Modified by ARQ, from © 2002 Prentice-Hall.Chap 3-1 Numerical Descriptive Measures Chapter 2 ...

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Modified by ARQ, from © 2002 Prentice-Hall. Chap 3-1 Numerical Descriptive Measures Chapter 2 http://www2.uta.edu/infosys/amer/ courses/3321%20ppts/c3.ppt

Transcript of Modified by ARQ, from © 2002 Prentice-Hall.Chap 3-1 Numerical Descriptive Measures Chapter 2 ...

Modified by ARQ, from © 2002 Prentice-Hall. Chap 3-1

Numerical Descriptive Measures

Chapter 2http://www2.uta.edu/infosys/amer/

courses/3321%20ppts/c3.ppt

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

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

Modified: 2002 Prentice-Hall, Inc. Chap 3-4

Measures of Central Tendency

Central Tendency

Average (Mean) Median Mode

1

1

n

ii

N

ii

XX

n

X

N

Modified: 2002 Prentice-Hall, Inc. Chap 3-5

Mean (Arithmetic Mean) Mean (arithmetic mean) of data values

Sample mean

Population mean

1 1 2

n

ii n

XX X X

Xn n

1 1 2

N

ii N

XX X X

N N

Sample Size

Population Size

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 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Mean = 5 Mean = 6

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 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Median = 5 Median = 5

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

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% 25% 25% 25%

1Q 2Q 3Q

Data in Ordered Array: 11 12 13 16 16 17 18 21 22

1 1

1 9 1 12 13Position of 2.5 12.5

4 2Q Q

1Q 3Q

2Q

1

4i

i nQ

Modified: 2002 Prentice-Hall, Inc. Chap 3-10

Measures of Variation

Variation

Variance Standard Deviation Coefficient of Variation

PopulationVariance (σ2)

Sample

Variance (S2)

Population Standard deviation (σ)

Sample Standard deviation (S)

Range

Inter-quartile Range

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

Largest SmallestRange X X

7 8 9 10 11 12

Range = 12 - 7 = 5

7 8 9 10 11 12

Range = 12 - 7 = 5

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

3 1Interquartile Range 17.5 12.5 5Q Q

Interquartile Range

Data in Ordered Array: 11 12 13 16 16 17 17 18 21

Modified: 2002 Prentice-Hall, Inc. Chap 3-13

2

2 1

N

ii

X

N

Important measure of variation Shows variation about the mean

Sample variance:

Population variance:

2

2 1

1

n

ii

X XS

n

Variance

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:

2

1

1

n

ii

X XS

n

2

1

N

ii

X

N

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

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

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

100%S

CVX

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:

$5100% 100% 10%

$50

SCV

X

$5100% 100% 5%

$100

SCV

X

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-Skewed Symmetric

Modified: 2002 Prentice-Hall, Inc. Chap 3-19

Coefficient of Correlation

Measures the strength of the linear relationship between two quantitative variables

1

2 2

1 1

n

i ii

n n

i ii i

X X Y Yr

X X Y Y

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

Modified: 2002 Prentice-Hall, Inc. Chap 3-21

Scatter Plots of Data with Various Correlation

CoefficientsY

X

Y

X

Y

X

Y

X

Y

X

r = -1 r = -.6 r = 0

r = .6 r = 1

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

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

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.