1

Measures Central Tendency

Last lecture we covered:

Measures

Central

Tendency

MODE MEDIANMEAN

OBJECTIVES

1. Which is the preferred measure? Mean or

Median

2. Shape

3

MfPitiPtil

3

.

M

easure o

f

P

os

iti

on <

P

ercen

til

e>

4. Measures of variation

How to compute mean?

Data

Raw Data

Example 3.1

(page 133)

Grouped Data

In the form of

Frequency

Distribution

Example 3.7

(page 136)

Weights

Examples 3.2, 3.3, 3.4,

3.5

(pages 133, 134,135 )

Total

Example 3.6

(page 135)

Median Salary by City - Job: Family Physician /

Doctor (Canada)

Which is the preferred

measure? Mean or Median

Page 139

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2

Illustration 1

The following data represents the prices of houses sold last week

in a particular neighbourhood. The data is:

$240,000 245,000 250,000 260,000 265,000

Mean house price = $252,000

You would get the impression as to the size and type of the typical

hithihbhd

h

ouse

i

n

th

e ne

i

g

hb

our

h

oo

d

.

Median house price = $250,000

You would get the same impression as to the size and type of the

typical house in the neighbourhood.

If the mean and median are CLOSE,

then the mean will give the correct impression and is the

preferred measure.

Illustration 2

The following data represents the prices of houses sold

last week in a particular neighbourhood. The data is:

$240,000 245,000 250,000 260,000 465,000

Mean house price = $292,000

You would

g

et the im

p

ression as to the size and t

yp

e of

gp yp

the typical house in the neighbourhood.

Median house price = $250,000

You would get the different impression as to the size and

type of the typical house in the neighbourhood.

If the mean and median are NOT CLOSE,

then the median will give the correct impression

Note: The median didnâ€™t change

Two issues

1. Median is less sensitive to extreme

scores

2. How do we conclude that the mean

and median are close or not close?

Median and Mean

â€¢The median is the middle of a distribution:

half the scores are above the median and

half are below the median.

â€¢The median is less sensitiveto extreme

scores than the mean and this makes it a

better measure than the mean for highly

skewed distributions. The median income

is usually more informative than the mean

income, for example

How do we conclude that the

mean and median are close or

not close?

not

close?

We need some rules.

Heuristic Rule

How do we conclude that the

mean and median are close or

not close?

not

close?

10% Rule

The "Ten Percent Rule" is a

general rule of thumb

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3

What is 10% Rule?page 140

Which is a better measure of tendency, mean or median?

To help you determine if the mean and median are really different, use the

following rule:

1: Calculate the â€˜difference' between the mean and median.

Difference = ~mean- median~

2: Calculate â€˜10% of the smaller' of the mean or median.

10% smaller = Minimum (10% umean , 10% umedian)

3: Compare â€˜Differenceâ€™ with â€˜10% smallerâ€™

The following decision rule is:

If Difference is less than 10% of smaller, you conclude that mean is

approximately equal to median, in which case the mean is the preferred

measure.

If Difference is greater than 10% of smaller, you conclude that mean is

NOTequal to median, in which case the median is the preferred measure.

Example 1

Consider the following:

Mean = 12 and Median = 14

Apply the 10% rule, you have

1: Difference = ~Mean-Median~= ~12 - 14~=2

2: 10% smaller = Minimum ( 10% u12 , 10%

u

14)

= Minimum ( 1.2 , 1.4) = 1.2

3: Since Difference of 2 is greater than 10%

smaller of 1.2, which indicate that mean is NOT

equal to median, in which case the median is the

preferred measure.

Illustration

Mean house price = $252,000

Median house price = $250,000

Difference = Mean â€“Median

Difference= 252,000-250,000

= 2,000

10% smaller

=

10%(250 000)

Mean house price = $292,000

Median house price = $250,000

Difference = Mean â€“Median

Difference= 292,000-250,000

= 42,000

10% smaller

=

10%(250 000)

10%

smaller

10%(250

,

000)

= 25,000

Conclusion: 2,000 < 25,000

If Difference < 10% of

smaller ; mean is equal to

median; mean is the

preferred measure

10%

smaller

10%(250

,

000)

= 25,000

Conclusion: 42,000 > 25,000

If Difference >10% of

smaller ; mean is NOT

equal to median; median

is the preferred measure

3.3 Descriptive Statistics III

Shape

Page 160

Shape

â€¢Describe a set of numerical data

â€¢Pattern of the distribution of data values

throughout the entire range of all the

values

values

â€¢3 shapes:

â€“(1) Symmetrical

â€“(2) Left-skewed (Negative Skewed)

â€“(3) Right-skewed (Positive Skewed)

Left-Skewed

Mostof the data are in the upper

Median

XX

Median

Right-Skewed

Most

of

the

data

are

in

the

upper

portion of the distribution

Mean < Median

Most of the data are in the

lower portion of the

distribution

A long tail and distortion to the left

that is caused by some extremely

small values.

These extremely small values pull

the mean downward so that

A long tail and distortion

to the right that is caused by

some extremely large

values.

These extremely large values pull

the mean downward so that

Mean > Median

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