1. Modified by ARQ, from © 2002 Prentice-Hall.Chap 3-1 Numerical Descriptive Measures Chapter 2 http://www2.uta.edu/infosys/amer/courses/3 321%20ppts/c3.ppt

2. Modified: 2002 Prentice-Hall, Inc. Chap 3-2 Chapter Topics Measures of central tendency Mean, median, mode, midrange Quartiles Measure of variation Range, interquartile range, variance and Standard deviation, coefficient of variation Shape Symmetric, skewed (+/-) Coefficient of correlation

3. Modified: 2002 Prentice-Hall, Inc. Chap 3-3 Summary Measures Central Tendency Arithmetic Mean Median Mode Quartile Geometric Mean Summary Measures Variation Variance Standard Deviation Coefficient of Variation Range

4. Modified: 2002 Prentice-Hall, Inc. Chap 3-4 Measures of Central Tendency Central Tendency Average (Mean)MedianMode

5. Modified: 2002 Prentice-Hall, Inc. Chap 3-5 Mean (Arithmetic Mean) Mean (arithmetic mean) of data values Sample mean Population mean Sample Size Population Size

6. Modified: 2002 Prentice-Hall, Inc. Chap 3-6 Mean (Arithmetic Mean) The most common measure of central tendency Affected by extreme values (outliers) (continued) 0 1 2 3 4 5 6 7 8 9 100 1 2 3 4 5 6 7 8 9 10 12 14 Mean = 5Mean = 6

7. Modified: 2002 Prentice-Hall, Inc. Chap 3-7 Median Robust measure of central tendency Not affected by extreme values In an Ordered array, median is the “middle” number If n or N is odd, median is the middle number If n or N is even, median is the average of the two middle numbers 0 1 2 3 4 5 6 7 8 9 100 1 2 3 4 5 6 7 8 9 10 12 14 Median = 5

8. Modified: 2002 Prentice-Hall, Inc. Chap 3-8 Mode A measure of central tendency Value that occurs most often Not affected by extreme values Used for either numerical or categorical data There may 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

9. Modified: 2002 Prentice-Hall, Inc. Chap 3-9 Quartiles Split Ordered Data into 4 Quarters Position of i-th Quartile and Are Measures of Noncentral Location = Median, A Measure of Central Tendency 25% Data in Ordered Array: 11 12 13 16 16 17 18 21 22

10. Modified: 2002 Prentice-Hall, Inc. Chap 3-10 Measures of Variation Variation VarianceStandard DeviationCoefficient of Variation Population Variance (σ 2 ) Sample Variance (S 2 ) Population Standard deviation (σ) Sample Standard deviation (S) Range Inter- quartile Range

11. Modified: 2002 Prentice-Hall, Inc. Chap 3-11 Range Measure of variation Difference between the largest and the smallest observations: Ignores the way in which data are distributed 7 8 9 10 11 12 Range = 12 - 7 = 5 7 8 9 10 11 12 Range = 12 - 7 = 5

12. Modified: 2002 Prentice-Hall, Inc. Chap 3-12 Measure of variation Also known as midspread Spread in the middle 50% Difference between the first and third quartiles Not affected by extreme values Interquartile Range Data in Ordered Array: 11 12 13 16 16 17 17 18 21

13. Modified: 2002 Prentice-Hall, Inc. Chap 3-13 Important measure of variation Shows variation about the mean Sample variance: Population variance: Variance

14. Modified: 2002 Prentice-Hall, Inc. Chap 3-14 Standard Deviation Most important measure of variation Shows variation about the mean Has the same units as the original data Sample standard deviation: Population standard deviation:

15. Modified: 2002 Prentice-Hall, Inc. Chap 3-15 Comparing Standard Deviations Mean = 15.5 S = 3.338 11 12 13 14 15 16 17 18 19 20 21 Data B Data A Mean = 15.5 S =.9258 11 12 13 14 15 16 17 18 19 20 21 Mean = 15.5 S = 4.57 Data C

16. Modified: 2002 Prentice-Hall, Inc. Chap 3-16 Coefficient of Variation Measures relative variation Always in percentage (%) Shows variation relative to mean Is used to compare two or more sets of data measured in different units

17. Modified: 2002 Prentice-Hall, Inc. Chap 3-17 Comparing Coefficient of Variation 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: Stock B:

18. Modified: 2002 Prentice-Hall, Inc. Chap 3-18 Shape of a Distribution Describes how data is distributed Measures of shape Symmetric or Skewed Mean = Median =Mode Mean < Median < Mode Mode < Median < Mean (+) Right-Skewed (-) Left-SkewedSymmetric

19. Modified: 2002 Prentice-Hall, Inc. Chap 3-19 Coefficient of Correlation Measures the strength of the linear relationship between two quantitative variables

20. Modified: 2002 Prentice-Hall, Inc. Chap 3-20 Features of Correlation Coefficient Unit free Ranges between –1 and 1 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

21. Modified: 2002 Prentice-Hall, Inc. Chap 3-21 Scatter Plots of Data with Various Correlation Coefficients Y X Y X Y X Y X Y X r = -1 r = -.6r = 0 r =.6 r = 1

22. Modified: 2002 Prentice-Hall, Inc. Chap 3-22 Chapter Summary Described measures of central tendency Mean, median, mode, midrange Discussed quartile Described measure of variation Range, interquartile range, variance and standard deviation, coefficient of variation Illustrated shape of distribution Symmetric, Skewed Discussed correlation coefficient

23. Modified: 2002 Prentice-Hall, Inc. Chap 3-23 Stem-n-leaf plot A class took a test. The students' got the following scores. 94 85 74 85 77 100 85 95 98 95 First let us draw the stem and leaf plot This makes it easy to see that the mode, the most common score is an 85 since there are three of those scores and all of the other scores have frequencies of either one or tow. It will also make it easier to rank the scores

24. Modified: 2002 Prentice-Hall, Inc. Chap 3-24 This will make it easier to find the median and the quartiles. There are as many scores above the median as below. Since there are ten scores, and ten is an even number, we can divide the scores into two equal groups with no scores left over. To find the median, we divide the scores up into the upper five and the lower five.