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Describing Data

Describing Data. Descriptive Statistics: Central Tendency and Variation. Lecture Objectives. You should be able to: Compute and interpret appropriate measures of centrality and variation . Recognize distributions of data .

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Describing Data

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  1. Describing Data Descriptive Statistics: Central Tendency and Variation

  2. Lecture Objectives • You should be able to: • Compute and interpret appropriate measures of centrality and variation. • Recognize distributions of data. • Apply properties of normally distributed data based on the mean and variance. • Compute and interpret covariance and correlation.

  3. Summary Measures 1. Measures of Central Location Mean, Median, Mode 2. Measures of Variation Range, Percentile, Variance, Standard Deviation 3. Measures of Association Covariance, Correlation

  4. Measures of Central Location:The Arithmetic Mean It is the Arithmetic Average of data values: The Most Common Measure of Central Tendency Affected by Extreme Values (Outliers) Sample Mean 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14 Mean = 5 Mean = 6

  5. Median • Important Measure of Central Tendency • In an ordered array, the median is the “middle” number. • If n is odd, the median is the middle number. • If n is even, the median is the average of the 2 middle • numbers. • Not Affected by Extreme Values 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14 Median = 5 Median = 5

  6. Mode A Measure of Central Tendency Value thatOccurs Most Often Not Affected by Extreme Values There May Not be a Mode There May be Several Modes Used for Either Numerical or Categorical Data 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0 1 2 3 4 5 6 No Mode Mode = 9

  7. Measures of Variability Range The simplest measure Percentile Used with Median Variance/Standard Deviation Used with the Mean

  8. 7 8 9 10 11 12 7 8 9 10 11 12 Range Range = 12 - 7 = 5 Difference Between Largest & Smallest Observations: Range = Ignores How Data Are Distributed: Range = 12 - 7 = 5

  9. Percentile 2008 Olympic Medal Tally for top 55 nations. What is the percentile score for a country with 9 medals? What is the 50th percentile?

  10. Percentile - solutions Order all data (ascending or descending). • Country with 9 medals ranks 24th out of 55. There are 31 nations (56.36%) below it and 23 nations (41.82%) above it. Hence it can be considered a 57th or 58th percentile score. • The medal tally that corresponds to a 50th percentile is the one in the middle of the group, or the 28th country, with 7 medals. Hence the 50th percentile (Median) is 7. Now compute the first and third quartile values.

  11. Smallest Largest Q1 Q3 Median Box Plot The box plot shows 5 points, as follows:

  12. 20 60 80 105 Outlier 40 50 Outliers Interquartile Range (IQR) = [Q3 – Q1] = 60-40 = 20 1 Step = [1.5 * IQR] = 1.5*20 = 30 Q1 – 30 = 40 - 30 = 10 Q3 + 30 = 60 + 30 = 90 Any point outside the limits (10, 90) is considered an outlier.

  13. Variance For the Population: For the Sample: Variance is in squared units, and can be difficult to interpret. For instance, if data are in dollars, variance is in “squared dollars”.

  14. Standard Deviation For the Population: For the Sample: Standard deviation is the square root of the variance.

  15. Computing Standard Deviation

  16. The Normal Distribution A property of normally distributed data is as follows:

  17. Data A 11 12 13 14 15 16 17 18 19 20 21 Data B 11 12 13 14 15 16 17 18 19 20 21 Data C Comparing Standard Deviations Mean = 15.5 s = 3.338 Mean = 15.5 s = .9258 Mean = 15.5 s = 4.57 11 12 13 14 15 16 17 18 19 20 21

  18. Outliers Typically, a number beyond a certain number of standard deviations is considered an outlier. In many cases, a number beyond 3 standard deviations (about 0.25% chance of occurring) is considered an outlier. If identifying an outlier is more critical, one can make the rule more stringent, and consider 2 standard deviations as the limit.

  19. Coefficient of Variation Standard deviation relative to the mean. Helps compare deviations for samples with different means

  20. Computing CV Stock A: Average Price last year = $50 Standard Deviation = $5 Stock B: Average Price last year = $100 Standard Deviation = $5 Coefficient of Variation: Stock A: CV = 10% Stock B: CV = 5%

  21. Standardizing Data Which of the two numbers for person 8 is farther from the mean? The age of 75 or the income of 200,000? Z scores tell us the distance from the mean, measured in standard deviations

  22. Measures of Association Covariance and Correlation Covariance measures the average product of the deviations of two variables from their means. Correlation is the standardized form of covariance (divided by the product of their standard deviations). Correlation is always between -1 and +1.

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