L2_Descriptives - Created by Dr Friedman Updated by Dr...

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Created by: Dr. Friedman Updated by: Dr. Fluture L2_descriptives p. 1 Descriptive Statistics For a Single Variable I. Numerical data A. Measures of Location 1. Measures of central tendency Mean; Median; Mode 2. Quantiles - measures of noncentral tendency Quartiles; Percentiles B. Measures of Dispersion Range; Interquartile range; Variance; Standard Deviation; Coefficient of Variation C. Measures of Shape Skewness 5-number summary Box-and-whisker Stem-and-leaf D. Standardizing Data II. Categorical data A. Frequencies (also useful for grouped numerical data) Frequency Distribution Percentage Distribution Cumulative Distribution Histogram Polygon Ogive B. Charts Bar chart Pie chart Pareto diagram
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A. Measures of Location 1. Measures of central tendency Mean; Median; Mode MEAN The sample mean is the sum of all the observations divided by the number of observations: X = X 1 + X 2 + X 3 + + X n n or n X i X = i = 1 where ∑X i is the same as X 1 + X 2 + X 3 + … + X n n Example: 1 2 2 4 5 10 X = 24 / 6 = 4.0 Example: 1 1 1 1 51 X = 55 / 5 = 11.0 Note that the mean is affected by extreme values. MEDIAN The median is the middle of the data (after data is arranged in ascending or descending order); half the observations are less than the median and half are more than the median. To get the median, we must first rearrange the data into an ordered array . Generally, we order the data from the lowest value to the highest value. The median is the data value such that half of the observations are larger than it and half are smaller. It is also the 50 th percentile (we will be learning about percentiles).
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If n is odd, the median is the middle observation of the ordered array. If n is even, it is midway between the two central observations. Example: 0 2 3 5 20 99 100 Median = 5; n=7 Since n is odd, the median is the (n+1)/2 ordered observation, or the 4 th observation. Example: 10 20 30 40 50 60 Median = 35 Example : Exam scores 0 0 0 0 100 What is the mean? Suppose the prof lets students know the following grading policy: Anyone who got the mean or better gets an A for the course; anyone who got below the mean fails. Note that the mean and median are UNIQUE for a given set of data. Advantage: the Median is not affected by extreme values. In the above example, if you change the 60 to 6,000, the median will still be 35. The mean, on the other hand will change by a great deal. Problem: Sometimes it is difficult to order data.
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MODE The m ode is the value of the data that occurs with the greatest frequency. EXAMPLE: 1 1 1 2 3 4 5 The mode is 1 since it occurs three times. The other values only appear once in the data set. EXAMPLE: 5 5 5 6 8 10 10 10 Mode = 5, 10 The modes for this data set are 5 and 10. This is a bi-modal dataset. Problems: The mode may not exist. The mode may not be unique. A. Measures of Location 2. Quantiles - measures of noncentral tendency Quartiles; Percentiles QUANTILES Measures of non-central location Quartiles Deciles Percentiles These are all commonly used quantiles.
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