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Univariate Analysis

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???

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Univariate Analysis

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  1. Univariate Analysis POLS 300 Butz

  2. The Normal Distribution • Unimodal • Symmetric • Bell-shaped • Mean=Median=Mode

  3. The Normal Distribution

  4. 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

  5. The Normal Distribution

  6. 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

  7. The Normal Distribution

  8. 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!!!

  9. 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

  10. 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)

  11. 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!

  12. 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

  13. 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”!

  14. 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%

  15. 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???

  16. 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

  17. 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.

  18. 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.

  19. 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!

  20. 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!

  21. 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)

  22. Example • Gays/Lesbian Feeling Thermometer (NES 2000) • Mean = 47.52, s.d. = 27.45, N = 1448

  23. Example • Gays/Lesbian Feeling Thermometer (NES 2000) • Mean = 47.52, s.d. = 27.45, N = 1448 • Standard Error = 27.45 / √1448 = .72

  24. 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

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