2. Distribution Summaries Measures of central tendency Mean Median Mode Measures of spread Standard Deviation Interquartile Range (IQR)

3. Distribution spread Range Standard deviation Variance

4. Range The range of a distribution is the difference between the highest value and the lowest value Length of Cohabitation in Months 0103

5. Range (cont.). sum cohbl Variable | Obs Mean Std. Dev. Min Max ---------+----------------------------------------------------- cohblnth | 626 11.74601 17.1347 0 103

6. Range (cont.). sum cohbl, d # Months Cohabited ------------------------------------------------------------- Percentiles Smallest 1% 0 0 5% 0 0 10% 0 0 Obs 626 25% 0 0 Sum of Wgt. 626 50% 5 Mean 11.74601 Largest Std. Dev. 17.1347 75% 17 97 90% 32 97 Variance 293.5978 95% 46 103 Skewness 2.304175 99% 79 103 Kurtosis 9.411293

7. Range (cont.) 103 97

8. Range The range of a distribution is the difference between the highest value and the lowest value

9. Variance The most commonly used measure of spread One of the most fundamental concepts in statistics

10. Variance Formula In words, the variance is the mean squared deviation (from the mean) A deviation is the difference between a score and the mean of all scores We square this deviation for all observations We then take the mean of all these

11. Variance Formula (cont.) Definitional Formula

12. Variance Formula (cont.) Computational Formula

13. Variance (example) Obs Square Dev Dev Sq 1 1 -2 4 2 4 -1 1 3 9 0 0 4 16 1 1 5 25 2 4 Sum 15 55 0 10 Mean 3 2 Variance = (55 - 225 / 5) / 5 = (55-45) / 5 = 2

14. Why sum the SQUARES? Recall that the sum of the deviations around the mean is zero Therefore the average deviation is zero Squaring a positive or negative number always creates a positive result This way we are assured of a sum that is greater than or equal to zero

15. Compare

16. Compare (cont.) Deviations Squared Deviations 10 - 12 = -2 11 - 12 = -1 12 - 12 = 0 13 - 12 = 1 14 - 12 = 2 10 - 12 = -2 11 - 12 = -1 12 - 12 = 0 13 - 12 = 1 14 - 12 = 2 4101441014 Sum Mean 60 60 0 10 12 12 0 2 Variance

17. Standard Deviation The second most commonly used measure of spread The square root of the variance Which brings us back to the original metric or units of measure Standard DeviationVariance

18. What are units? Consider age Units are years Deviations are years Squared deviations are years squared Summing and taking mean leaves squared years Taking square root yields years again

19. So we have the sd? The standard deviation is about 1/6 of the range For a normal distribution, about 70% of observations are ± 1 σ from the mean. And, about 90% are ± 2 σ from the mean And, about 99% are ± 3 σ from the mean

20. Variance (example) 12345 Variance = 2 Std. Dev. = 1.414 Mean

21. Variability of the scores Variability and spread of the scores indicate the second characteristic of a distribution that we need to know. The first was the mean or central location of the distribution

22. The mean and variance are independent Means can change without affecting the variance (or standard deviation) Standard deviation (or variance) can change without affecting the mean Two distributions may differ on means or on standard deviations or both (or neither)

23. What makes scores variable? Why are some scores high and others low? Why does the variance change?. tab sex, sum(income1) | Summary of income1 sex | Mean Std. Dev. Freq. ------------+------------------------------------ female | 16.207224 10.82088 263 male | 22.371972 13.304104 289 ------------+------------------------------------ Total | 19.434783 12.557429 552