# Numerical Descriptive Measures Chapter 2 http www 2

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

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

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

Measures of Central Tendency Average (Mean) Modified: 2002 Prentice-Hall, Inc. Median Mode 4

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

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 Modified: 2002 Prentice-Hall, Inc. 0 1 2 3 4 5 6 7 8 9 10 12 14 Mean = 6 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, median is the “middle” number n n 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 Modified: 2002 Prentice-Hall, Inc. 7

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 Modified: 2002 Prentice-Hall, Inc. 0 1 2 3 4 5 6 No Mode 8

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

Measures of Variation Variance Standard Deviation Range Interquartile Range Modified: 2002 Prentice-Hall, Inc. Population Variance (σ2) Sample Variance (S 2) Coefficient of Variation Population Standard deviation (σ) Sample Standard deviation (S) 10

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 Modified: 2002 Prentice-Hall, Inc. 10 11 12 7 8 9 10 11 12 11

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

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

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

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 Modified: 2002 Prentice-Hall, Inc. 13 14 15 16 17 18 19 Mean = 15. 5 S = 3. 338 15

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

Comparing Coefficient of Variation n Stock A: n n n Stock B: n n n Average price last year = \$50 Standard deviation = \$5 Average price last year = \$100 Standard deviation = \$5 Coefficient of variation: n Stock A: n Stock B: Modified: 2002 Prentice-Hall, Inc. 17

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 Modified: 2002 Prentice-Hall, Inc. (+) Right-Skewed Mode < Median < Mean 18

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

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 Modified: 2002 Prentice-Hall, Inc. 20

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

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

Stem-n-leaf plot n n n A class took a test. The students' got the following scores. 94 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 Modified: 2002 Prentice-Hall, Inc. 23

n This will make it easier to find the median and the quartiles. There 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. Modified: 2002 Prentice-Hall, Inc. 24