Distribution Summaries

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

2. Distribution spread • Range • Standard deviation • Variance

3. Range • The range of a distribution is the difference between the highest value and the lowest value Length of Cohabitation in Months 0 103

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

5. 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

6. Range (cont.) 103 97

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

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

9. 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

10. Variance Formula (cont.) Definitional Formula

11. Variance Formula (cont.) Computational Formula

12. 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

13. 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

14. Compare

15. Compare (cont.) Squared Deviations Deviations 4 1 0 1 4 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 60 60 0 Sum 60 60 0 10 12 12 0 Mean 12 12 0 2 Variance

16. 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 Variance Standard Deviation

17. 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

18. 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

19. 1 2 3 4 5 Variance (example) Variance = 2 Std. Dev. = 1.414 Mean

20. 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

21. 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)

22. 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