratio Freq. distribution of midterm grades

Example – Interval/Ratio

Example – Interval/Ratio Freq. / Total

Example – Interval/Ratio Freq. / Total*100

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

Example - Nominal

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

Example – Bar Chart

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

Histogram of County Unemployment Rates in Fla

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

Measures of Central Tendency Median: Defined as the value of the variable in the “middle” of the distribution. Odd# of obs: 2 2 5 9 11 median=5 Even# of obs: 2 2 5 9 11 15 median=(5+9)/2 = 7 Appropriate for ordinal, interval and ratio

Measures of Central Tendency Mode: Defined as the value that occurs most often. 2 2 5 9 11 15 Mode=2 Appropriate for all levels of measurement

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|

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

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

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

The Normal Distribution

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

Histogram of County Unemployment Rates in Fla

Descriptive Statistics for County Unemployment Rates in Fla . sum unemp, detail unemp ------------------------------------------------------------- Percentiles Smallest 1% 2 1.7 5% 2.4 1.7 10% 2.7 1.7 Obs 3149 25% 3.4 1.7 Sum of Wgt. 3149 50% 4.4 Mean 4.809908 Largest Std. Dev. 2.129031 75% 5.5 19.5 90% 7.2 19.5 Variance 4.532774 95% 8.6 19.6 Skewness 2.30285 99% 13 19.7 Kurtosis 12.11621

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

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.

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

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)

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)

Example Hillary Clinton Feeling Thermometer (NES 2004)

Example Hillary Clinton Feeling Thermometer (NES 2004) Mean = 64.137, s.d. = 88.408, N = 1212

Example Hillary Clinton Feeling Thermometer (NES 2004) Mean = 64.137, s.d. = 88.408, N = 1212 Standard Error = 88.408 / √1212 = 2.539

Example Hillary Clinton Feeling Thermometer (NES 2004) Mean = 64.137, s.d. = 88.408, N = 1212 Standard Error = 88.408 / √1212 = 2.539 95% CI = 64.137 ± 1.96 * 2.539 = 59.158, 69.116