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Descriptive Statistics

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- 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)

A frequency distribution tabulates all the values of a variable. The table usually provides frequency counts, percentages, cumulative counts, and cumulative percentages.

- 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

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

- 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

Sum the cases = 149

Divide by number of cases, 20

149/20 = 7.45

Example: Calculating a Median

20 cases for the variable PERSONS

- Identify the variable
- Sort the values of the variable
- Find the case that is at the half way point or the 50th percentile.

- Median = 7.5
- (with even number of cases, average the 2 middle cases)

- 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

- 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

- 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.

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

- 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.

- Mean
- Variance
- Standard Deviation
- 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.

- 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
- The Sum of the squared deviations = 198.950
- Variance = 198.950/20 = 9.948

- 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

- 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.

- 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

- 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.

- 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