MEASURE OF CENTRAL TENDENCY (QUARTILES AND DECILES)

QUARTILES

The three values which divide the distribution into four equal parts are called quartiles. These values are denoted by Q1, Q2 and Q3 respectively. Ql is called lower quartile, Q2 is also called median and Q3 is called upper quartile.

Q1 = (n/4)th value of the data

Q2= (n/2)th value of the data

Q3= (3n / 4)^th value of the data

It is important to note that n is the total number of frequency. i.e. n=∑f

DECILE AND PERCENTILE

Deciles divide the data into ten equal parts:

D1 = (n / 10)th value of the data

D7= (7n / 10)^th value of the data

Percentiles divide the data into hundred equal parts.

P10 = (10n/100)

th value of the data

P₇₃= (73n /100)th value of the data

Note: For odd Number of observations we use n + 1 in the place of n, above mentioned

Formulas.

Example

Data Set: 6, 47, 49, 15, 42, 41, 7, 39, 43, 40, 36

For finding

Median i.e. Q2, Q1 and Q3 first we have to arrange set of data into ascending order.
Ordered Data Set: 6, 7, 15, 36, 39, 40, 41, 42, 43, 47, 49

Q1 = (n+1/4)th value of the data ( n+1 because of odd number we add 1)

Q1=11+1/4

Q1=3^rd value that is 15

Q1=15

Q2= (n+1/2)th value of the data

Q2=11+1/2

Q2=6^th value that is 40

Q2=40

Q3= (3n +1/ 4)^th

Q3=43

THE RANGE

It is the simplest way to measure of dispersion. The range is defined as the difference b the largest and the smallest observations in a set of data.

Range = largest value - smallest value

THE INTER QUARTILE RANGE

The inter-quartile range is a measure of dispersion; define by the difference between the third and the first quartiles.

Inter Quartile Range=Q3-Q1

SEMI INTER-QUARTILE RANGE

The half difference between the third and first quartiles is called semi inter-quartile range.

SEMI INTER-QUARTILE RANGE= (Q3-Q1)/2

From the above Example:

Data Set: 6, 47, 49, 15, 42, 41, 7, 39, 43, 40, 36

For finding

median i.e. Q2, Q1 and Q3 first we have to arrange set of data into ascending order.
Ordered Data Set: 6, 7, 15, 36, 39, 40, 41, 42, 43, 47, 49

Q1=15, Q2=40, Q3=43

Range =largest-smallest

Range=49-6

Range=43

Inter Quartile Range=Q3-Q1

=43-15

=28

SEMI INTER-QUARTILE RANGE= (Q3-Q1)/2

=28/2

=14

GRAPHICAL METHOD

http://www.onlinemathlearning.com/image-files/percentile_clip_image002.gif

upper quartile=3n/4

upper quartile=30^th value

upper quartile=52

In the same way we will find Q2, Q1 and so interquartile range.

MODE (MEASURE OF CENTRAL TENDENCY) STATISTICS FOR GCE O LEVELS/IGCSE

The mode is defined as the value which occurs most frequently in a set of data that is indicating the most common result.

Example: find the mode of the following set of numbers. 146, 167, 164, 157, 171, 167, 182, 197.

Mode = 167

Solution:

MODE FREQUENCY DISTRIBUTION EXAMPLE

Class Intervals	Frequency
10-14	2
15-19	8
20-24	20
25-29	15
30-34	3
35-39	6

Modal Class is 20-24

ADVANTAGES OF MODE

1. It is simply defined and easily calculated. It is very easy to locate.
2. It is not affected by large or small observations.
3. It can be determined for both the qualitative and quantitative data.
4. It is capable of being ascertained graphically

DISADVANTAGES OF MODE

1. It represents the most frequent value and hence it is very often in practice. The arrangement of data is not necessary.
2. It is not based on all observations made.
3. It is often indeterminate and indefinite.
4. It is not capable of lending itself to further statistical treatment. • It cannot be subjected to algebraic treatments.
5. It is used for the study of most popular fashions.
6. It is extremely used by businessmen and commercial managements.

In histogram, frequency curve or polygon, the mode can be located easily. Class interval/boundary with highest frequency is the mode.

EXAMPLE WORKING THROUGH GRAPH

MEASURE OF CENTRAL TENDENCY ( MEDIAN) FOR STATISTICS O LEVEL/IGCSE

The median is defined as a value which divides the data set that have been ordered, into two

equal parts, one part compromising of observations greater than and the other part smaller than it.

Median=(n/2)th value ( for single set of observations)

Median= L+h/f(n/2-C) for frequency distribution

Note: For odd number of observations we use n+1 in the place of n, above mentioned formulas.

Example: find the median

3, 4, 5, 1, 6, 4,1,8

To find the median first we have to arrange it in ascending order

1,1,3,4,4,5,6,8

Solution:

Median=(8/2)th value

Median=4th value

Median=4

Median frequency distribution example

Class Intervals	Class Boundaries	Frequency	CF
16-19	15.5-19.5	2	2
20-23	19.5-23.5	8	10
24-27	23.5-27.5	18	28
28-31	27.5-31.5	15	43
32-35	31.5-35.5	6	49
36-39	35.5-39.5	1	50

Ef=50

Median value= 50/2= 25^th value

Median=L+h/f(n/2-C)

Median=23.5+4/18(25-10)

Median=26.83

Graphical Determination of the Median

1. With the help of cumulativee frequency curve

Median=n/2 where n=sum of all frequencies.

200/2=100th

Median is between 439.5 and 449.5. Median in statistics can also be called Q2 or Median Quartile.

Advantages of Median

1. It is easily calculated and understood
2. It is located even when the values are not capable of quantitative measurement.
3. It is not affected by extreme values
4. It can be located graphically
5. It can be easily located even if the class intervals in the series are unequal

Disadvantages of Median

1. It is not subjected to algebraic treatments.
2. It cannot be used in further statistical treatment
3. It does not have sampling stability
4. It does not take into account the values of all items in the series.

USES

• It is useful in those cases where numerical measurements are not possible.
• It is also useful in those cases where mathematical calculation cannot be made in order to obtain the mean.
• It is generally used in studying phenomenon like skills, honesty, intelligence etc.