# Inhalt - 1 DATA Variables Data Set Elements 1.1 Scales of...

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DATA Qualitative Numeric Nominal Ordinal Nonnumeric Nominal Ordinal Quantitative Numeric Interval Ratio 1 - DATA 1.1. Scales of Measurement Nominal - Data are labels or names used to identify attributes of the element - Numeric or nonnumeric The university course of 100 students (Business, Science, Law…) Ordinal - The same as nominal, but: ranking is meaningful - Numeric or nonnumeric The course performance of 100 students (Sehr gut, Gut, Befriedigend…) Interval - The same as ordinal, but: interval (difference) between observations is expressed in a unit - Always numeric Eva’s GMAT: 605, Adam’s GMAT: 490. Eva scored 115 points more than Adam Ratio - The same as interval, but: ratio between values is important - Scale must contain a 0 value Ernie’s GMAT: 36, Bert’s GMAT: 72. Ernie has twice as many credits as Bert 1.2. Qualitative and Quantitative Data Qualitative = “categorical” = labels/names = Numeric or nonnumeric Quntitative = Always numeric Discrete Continuous Elements Variables Data Set
1.3. Cross-Sectional and Time Series Cross-sectional data: Collected at about the same time Time series: Collected over several time periods
2 SUMMARIZING DATA 2.1. Frequency Distributions Classes Frequency Cumulated Frequency Relative Frequency % Frequency Cumulated Relative Frequency 0 - 4 4 6 4/20 = 0,20 20% 0,20 5 - 9 8 13 8/20 = 0,40 40% 0,60 10 - 14 5 18 5/20 = 0,25 25% 0,85 15 - 19 2 20 2/20 = 0,1 10% 0,95 20 - 24 1 1/20 = 0,05 5% 1 Σ 20 1 100% 2.2. Histograms Moderately skewed left Moderately skewed right Symmetric Highly skewed right 2.3. Ogive, Cumulative (Relative) Frequency Curve
3 NUMERICAL MEASURES 3.1. Measures of Location Mean The mean of a data set is the average of all its values Population: Sample: Example School classes: = n = n = 46 54 42 46 32 5 = 44 Mode The mode is the value that occurs with the highest frequency Median The median of a data set is the value in the middle when data is arranged in ascending order Example: 12 14 18 19 26 27 27 Median = 19 12 14 18 19 26 27 27 30 Median = = 22,5 Percentiles and Quartiles Quartiles are specific percentiles Q 1 : First Quartile = 25 th percentile Q 2 : Second Quartile = 50 th percentile = the median Q 3 : Third Quartile = 75 th percentile = ( 100 ) ? = ? 𝑖 ? ? = ? 𝑖 ?
Example: Consider a sample with data values of 27, 25, 20, 15, 30, 34, 28 and 25. Compute the 20th, 25th, 65th and 75th percentiles. Step 1: Arrange data in ascending order 15, 20, 25, 25, 27, 28, 30, 34 Step 2: i = ( ) = 1,6 2nd position = 20 i = ( ) = 2 = 22,5 i = ( ) = 5,2 6th position = 28 i = ( ) = 6 = 29 3.2. Measures of Variability Interquartile Range The interquartile range is the difference between the 1. and the 3. quartile The middle 50% = Q 3 Q 1 Example Salary: The monthly starting salaries for a sample of 12 business school graduates are repeated here in ascending order. 2710 2755 2850 2880 2880 2890 2920 2940 2950 3050 3130 3325 The quartiles are Q 3 = 3000 and Q 1 = 2865. Thus, the interquartile range is 3000 2865 = 135 Variance The variance is the measure of variability that uses all the data Population: Sample: Example School Classes: s = ∑( ) n 1 = 256 4 = 64 Standard Deviation The standard deviation is the average distance from the mean Population: = √ Sample: = √ Example School Classes: Recall that the sample variance for the sample of class sizes is s 2 = 64 Thus, the sample standard deviation s is √64 = 8 𝜎 = ∑( ? 𝑖 ? ) N ? = ∑( ? 𝑖 ? ) n 1
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