Summary Measures Variation Central Tendency Mean Median Mode

Mean (Arithmetic Mean) (continued) n n The most common measure of central tendency Affected

Median n n Robust measure of central tendency Not affected by extreme values 0

Mode n n n A measure of central tendency Value that occurs most often

Shape of a Distribution n Describes how data is distributed n Measures of shape

Measures of Variation Variance Range Population Variance Sample Variance Standard Deviation Population Standard Deviation

Range n Measure of variation Difference between the largest and the smallest observations: n

Comparing Standard Deviations Data A 11 12 13 14 15 16 17 18 19

Features of Correlation Coefficient n Unit free n Ranges between – 1 and 1

Scatter Plots of Data with Various Correlation Coefficients Y Y r = -1 X

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Summary Measures Variation Central Tendency Mean Median Mode Range Variance Coefficient of Variation Standard Deviation

Mean (Arithmetic Mean) (continued) n n The most common measure of central tendency Affected by extreme values (outliers) 0 1 2 3 4 5 6 7 8 9 10 Mean = 5 0 1 2 3 4 5 6 7 8 9 10 12 14 Mean = 6

Median n n Robust measure of central tendency Not affected by extreme values 0 1 2 3 4 5 6 7 8 9 10 Median = 5 n 0 1 2 3 4 5 6 7 8 9 10 12 14 Median = 5 In an ordered array, the median is the “middle” number n n If n or N is odd, the median is the middle number If n or N is even, the median is the average of the two middle numbers

Mode n n n A measure of central tendency Value that occurs most often Not affected by extreme values Used for either numerical or categorical data There may be no mode There may be several modes 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Mode = 9 0 1 2 3 4 5 6 No Mode

Shape of a Distribution n Describes how data is distributed n Measures of shape n Symmetric or skewed Left-Skewed Symmetric Mean < Median < Mode Mean = Median =Mode Right-Skewed Mode < Median < Mean

Measures of Variation Variance Range Population Variance Sample Variance Standard Deviation Population Standard Deviation Sample Standard Deviation Coefficient of Variation

Range n Measure of variation Difference between the largest and the smallest observations: n Ignores the way in which data are distributed n Range = 12 - 7 = 5 7 8 9 10 11 12

Comparing Standard Deviations 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 Mean = 15. 5 s =. 9258 20 21 Mean = 15. 5 s = 4. 57 Data C 11 12 13 14 15 16 17 18 19 Mean = 15. 5 s = 3. 338

Features of Correlation Coefficient n Unit free n Ranges between – 1 and 1 n n n The closer to – 1, the stronger the negative linear relationship The closer to 1, the stronger the positive linear relationship The closer to 0, the weaker any positive linear relationship

Scatter Plots of Data with Various Correlation Coefficients Y Y r = -1 X Y r = -. 6 Y X r=0 Y r =. 6 X r=1 X X