table 1. number of minutes 20 clients waited to …...chapter 6 measures of variability measures of...

Table 1. Number of Minutes 20 Clients Waited to See a Consultant

Consultant X Consultant Y

05 15 11 12

12 03 10 13

04 19 11 10

37 11 09 13

06 34 09 11

Something to think about…

•What can you say to the

time allotted by the clients

to wait for the Consultant

X? How about for the

Consultant Y?

Table 1. Number of Minutes 20 Clients Waited to See a Consultant

Consultant X Consultant Y

05 15 11 12

12 03 10 13

04 19 11 10

37 11 09 13

06 34 09 11

In Consultant X:

•Sees some clients

almost immediately

•Others wait over 30

minutes

In Consultant Y:

•Clients wait about 10

minutes

•9 minutes least wait

and 13 minutes most

Something to think about…

•What can you say to the

time the Consultant X let

the clients wait? How

about for the Consultant

Y?

In Consultant X:

•It is highly

inconsistent

In Consultant Y:

•It is highly

consistent

CHAPTER 6MEASURES OF VARIABILITY

Measures of Variability

•It is a single number that

describes how the data

are scattered or how

much they are bunched.

Going Back:

•In Consultant X, we can say

that the data are scattered.

While, in Consultant Y the

data are bunched.

Furthermore,

•Measures of

Dispersion

•Measures of Spread

Note:

•Bunched data/closely

grouped data will have

relatively small values of

Measures of Variability

Note:

•Scattered data/more widely

distributed data will have

relatively small values of

Measures of Variability

Measures of Variability

•It is an indicator of

consistency among

a set of data

Measures of Variability

•It indicates how close

data are clustered

about Measures of

Central Tendency

Measures of Variability

COMMONLY USED TYPES

Measures of Variability•Index of Qualitative Variation (IQV)

•The Range

•The Interquartile Range (IQR)

•The Standard Deviation (𝝈)

Additional:

•The Variance (𝝈𝟐)

•The Mean Deviation (MD)

Index of Qualitative

Variation (IQV)

MEASURES OF VARIABILITY

Index of Qualitative Variation (IQV)

•A measure of variability for nominal

variables.

•It is based on the ratio of the total

number of differences in the

distribution to the maximum number

of possible differences within the

same distribution.

Index of Qualitative Variation (IQV)

•Since the mode is the preferred

measure of central tendency for a

nominal variable, the measure of

dispersion for a nominal variable

would indicate the degree to which

cases fall in the non-modal

categories.

Formula for IQV:

•𝑰𝑸𝑽 = 𝒕𝒐𝒕𝒂𝒍 𝒐𝒃𝒔𝒆𝒓𝒗𝒆𝒅 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆𝒔

𝒎𝒂𝒙𝒊𝒎𝒖𝒎 𝒑𝒐𝒔𝒔𝒊𝒃𝒍𝒆 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆𝒔

•𝑰𝑸𝑽 = 𝒇𝒊𝒇𝒋

𝑲(𝑲−𝟏)

𝟐

𝑵

𝑲

𝟐

In the formula:

𝑲 = 𝒕𝒉𝒆 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒄𝒂𝒕𝒆𝒈𝒐𝒓𝒊𝒆𝒔

𝒊𝒏 𝒕𝒉𝒆 𝒗𝒂𝒓𝒊𝒂𝒃𝒍𝒆

𝑵 = 𝒕𝒉𝒆 𝒕𝒐𝒕𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓

𝒐𝒇 𝒄𝒂𝒔𝒆𝒔

Understanding the IQV:

•The IQV is a single

number that expresses

the diversity of a

distribution.

Understanding the IQV:

•The IQV ranges

from 0 to 1.

Understanding the IQV:

•An IQV of 0 would

indicate that the

distribution has NO

diversity at all.

Understanding the IQV:

•An IQV of 1 would

indicate that the

distribution is maximally

diverse.

Further Our Knowledge:

•Let’s have IQV in Real-Life.

The next example will show

the ethnical and racial

diversity in the US.

Something to think about…

•Based on the IQV of

Diversity in the US,

what can we therefore

conclude?

The Range

MEASURES OF VARIABILITY

The Range

•It is the least

complicated measure of

describing the dispersion

of a set of data.

The Range

•It is the distance given by

the highest observed value

minus the lowest observed

value in the distribution.

