Measures of central tendencyMeanModeMedian

Mode

Distribution of application for admission to M.B.A. by discipline,

Findings,The commerce category is the most

predominant.

Discipline No. of studentsScience 55Arts 60

Commerce 101Engineering 45Medical 5Total 266

Mode – Grouped data.Procedure,1. Find the category or the class

interval which has the greatest frequency.

2. The midpoint on this category is the mode.

Age Group Frequency1--20 1521--40 3241--60 5461--80 3081--100 19Total 150

Mode frequency is 54 which is associated with the modal class interval of 41-60. The midpoint of this class interval is 50.5 i.e. (41+60)/2 = 50.5

Mode of this distribution is 50.5Age Group Frequency

1--20 1521--40 3241--60 5461--80 3081--100 19Total 150

Use:Useful measure for qualitative data.Appropriate measure for nominal

(qualitative) level of data.E.g. include gender, nationality,

ethnicity, language, genre, style, biological species, and form.

Application of data can be in distribution like ethnic classification of occupational classification.

For finding out the typical category.

MedianGrouped dataProcedure,= l + ((n/2 – cf) / f )*iFind out mid point (n/2)Find the class interval of mid point from

the cumulative frequency column (cf)l is the lower limit of the median category.cf is the cumulative frequency up to but

not median category.i is the size(range) of the median class

interval.

Age Group Frequency Cumulative Frequencies1--20 15 1521--40 32 4741--60 54 10161--80 30 13181--100 19 150Total 150

n/2= 150/2=75, Class interval= 41-60, f=54

l= 41, Cf=47, i=20

= l + ((n/2 – cf) / f )*i =41+((75-47)/54)*20Mean =51.4Use:Used for ordinal (where the order matters

but not the difference between values)or interval (where the difference between two values is meaningful)level data but not for nominal level data.

Arithmetic mean

= ∑ (fx) / nX=any valuef= the frequency of a value∑=sum n=the number of value

Daily wages (f) Frequency (x) Fx6 4 247 8 568 6 489 12 10810 7 7012 4 4815 2 30

Total 43 384

Arithmetic mean= = ∑ (fx) / n=384/43=8.9

Grouped data = ∑ (fm) / n m=midpoint of each class interval f=frequency of a value fm=midpoint multiplied by its

frequency n=number of cases

=460/20=23.Score Frequency(f) Midpoint(m) fm11--15 3 13 3916--20 4 18 7221--25 7 23 16126--30 3 28 8431--35 2 33 6636--40 1 38 38Total 20 460

Choice of an appropriate average

The choice depends upon the consideration of several factors:

The level of measurementThe research objective

Level of measurement:Mode requires only a frequency count, it can

be applied to any set of data at the nominal, ordinal or interval level of data.

Median requires an ordering of items form the highest to the lowest or vice versa. Hence it can be obtained from an ordinal or interval level of data and not from nominal data like party affiliation, caste or religion.

Mean is exclusively restricted to interval data such as income, age, wage rate & test score.

Research objective:Mode is useful to find out most

common category. E.g. test score, caste, age

Mean is useful for further mathematical manipulations.

Median is useful to find out mid values.