Central Tendency

Measures of Central Tendency Measures of Location Mean Median Mode Geometric Mean

Measures of Central Tendency: Summary Central Tendency

Arithmetic Mean

Median

Mo d e

n

X X

Geometric Mean

XG i



( X1  X2    Xn )1/ n

i 1

n

Middle value in the ordered array

Most frequently observed value

Rate of change of  a variable over time

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 (or not) the distribution of data values is.

Measures of Central Tendency: The Mean 

The arithmetic mean (often just called “mean”) is the most common measure of central tendency

Pronounced x-bar  

The ith value

For a sample of size n: n

X X

Sample size

i1

n

i



X1  X2    Xn n

Observed values

Measures of Central Tendency: The Mean (continued) 

The most common measure of central tendency



Mean = sum of values divided by the number of values



 Affected by extreme values (outliers)

0 1 2 3 4 5 6 7 8 9 10

Mean = 3 1 2  3  4  5 5

Mean = 4 15



0 1 2 3 4 5 6 7 8 9 10

5



3

1  2  3  4  10 5

20 

5



4

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  Where

i1

N

i



X1  X 2

   XN N

μ = population mean N = population size Xi = ith value of the variable X

 Approximating the Mean from a Frequency Distribution 

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

 x j f   j  X   Where

 j1

n

n = number of values or sample size

c = number of classes in the frequency distribution  x  j = midpoint of the j th class f  j = number of values in the j th class

Mean of Wealth The Distribution of Marketable Wealth, UK, 2001 Wealth Lower 0 10000 25000 40000 50000 60000 80000 100000 150000 200000 300000 500000 1000000 2000000

Boundaries Mid interval(000) Frequency(000) Upper   x f 9999 5.0 3417 24999 17.5 1303 39999 32.5 1240 49999 45.0 714 59999 55.0 642 79999 70.0 1361 99999 90.0 1270 149999 125.0 2708 199999 175.0 1633 299999 250.0 1242 499999 400.0 870 999999 750.0 367 1999999 1500.0 125 4000000 3000.0 41

Total

Mean

= 2225722.5 16933

fx

 

17085.0 22802.5 40300.0 32130.0 35310.0 95270.0 114300.0 338500.0 285775.0 310500.0 348000.0 275250.0 187500.0 123000.0

16933

2225722.5

=

131.443

Measures of Central Tendency: The Median



In an ordered array, the median is the “middle” number (50% above, 50% below)

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

Median = 3

Median = 3



Not affected by extreme values

Measures of Central Tendency: Locating the Median 

The location of the median when the values are in numerical order (smallest to largest):

Median position  



n 1 2

position in the ordered data

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 n  1 is not the value of the median, only the  position of 2

the median in the ranked data

Median of Wealth Median is £76 907

There are 16933 people 16933 / 2 = 8466 Person 8466 is shown by a yellow line. This person has £76 907 of marketable wealth.

Median of Wealth Jan Feb Mar  Apr May Jun

12 17 22 14 12 19

17

10

17 15

10 20

11 21 Cumulative Frequency Wealth Distribution, 29 14 10 17 UK,2001

18000 16000 14000 12000 0

)

10000 F

8000 C

0(

0

6000 4000 2000 0 0

50000

100000

150000

Wealth

200000

250000

300000

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

No Mode

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 = £100,000

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 tendency.

Central Tendency Solution* Mean n

  X 

i

 X 





i 1

n



 X 1

 X 2  …  X 8

8

17  16  21  18  13  16  12  11

 15 .5

8

Central Tendency Solution* Median Remember to order the data first 

Ordered:

11 12 13 16 16 17 18 21



Position:

1

2

3

4 n

Positioning Point Median 

16  16 2

1

2

 16

5



6

8 1 2

7

 4.5

8

Central Tendency Solution* Mode Raw Data:

17 16 21 18 13 16 12 11

Mode = 16

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 the mean and the median.

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

XG 

RG



1/ n

(X1  X 2   X n )

Geometric mean rate of return 

Measures the status of an investment over time



[(1  R1 )  (1  R 2 )    (1  Rn )]1/ n



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



1

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:

X1



$100,000

X2



50% decrease

$50,000

X3



$100,000

100% increase

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

The Geometric Mean Rate of Return: Example

(continued)

Use the 1-year returns to compute the arithmetic mean and the geometric mean: Arithmetic mean rate of return:

Geometric mean rate of return:

 X 

 R G





( .5)  (1) 2



Misleading result

.25  25%

[(1   R1 )  (1   R2 )    (1   Rn )]1 / n



[(1  (.5))  (1  (1))]1 / 2



[(.50)  (2)]1 / 2





1  11 / 2

1



1  0%



1

More representative result

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 

Should be done in fair, neutral and clear manner 

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 measures to distort facts

Measures of Central Tendency: Summary Central Tendency

Arithmetic Mean

Median

Mode

n

X X

Geometric Mean

XG i



( X1  X2    Xn )1/ n

i 1

n

Middle value in the ordered array

Most frequently observed value

Rate of change of  a variable over time