Chapter 3

Transcript Chapter 3

Chapter 3 Descriptive Statistics Slide 3-1

Learning Objectives • Distinguish between measures of central tendency, measures of variability, and measures of shape • Understand the meanings of mean, median, mode, quartile, percentile, and range • Compute mean, median, mode, percentile, quartile, range, variance, standard deviation, and mean absolute deviation © 2002 Thomson / South-Western

Slide 3-2

Learning Objectives -- Continued • Differentiate between sample and population variance and standard deviation • Understand the meaning of standard deviation as it is applied by using the empirical rule • Understand box and whisker plots, skewness, and kurtosis © 2002 Thomson / South-Western

Slide 3-3

Measures of Central Tendency • Measures of central tendency yield information about “particular places or locations in a group of numbers.” • Common Measures of Location –Mode –Median –Mean –Percentiles –Quartiles © 2002 Thomson / South-Western

Slide 3-4

Mode • The most frequently occurring value in a data set • Applicable to all levels of data measurement (nominal, ordinal, interval, and ratio) • Bimodal -- Data sets that have two modes • Multimodal -- Data sets that contain more than two modes © 2002 Thomson / South-Western

Slide 3-5

Mode -- Example • The mode is 44.

• There are more 44s than any other value.

35 37 41 44 45 41 44 46

37 39 40 40 43 44 46 43 44 46 43 44 46 43 45 48 Slide 3-6

Median ( ΔΙΑΜΕΣΟΣ) • Middle value in an ordered array of numbers.

• Applicable for ordinal, interval, and ratio data • Not applicable for nominal data • Unaffected by extremely large and extremely small values.

Slide 3-7

Median: Computational Procedure • First Procedure – Arrange observations in an ordered array.

– If number of terms is odd, the median is the middle term of the ordered array.

– If number of terms is even, the median is the average of the middle two terms.

• Second Procedure – The median’s position in an ordered array is given by (n+1)/2.

Slide 3-8

Median: Example with an Odd Number of Terms Ordered Array includes: 3 4 5 7 8 9 11 14 15 16 16 17 19 19 20 21 22 • There are 17 terms in the ordered array.

• Position of median = (n+1)/2 = (17+1)/2 = 9 • The median is the 9th term, 15.

• If the 22 is replaced by 100, the median remains at 15.

• If the 3 is replaced by -103, the median remains at 15.

Slide 3-9

Mean (ΜΕΣΟΣ) • Is the average of a group of numbers • Applicable for interval and ratio data, not applicable for nominal or ordinal data • Affected by each value in the data set, including extreme values • Computed by summing all values in the data set and dividing the sum by the number of values in the data set © 2002 Thomson / South-Western

Slide 3-10

Population Mean   

X N

   93 5  5  

X

Slide 3-11

Sample Mean

 

n X

  

X X X

3    6  379 6 

X

Slide 3-12

Quartiles Measures of central tendency that divide a group of data into four subgroups • Q • Q • Q 1 2 3 : 25% of the data set is below the first quartile : 50% of the data set is below the second quartile : 75% of the data set is below the third quartile © 2002 Thomson / South-Western

Slide 3-13

Quartiles,

continued

• Q 1 is equal to the 25th percentile • Q 2 is located at 50th percentile and equals the median • Q 3 is equal to the 75th percentile Quartile values are not necessarily members of the data set © 2002 Thomson / South-Western

Slide 3-14

Slide 3-15

Quartiles: Example • Ordered array: 106, 109, 114, 116, 121, 122, 125, 129 • Q • Q • Q 1 2 3 : : :

 25 100 ( )

 50 100 ( )

 75 100 ( )

1 

2 

Slide 3-16

Measures of Variability • Measures of variability describe the spread or the dispersion of a set of data.

