Central Tendency

(1)

Measures of Central Tendency

Measures of Location

Mean Mean Median Median Mode Mode Geometric Mean Geometric Mean

(2)

Measures of Central Tendency:

Summary

Central

Central TTendencyendency

Arithmetic Arithmetic

Mean Mean

Median

Median MMoodde e GGeeoommeettrriic c MMeeaann

n n X X X X n n ii ii       11 n n // 1 1 n n 2 2 1 1 G G ((XX XX XX )) X X   __ Middle value Middle value in the ordered in the ordered array array Most Most frequently frequently observed observed value value Rate of Rate of change of change of a variable a variable over time over time

(3)

Summary Definitions

 The measure of location or central tendency

is a central value that the data values group around. It gives an average value.

 The measure of dispersion shows how the

data is spread or scattered around the mean.

 The measure of skewness is how symmetrical

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Measures of Central Tendency:

The Mean

 The arithmetic mean (often just called “mean”)

is the most common measure of central tendency

 For a sample of size n:

Sample size n X X X n X X 1 2 n n 1 i i _ _ _  







Observed values The ith _value Pronounced x-bar

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Measures of Central Tendency:

The Mean

 The most common measure of central tendency

 Mean = sum of values divided by the number of values  Affected by extreme values (outliers)

(continued) 0 1 2 3 4 5 6 7 8 9 10 Mean = 3 0 1 2 3 4 5 6 7 8 9 10 Mean = 4 3 5 15 5 5 4 3 2 1       4 5 20 5 10 4 3 2 1      

(6)

Numerical Descriptive Measures

for a Population: The mean µ

 The population mean is the sum of the values in

the population divided by the population size, N

N X X X N X N 2 1 N 1 i i

_











μ = population mean N = population size

X_i = ith _{value of the variable X}

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Approximating the Mean from a

Frequency Distribution

 Use the midpoint of a class interval to approximate the values in that class

Where n = number of values or sample size

c = number of classes in the frequency distribution x _j = midpoint of the jth _class

f _j = number of values in the jth _class

n

c 1 j j j



 

f

x

X

(8)

Mean of Wealth

The Distribution of Marketable Wealth, UK, 2001

Wealth Boundaries Mid interval(000) Frequency(000)

Lower Upper x f fx 0 9999 5.0 3417 17085.0 10000 24999 17.5 1303 22802.5 25000 39999 32.5 1240 40300.0 40000 49999 45.0 714 32130.0 50000 59999 55.0 642 35310.0 60000 79999 70.0 1361 95270.0 80000 99999 90.0 1270 114300.0 100000 149999 125.0 2708 338500.0 150000 199999 175.0 1633 285775.0 200000 299999 250.0 1242 310500.0 300000 499999 400.0 870 348000.0 500000 999999 750.0 367 275250.0 1000000 1999999 1500.0 125 187500.0 2000000 4000000 3000.0 41 123000.0 Total 16933 2225722.5 Mean = 2225722.5 = 131.443 16933

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Measures of Central Tendency:

The Median

 In an ordered array, the median is the “middle”

number (50% above, 50% below)

 Not affected by extreme values

0 1 2 3 4 5 6 7 8 9 10

Median = 3

0 1 2 3 4 5 6 7 8 9 10

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Measures of Central Tendency:

Locating the Median

 The location of the median when the values are in numerical order

(smallest to largest):

 If the number of values is odd, the median is the middle number  If the number of values is even, the median is the average of the

two middle numbers

Note that is not the value of the median, only the position of the median in the ranked data

data ordered the in position 2 1 n position Median   2 1 n 

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Median of Wealth

Median is £76 907

There are 16933 people 16933 / 2 = 8466

Person 8466 is shown by a yellow line.

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Median of Wealth

Jan 12 17 10 Feb 17 11 21 Mar 22 29 14 Apr 14 10 17 May 12 17 10 Jun 19 15 20

Cumulative Frequency Wealth Distribution, UK,2001 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 0 50000 100000 150000 200000 250000 300000 Wealth C F (0 0 0 )

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Measures of Central Tendency:

The Mode

 Value that occurs most often  Not affected by extreme values

 Used for either numerical or categorical

(nominal) 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

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Measures of Central Tendency:

Review Example

House Prices: £2,000,000 £500,000 £300,000 £100,000 £100,000 Sum £3,000,000  Mean: (£3,000,000/5) = £600,000

 Median: middle value of ranked

data

= £300,000

 Mode: most frequent value

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Thinking Challenge

You’re a financial analyst for Prudential-Bache Securities. You have collected the

following closing stock prices of new stock issues: 17, 16, 21, 18, 13, 16, 12, 11.

Describe the stock prices in terms of central

(16)

Central Tendency Solution*

Mean X X n X X X i i n               



1 1 2 8

8

17

16 21 18

13

16

12

11

8 15 5

…

.

(17)

Central Tendency Solution*

Median

Remember to order the data first

 Ordered: 11 12 13 16 16 17 18 21  Position: 1 2 3 4 5 6 7 8

Positioning Point

Median

        n

1

2

8

1

2 4 5

16

2

16 .

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Central Tendency Solution*

Mode

Raw Data:

17

16 21 18 13

16 12 11

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Measures of Central Tendency:

Which Measure to Choose?

 The mean is generally used, unless extreme

values (outliers) exist.

 The median is often used, since the median is

not sensitive to extreme values. For example, median home prices may be reported for a

region; it is less sensitive to outliers.

 In some situations it makes sense to report both

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Measure of Central Tendency For The Rate Of Change Of A Variable Over Time:

The Geometric Mean & The Geometric Rate of Return

 Geometric mean

 Used to measure the rate of change of a variable over

time

 Geometric mean rate of return

 Measures the status of an investment over time

 Where R _i is the rate of return in time period i

n / 1 n 2 1 G

(

X

)

X

  

_

 1 )] R 1 ( ) R 1 ( ) R 1 [( RG   ₁   ₂ 

_

  _n 1/n 

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The Geometric Mean Rate of

Return: Example

An investment of $100,000 declined to $50,000 at the end of year one and rebounded to $100,000 at end of year two:

The overall two-year return is zero, since it started and ended at the same level.

000 , 100 $ X 000 , 50 $ X 000 , 100 $ X₁  ₂  ₃  50% decrease 100% increase

(22)

The Geometric Mean Rate of

Return: Example

Use the 1-year returns to compute the arithmetic mean and the geometric mean:

% 25 25 . 2 ) 1 ( ) 5 . (      X Arithmetic mean rate of return: Geometric mean rate of return: % 0 1 2 / 1 1 1 2 / 1 )] 2 ( ) 50 [(. 1 2 / 1 ))] 1 ( 1 ( )) 5 . ( 1 [( 1 / 1 )] 1 ( ) 2 1 ( ) 1 1 [(                     R R R_n n G R _ Misleading result More representative result (continued)

(23)

Pitfalls in Numerical

Descriptive Measures

 Data analysis is objective

 Should report the summary measures that best

describe and communicate the important aspects of the data set

 Data interpretation is subjective

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Ethical Considerations

Numerical descriptive measures:

 Should document both good and bad results  Should be presented in a fair, objective and

neutral manner

 Should not use inappropriate summary

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Measures of Central Tendency:

Summary

Central Tendency

Arithmetic Mean

Median Mode Geometric Mean

n X X n i i    1 n / 1 n 2 1 G (X X X ) X   _ Middle value in the ordered array Most frequently observed value Rate of change of a variable over time