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Descriptive Statistics. Frequency Distributions Measures of Central Tendency Measures of Dispersion Shape of the Distribution Introducing the Normal Curve (Today’s data file for calculations: 20 cases from the Simon data set for the 14 th ward). Segment of Simon Data Set.

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## Descriptive Statistics

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**Descriptive Statistics**• Frequency Distributions • Measures of Central Tendency • Measures of Dispersion • Shape of the Distribution • Introducing the Normal Curve (Today’s data file for calculations: 20 cases from the Simon data set for the 14th ward)**Frequency Distribution**A frequency distribution tabulates all the values of a variable. The table usually provides frequency counts, percentages, cumulative counts, and cumulative percentages.**Examples of Frequency Distributions**• Cum Cum • Count Count Pct Pct FAMILIES • 10. 10. 50.0 50.0 1 • 10. 20. 50.0 100.0 2 • Cum Cum • Count Count Pct Pct OCC$ • 7. 7. 35.0 35.0 skilled • 13. 20. 65.0 100.0 unskilled • Cum Cum • Count Count Pct Pct OWN • 3. 3. 15.8 15.8 0 • 16. 19. 84.2 100.0 1**Frequency Distributionfor Persons**Cum Cum Count Count Pct Pct PERSONS 2. 2. 10.0 10.0 2 1. 3. 5.0 15.0 3 1. 4. 5.0 20.0 4 1. 5. 5.0 25.0 5 3. 8. 15.0 40.0 6 2. 10. 10.0 50.0 7 2. 12. 10.0 60.0 8 2. 14. 10.0 70.0 9 2. 16. 10.0 80.0 10 2. 18. 10.0 90.0 11 1. 19. 5.0 95.0 12 1. 20. 5.0 100.0 13**Measures of Central Tendency**• Mean (X with a bar on top) - the sum of the values for a variable divided by the number of values (N). Used for interval level data. • Median - the point at which half of values are greater than and half the values are less than the point. A good measure of central tendency for skewed interval level data (such as income) and for ordinal data. • Mode - the value occurring most frequently. A good measure of central tendency for small ordinal and nominal scales.**Example: Calculating a Mean**20 cases for the variable PERSONS**Steps for Calculating a Mean**Sum the cases = 149 Divide by number of cases, 20 149/20 = 7.45**Example: Calculating a Median**20 cases for the variable PERSONS**Steps for Calculating a Median**• Identify the variable • Sort the values of the variable • Find the case that is at the half way point or the 50th percentile.**20 cases sorted and midpoints marked**• Median = 7.5 • (with even number of cases, average the 2 middle cases)**Steps for Finding the Mode**• Identify the variable • Create a Frequency Distribution of Values • A frequency distribution tabulates all the values of a variable. The table usually provides frequency counts, percentages, cumulative counts, and cumulative percentages. • Find the value that occurs most frequently**Finding the Mode**• Cum Cum • Count Count Pct Pct FAMILIES • 10. 10. 50.0 50.0 1 • 10. 20. 50.0 100.0 2 • Cum Cum • Count Count Pct Pct OCC$ • 7. 7. 35.0 35.0 skilled • 13. 20. 65.0 100.0 unskilled • Cum Cum • Count Count Pct Pct OWN • 3. 3. 15.8 15.8 0 • 16. 19. 84.2 100.0 1**Measures of Dispersion**• Minimum - lowest score • Maximum - highest score • Range - the difference between the highest and lowest score • Ntiles - Percentiles of cases in the frequency distribution. The median is the 50th percentile. Other common percentiles are quartiles, quintiles, thirds, deciles.**Frequency Distributionfor Persons**Cum Cum Count Count Pct Pct PERSONS 2. 2. 10.0 10.0 2 1. 3. 5.0 15.0 3 1. 4. 5.0 20.0 4 1. 5. 5.0 25.0 5 3. 8. 15.0 40.0 6 2. 10. 10.0 50.0 7 2. 12. 10.0 60.0 8 2. 14. 10.0 70.0 9 2. 16. 10.0 80.0 10 2. 18. 10.0 90.0 11 1. 19. 5.0 95.0 12 1. 20. 5.0 100.0 13**Measures of Dispersion, cont.**• Variance - the mean of the squared deviations of values from the mean. • Standard deviation (s) - the square root of the sum of the squared deviations from the mean divided by the number of cases. (Variance is the standard deviation squared) • Coefficient of variation – standard deviation divided by the mean.**Equations**• Mean • Variance • Standard Deviation • Coefficient of Variation**Steps for calculating variance, the standard deviation and**coefficient of variation • 1. Calculate the mean of a variable • 2. Find the deviations from the mean: subtract the variable mean from each case • 3. Square each of the deviations of the mean • 4. The variance is the mean of the squared deviations from the mean, so sum the squared deviations from step 3 and divide by the number of cases • 5. The standard deviation is the square root of the variance, so take the square root of the result of step 4. • 6. The coefficient of variation is the standard deviation divided by the mean, so take the result of step five and divide by the result of step 1.**Calculating Variance**• 1. Calculate the mean of a variable • 2. Find the deviations from the mean: subtract the variable mean from each case**Calculating Variance, cont.**• 3. Square each of the deviations of the mean • 4. The variance is the mean of the squared deviations from the mean, so sum the squared deviations from step 3 and divide by the number of cases • The Sum of the squared deviations = 198.950 • Variance = 198.950/20 = 9.948**Calculating the Standard Deviation and the Coefficient of**Variation • Standard Deviation = Square root of the Variance, so (SQR)9.948 = 3.2 • Coefficient of Variation = Standard Deviation/Mean, so 3.2/7.45 = .43**Shape of the Distribution**• Skewness. A measure of the symmetry of a distribution about its mean. If skewness is significantly nonzero, the distribution is asymmetric. A significant positive value indicates a long right tail; a negative value, a long left tail. • Kurtosis: A value of kurtosis significantly greater than 0 indicates that the variable has longer tails than those for a normal distribution; less than 0 indicates that the distribution is flatter than a normal distribution.**Normal Curve**• A bell shaped frequency curve defined by 2 parameters: the mean and the standard deviation. • For more information see: http://www.psychstat.smsu.edu/introbook/sbk11m.htm**Actual and Theoretical Distributions: Assessing the**“normality” of a distribution**Properties of the Normal Curve**• The normal curve has a special quality that gives tangible meaning to the standard deviation. In a normal distribution: • 68.26% of cases will have values within one standard deviation below or above the mean. • About 95.46% of cases will have values within two standard deviations below or above the mean. • And about 99.74% of cases will have values within three standard deviations below or above the mean.**Z Score**• Converts the values of a variable with its standard score (z score). Subtract the variable’s mean from each value and then divide the difference by the standard deviation. The standardized values have a mean of 0 and a standard deviation of 1. • Z score = (x – μ)/ sd

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