Slide 3-17

Variability

No Variability in Cash Flow Variability in Cash Flow

Slide 3-18

Variability

No Variability Slide 3-19

Range • The difference between the largest and the smallest values in a set of data • Simple to compute • Ignores all data points

35 37 41 41 44 44 45 46

except the

37 43 44 46

two extremes • Example: Range Largest - Smallest

39 40 43 43 44 44

46 46

= 48 - 35 = 13

40 43 45 48

Slide 3-20

Interquartile Range • Range of values between the first and third quartiles • Range of the “middle half” • Less influenced by extremes 3 

Q

Slide 3-21

Deviation from the Mean • Data set: 5, 9, 16, 17, 18 • Mean:   

N X

 65 5  13 • Deviations from the mean: -8, -4, 3, 4, 5

+5 +3 +4

-8 -4

0 5 10



15 20

Slide 3-22

Mean Absolute Deviation • Average of the absolute deviations from the mean

5 9 16 17 18

 

-8 -4 +3 +4 +5 0

 

+8 +4 +3 +4 +5 24

   24 5 

X N

Slide 3-23

Population Variance • Average of the squared deviations from the arithmetic mean

X

  

   2

5 9 16 17 18 -8 -4 +3 +4 +5 0 64 16 9 16 25 130

 2     1 3 0 5 

X N

Slide 3-24

Population Standard Deviation • Square root of the variance

X

5 9 16 17 18

  

   2

-8 -4 +3 +4 +5 0

64 16 9 16 25 130

 2         

X N

   2 1 3 0 5  2

Slide 3-25

Empirical Rule • Data are normally distributed (or approximately normal) Distance from the Mean     1 2   3    © 2002 Thomson / South-Western Percentage of Values Falling Within Distance 68 95 99.7

Slide 3-26

Sample Variance • Average of the squared deviations from the arithmetic mean

2,398 1,844 1,539 1,311 7,092



625 71 -234 -462 0





 2

390,625 5,041 54,756 213,444 663,866

2   

X n

  1

Slide 3-27

Sample Standard Deviation • Square root of the sample variance

X X







 2

2   

X n

  1

 2 

2,398 1,844 1,539 1,311 7,092 625 71 -234 -462 0 390,625 5,041 54,756 213,444 663,866



  

Slide 3-28

Coefficient of Variation • Ratio of the standard deviation to the mean, expressed as a percentage • Measurement of relative dispersion

C V

Slide 3-29

Coefficient of Variation   1 1   29

C V

2  84   10   2 2  100   10 84  100  

Slide 3-30

Measures of Shape • • •

Skewness

– Absence of symmetry – Extreme values in one side of a distribution

Kurtosis

– Peakedness of a distribution

Box and Whisker Plots

Slide 3-31

Skewness

Negatively Skewed Symmetric (Not Skewed) Positively Skewed

Slide 3-32

Skewness

Mean Median Negatively Skewed Mode Mean Median Mode Symmetric (Not Skewed) Mode Median Positively Skewed Mean

Slide 3-33

Coefficient of Skewness • Summary measure for skewness

 3    

M d

 • If S < 0, the distribution is negatively skewed (skewed to the left).

• If S = 0, the distribution is symmetric (not skewed).

• If S > 0, the distribution is positively skewed (skewed to the right).

Slide 3-34

Coefficient of Skewness  1  23

M d

 1 1   26

1     3   1  1

M d

 1   2  26

M d

 2 2   26

2     3   2  2 

M d

 2  3  29

M d

 3 3   26

3     3   3  3

M d

 3    .

Slide 3-35

Kurtosis • Peakedness of a distribution – Leptokurtic: high and thin – Mesokurtic: normal in shape – Platykurtic: flat and spread out

Leptokurtic Mesokurtic Platykurtic

Slide 3-36

Box and Whisker Plot •

Five specific values are used:

–

Median, Q 2

–

First quartile, Q 1

–

Third quartile, Q 3

–

Minimum value in the data set

–

Maximum value in the data set

Slide 3-37

Box and Whisker Plot,

continued

•

Inner Fences

–

IQR = Q 3 - Q 1

–

Lower inner fence = Q 1

–

Upper inner fence = Q 3 - 1.5 IQR + 1.5 IQR

•

Outer Fences

–

Lower outer fence = Q 1

–

Upper outer fence = Q 3 - 3.0 IQR + 3.0 IQR

Slide 3-38

Box and Whisker Plot Minimum

Q 1 Q 2 Q 3

Slide 3-39

Skewness: Box and Whisker Plots, and Coefficient of Skewness

S < 0 S = 0 S > 0 Negatively Skewed

Symmetric (Not Skewed) Positively Skewed Slide 3-40

Chapter 3

Transcript Chapter 3

X

X X X

X

Q

X

X

Directory