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### Example: Calculating a Mean

### Example: Calculating a Median

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.

20 cases for the variable PERSONS

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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