Univariate Analysis

POLS 300

Butz

The Normal Distribution

• Unimodal

• Symmetric

• Bell-shaped

• Mean=Median=Mode

The Normal Distribution

The Normal Distribution

• A Fixed Proportion of cases lies between the mean and any distance from the mean…distance measured in terms of???

• Standard Deviation

The Normal Distribution

Normal Distribution

• 68.26% of “total area”/ “total cases” falls 1 SD above/below the mean --- (34.13)

• 95.44% --- 2 SD from mean (47.72)

• 95.00% --- 1.96 SD from mean (47.5)

• 99.00% --- 2.58 SD from mean

The Normal Distribution

Standard Normal Distribution and Z-Score

• For any value/score of a variable…Find how many standard deviations above/below the mean

• Find “areas” (i.e. percentage of values) that fall above and below a value/score…and find areas that fall between 2 values/scores!!!

Standard Normal Distribution and Z-Score

• Normal distribution with Mean of Zero and Standard Deviation of 1!

• Distribution of Z-Scores – Standardizing!!!

• Z-Score – Number of Standard Deviations any value/score is above/below the Mean

Calculate Z-Score

• How do we calculate a z-score for a given case in a distribution?

• Need to know sample Mean and SD…then…

• (Yi – Y)/ SD = Z-score• Find Z-score then go to Appendix D (old book)

or Appendix A (new book) to find Area Between Midpoint and Z-score…then subtract from .5 (Find out the proportion of values that are above that value)

Z-Score

• Find Z-score then go to Appendix D (old book; p. 484) to find Area Between Midpoint (.5) and Z-score…then subtract that area from .5

• Appendix A (new book; p. 575) gives you the Area in the “tail”… i.e. the proportion of cases falling above/below the mean!

Examples

• 100 point scale examples…

• Distribution of test scores with Mean of 50 and SE of 10…How did you do versus other students if you received a 70%? What proportion of students did better than you?

• Z = (70 – 50)/10 = 2 --- .4772 --- (.5 - .4772 = 2.28%)…only 2% basically did better

Z- Score Cont.

• What percent scored below a 40%• (40-50)/10 = -1.00 = Z• Go to Appendix D; p. 484• Area corresponding to 1.0 = .3413• .5 - .3413 = .1587 --- 15.87% did worse!

• New Book… Go to Appendix A; p. 575… gives you the area in the “tail”!

Z- Score Cont.

• How many of the scores fall between 40% and 70%

• Add the Area above 70 and Area below 40

• .1587 + .228 = .1815

• Subtract this area from total area = 1.0

• 1.0 - .1815 = .8185

• Thus 81.85% of values fall between 40 and 70%

Normal Distribution and Statistical Inference

• Using the principles of Normal Distribution and the Sampling Distribution of Sample Means…to make inferences about Unknown POPULATION Parameters!!!

• How certain are we that any one sample mean reflects the true population mean??

• Can we construct a range in which the population mean is likely to fall???

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, and can use sampling standard deviation to get standard error of the sampling distribution

• The standard error is simply a standard deviation applied to a sampling distribution

• How the sample proportions/means vary from sample to sample (i.e. within the sampling distribution) is expressed statistically by the value of the Standard Error of the sampling distribution.

Standard Error

• SE of the sampling distribution can be reliably estimated as (where sY = sample standard deviation for Y and N= sample size).

sY /√N

• Use SE to estimate the true Mean of Population from a Sample!!!

• Most important Can use Standard Error to Calculate Confidence Intervals

• How confident we are that the population mean falls with a certain range!

Confidence Intervals

• How do we get these estimates???

• Standard Error Used to Calculate Confidence Intervals

• How confident we are that the population mean falls with a certain range!

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 with sample mean of Y:

_95% CI = Y ± 1.96*(standard error)

Example

• Gays/Lesbian Feeling Thermometer (NES 2000)

• Mean = 47.52, s.d. = 27.45, N = 1448

Example

• Gays/Lesbian Feeling Thermometer (NES 2000)

• Mean = 47.52, s.d. = 27.45, N = 1448

• Standard Error = 27.45 / √1448 = .72

Example

• Gays/Lesbian Feeling Thermometer (NES 2000)

• Mean = 47.52, s.d. = 27.45, N = 1448

• Standard Error = 27.45 / √1448 = .72

• 95% CI = 47.52 ± 1.96 * .72 = 1.41• = 46.11, 48.93