B.A.III(HONS)

Unit-4

Measures of Central Tendency

B.A.III(HONS)

INTRODUCTION� Measures that reflect the average characteristics of a

frequency distribution are referred to as ‘measures of central tendency.’

� Mean , median and mode are three most commonly used measures of central tendency.measures of central tendency.

� Mean is the mathematical measure while median and mode are the positional measures.

MEAN� THE MEAN IS OF FOUR TYPE:

1 ARITHMETIC MEAN ( X )

2 GEOMETRIC MEAN (GM)

3 HARMONIC MEAN (HM)

4 QUADRACTIC MEAN (QM)4 QUADRACTIC MEAN (QM)

. ARITHMETIC MEAN IS MORE FREQUENTLY USED.

ADVANTAGES OF ARTHMETIC MEAN� It makes information so simple that even a common man

understands its meaning.

� Its calculation is easy.

� It is not necessary that given units may be orderly ranked.

� It takes into consideration all the scores in distribution.� It takes into consideration all the scores in distribution.

� Mean of different distributions are useful for comparative purposes.

DISADVANTAGES OF ARTHEMETHIC

MEAN

� It is difficult to assume arthemetic mean merely by seeing the frequencies of the units.

� It cannot be used for qualitative analysis

� If the frequency of any one unit is missing, mean cannot be calculatedcalculated

� The mean is usually outside the given units.

� Mean gives more importance to large frequencies than smaller ones.

� Mean is not useful in calculating ratio.

MEDIAN

MEANING� IT IS THE MIDDLE VALUE IN A SERIES OF VALUES THAT

DIVIDES DISTRIBUTION INTO TWO EQUAL PARTS, i.e. , half of the values lies above the median and half below it.

Median in different types of series� Individual series

� Formula:

� Median= size of(N+1)/2th item (where N is the number of items)

In continious series

� Mdn=l1+l2-l1/f*(m-c) or� =l1+i/f*(m-c) where:� l1=lower limit of the median group� L2=upper limit of the median group� L2=upper limit of the median group� f=frequency of the median group� m =middle item� c =cumulative frequency of the group prior to the median

group� i =l2-l1

ADVANTAGES OF MEDIAN � Median can be calculated in all distributions.

� If the frequencies in observations are arranged in ascending order , the median can be calculated merely by looking at the extreme items.

� Median can be understood even by common people.� Median can be understood even by common people.

� It is very useful in quantitative analysis.

LIMITATION OF MEDIAN� Its use is limited as it is not used in the context of qualitative

phenomena.(E.G. IQ of individuals)

� It cannot be used where items are assigned weights.

MODE

MEANING� THE MODE IS THE MOST FREQUENT VALUE OR

SCORE IN THE DISTRIBUTION , i.E., It is the score with the highest number of points on the score scale

OBJECTIVE

�The objective of mode is descriptive frequent value.

INDIVIDUAL SERIES� Distribution can have one mode(uni-mode) , two mode (bi-

modal),more than two modes (multi-modal) , or even no mode at all (non-modal)

� The mode is not calculated mathematically but is identified logically on the basis of its relationship with other values.logically on the basis of its relationship with other values.

� It is a measure to ‘see’ rather than to ‘calculate’

Continious series� Formula:

� Z =l1+ f1-fo/2f1 – fo –f2 * (l2 –l1)

� where:

� Z= is the mode

l1 = lower limit of the modal class interval or group� l1 = lower limit of the modal class interval or group

� l2 =upper limit of the modal class interval or group

� f1 = frequency of the modal group

� fo = frequency of the group prior to the modal group

� f2 = frequency of the group just after the modal group

ADVANTAGES OF MODE

� In simple series the mode can be easily defined by observation.

� It can also be identified by a graph.

� It is easy to calculate.� It is easy to calculate.

� It is useful in the study of popular sizes.

LIMITATIONS OF MODE� It is not stable measure of central tendency as its position

might shift whenever the manner of the distribution’s divisions into categories is altered.

� It is not amenable to algebric treatment.

� It remains indeterminate when have two or more modal � It remains indeterminate when have two or more modal values in a series.

� It is unsuitable in cases where we want to give relative importance to items under consideration.

CONCLUSION

� Mean (average of all the values) used at interval level of measurement

� Median (distribution’s mid-point) used at ordinal level of measurement.

� Mode (highest density in the distribution)used at � Mode (highest density in the distribution)used at nominal level of measurement.

LEVELS OF MEASUREMENT

NOMINAL SCALE� This scale classifies individuals into two or more categories ,

the members of which differ with respect to the specific characteristic.

� The categories have no rank order.

� Example:hindus and non – hindus , male and females.� Example:hindus and non – hindus , male and females.

� All the members of a set are assigned the same numerals and no two sets are assigned the same numeral

ORDINAL SCALE� This scale ranks individuals along the continuum of the

charactreristic being scaled ,say, from highest to lowest , greater to least , first to last and so on.

� Suppose a,b and c are three students and a is the first divisioner ,b is second divisioner and c is third divisioner.divisioner ,b is second divisioner and c is third divisioner.

� Ordinal number indicates only rank order and nothing more.

INTERVAL SCALE

�This scale has equal units of measurement which enables the researcher to interpret the distance between them.

RATIO SCALE

�This is the scale which has absolute zero point of origin and which explains proportion of one value to another.

�Example: the ratio of female crime to male �Example: the ratio of female crime to male crime is 1:19

�Ratio scales are also referred to as absolute scales.

CONCLUSION TYPE OF SCALE RANGE Central Tendency

Nominial Number of categories Mode

Ordinal Number of ScalerPositions

Median

Interval or Ratio Top score minus MeanInterval or Ratio Top score minus Bottom Score

Mean