The Range

•It indicates how spread out

the data are

•It is dependent on two

extreme values

The Range

•Somewhat dependent on

the number of values in the

data set (i.e., more: the

larger the range)

Formula for Range:

•𝑹𝒂𝒏𝒈𝒆 = 𝑯𝑶𝑽 − 𝑳𝑶𝑽where:

𝑯𝑶𝑽 = 𝒉𝒊𝒈𝒉𝒆𝒔𝒕 𝒐𝒃𝒔𝒆𝒓𝒗𝒆𝒅 𝒗𝒂𝒍𝒖𝒆

𝑳𝑶𝑽 = 𝒍𝒐𝒘𝒆𝒔𝒕 𝒐𝒃𝒔𝒆𝒓𝒗𝒆𝒅 𝒗𝒂𝒍𝒖𝒆

Table 1. Number of Minutes 20 Clients Waited to See a Consultant

Consultant X Consultant Y

05 15 11 12

12 03 10 13

04 19 11 10

37 11 09 13

06 34 09 11

On the same example:

•The range of Consultant X is 34

minutes

•The range of Consultant Y is 4

minutes

“What can you conclude?”

Understanding the Range

•The range is the preferred

measure of variability for ordinal

level variables, and for interval

level variables that have a badly

skewed distribution.

Understanding the Range

•The range can be computed

for interval level variables, but

is not an appropriate statistic

for nominal or dichotomous

variables.

Understanding the Range

•The range is usually

described as the total

spread in the

distribution.

Understanding the Range

•The range is based only on two

scores in the distribution, the

highest and the lowest, and it tells

us nothing about the distribution

of the majority of scores in

between.

Understanding the Range

•The range is most useful when we

are comparing groups and can

describe one group as having a

larger or smaller range, or

spread, than the other groups.

The

Interquartile Range (IQR)

MEASURES OF VARIABILITY

Something to think about...

•Since many variables contain

one or more extremely large

or extremely small scores, the

range may be misleading.

Solution:

•That problem is

avoided with IQR.

Interquartile Range (IQR)

•It is the difference between the

3rd quartile and the 1st quartile.

The 3rd quartile is the value below

which 75% of the cases fall. The

1st quartile is the value below

which 25% of the cases fall.

Interquartile Range (IQR)

•While less subject to the influence

of extreme cases than the range,

the interquartile range still uses

information for only two cases or

values in the distribution.

Understanding the IQR

•It is the modified

version of the

range

Understanding the IQR

•It is the positional

measure of the

variability

Understanding the IQR

•It is the range of

the middle 50% of

scores or ranks.

Understanding the IQR

•It is not sensitive

to extreme values

in a data set.

Understanding the IQR

•It is not

sensitive to the

sample size.

Formula:

•𝑰𝑸𝑹 = 𝑸𝟐−𝑸𝟏

𝟐

•𝑰𝑸𝑹 = 𝑷𝟕𝟓−𝑷𝟐𝟓

𝟐

The

Standard

Deviation (𝛔)MEASURES OF VARIABILITY

The Standard Deviation•The standard deviation

measures the deviations

between the mean of the

distribution and each of

the individual scores.

The Standard Deviation•It is the preferred measure of

variability for interval level

variables, unless the distribution

is badly skewed. For badly skewed

distribution, the range is a

preferred measure of variability.

The Standard Deviation

•It is most frequently

used measure of

dispersion.

The Standard Deviation

•It is the average of the

distances of the observed

values from the mean

value for a set of data.

The Standard Deviation

•Here, the basic rule:

more spread will

yield a larger SD.

Formula:

•𝝈 = (𝑿−𝒙)

𝒏−𝟏

•𝝈 =𝒏 𝑿𝟐 − 𝑿 𝟐

𝒏(𝒏−𝟏)

Interpreting the Standard Deviation

•The standard deviation does not

have any inherent or intuitive

meaning; it is a statistical

measure of the variability of cases

around the mean for an interval

level variable.

Interpreting the Standard Deviation

•Standard deviation is commonly

presented in terms of the

proportion of cases that fall

between the mean plus/minus 1,

2, or 3 standard deviation

measures.

The Mean Deviation

(MD)MEASURES OF VARIABILITY

Mean Deviation

•It is the average

distance between the

mean and the scores

in the distribution.

Mean Deviation

•This technic provides

a reasonably stable

estimate variation.

Mean Deviation

•It is also called

Average Deviation.

Formula for MD:

•𝑴𝑫 = 𝑿−𝒙

𝒏

The Variance

(𝝈𝟐)MEASURES OF VARIABILITY

The Variance•The variance is another

measure of variability

that is equal to the

square the standard

deviation.

The Variance

•The variance is the

average of the squared

deviations from the

mean.

The Variance

•It is expected value of

the square of the

deviation from the

mean.

The Variance•In describing distributions, the

standard deviation is the more

commonly cited statistics. Variance is

used primarily in inferential statistics

such as the analysis of variance and

correlation, which you will study later

in Advanced Statistics.

Formula:

•𝝈𝟐 = (𝑿−𝒙)

𝒏−𝟏

•𝝈𝟐 =𝒏 𝑿𝟐 − 𝑿 𝟐

𝒏(𝒏−𝟏)