# Descriptive Statistics - PowerPoint PPT Presentation

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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.

### 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