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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 l.jpg
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 l.jpg
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 l.jpg
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 distribution for persons l.jpg
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 l.jpg
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 l.jpg

Example: Calculating a Mean

20 cases for the variable PERSONS


Steps for calculating a mean l.jpg
Steps for Calculating a Mean

Sum the cases = 149

Divide by number of cases, 20

149/20 = 7.45


Example calculating a median l.jpg

Example: Calculating a Median

20 cases for the variable PERSONS


Steps for calculating a median l.jpg
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 l.jpg
20 cases sorted and midpoints marked

  • Median = 7.5

  • (with even number of cases, average the 2 middle cases)


Steps for finding the mode l.jpg
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 l.jpg
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 l.jpg
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 distribution for persons17 l.jpg
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 l.jpg
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 l.jpg
Equations

  • Mean

  • Variance

  • Standard Deviation

  • Coefficient of Variation


Steps for calculating variance the standard deviation and coefficient of variation l.jpg
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 l.jpg
Calculating Variance coefficient of variation

  • 1. Calculate the mean of a variable

  • 2. Find the deviations from the mean: subtract the variable mean from each case


Calculating variance cont l.jpg
Calculating Variance, cont. coefficient of variation

  • 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 l.jpg
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 l.jpg
Shape of the Distribution Variation

  • 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 l.jpg
Normal Curve Variation

  • 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 l.jpg
Actual and Theoretical Distributions: Assessing the “normality” of a distribution


Assessing the normality of a distribution l.jpg
Assessing the Normality of a Distribution “normality” of a distribution


Properties of the normal curve l.jpg
Properties of the Normal Curve “normality” of a distribution

  • 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. 


Normal curve31 l.jpg
Normal Curve “normality” of a distribution


Z score l.jpg
Z Score “normality” of a distribution

  • 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