Measures of central tendency and relative position are used to summarize and compare datasets. The three main measures of central tendency are the mean, median, and mode. The mean is the average value and is calculated by summing all values and dividing by the total number of data points. The median is the middle value when data is arranged in order. The mode is the most frequent occurring value. Quantiles like quartiles, deciles, and percentiles divide a dataset into equal parts to indicate what percentage of values are below certain points.

1. Measures of central tendency
* To compare the central value of a character of
two or more series of data
Average
It is a central value of an attribute around which
other observations dispersed.
Uses
* To compare any observation with the central
value of an attribute in the same series of data

2. Mean
Median
Mode
Geometric Mean
Harmonic mean

3. X1+ X2+X3+. … +Xn
Mean = ----------------------------
n
Mean
If X1, X2, X3,... …. ,Xn are n observations

4. Merits
It is based on all the observations
Demerit
In case of extreme observations, it is not a
representative of a data.
e.g. 10,000, 250, 350, 400 mean=2750

5. Median
If X1, X2, X3,... …. ,Xn are n observations
arranged in ascending order of magnitude.
X(n+1)/2 If n is odd
Median = {
Xn/2 + X(n/2+1) If n is even
------------------
2

6. 74+75
Median = ---------- = 74.5
2
The following are the pulse rate per minute of
10 healthy individuals
82, 79, 60, 76, 63,81, 68, 74, 60, 75.
60, 60, 63, 68, 74,75, 76, 79, 81, 82

7. 162, 164, 166, 168, 170, 172, 174, 178,184, 188, 188
Median = 172
The following are the BP(S) of 11 hypertensive
patients of aged 50yrs
184, 170, 168, 188 162, 164, 174, 172, 178, 166, 188

8. Merits
1. It is unique for the given set of data.
2. It is not affected by presence of extreme
observations.
3. Median can be calculated even if values
of extremes are not known, if n is known.

9. Demerits
1. It is not based on all the observations.

10. Mode
It is most frequently occurring observation in
a given series of data.
Demerits
1. It is not based on all the observations.
2. It may not exist.

11. The following are the BP(S) of 11 hypertensive
patients of aged 50yrs.
184,170,168,188,162,164,174,172,178,166,188
Mode = 188

12. The following are the BP(S) of 10
hypertensive patients of aged 50yrs.
184,170,168,188,162,164,174,172,178,166
All the values have got the same frequency.
hence,
Mode does not exits.

13. The following are the BP(S) of 10
hypertensive patients of aged 50yrs.
188,170,166,188,162,164,174,172,178,166
Two values have got the same maximum
frequency.
hence,
Mode =188 and 166

14. Geometric Mean(G.M.)
If x1, x2, …………xn are n observations then G.M. is defined as
G.M. = (x1x2…..xn )1/n
Merits :
1. It is unique for the given set of data.
2. It is based on all the observations.
3. It less affected by extreme values.
Demerits:
1. It can not be used if any observation is either negative or
zero.

15. Harmonic Mean(H.M.)
If x1, x2, …………xn are n observations then
H.M. is defined as
1
H.M. = --------------------------------
(1/x1+ 1/x2…….+ 1/xn )/n
H.M. it is reciprocal of mean of reciprocals.

16. Merits :
1. It is unique for the given set of data.
2. It is based on all the observations.
3. It is least affected by the fluctuation of the
sampling.
Demerits:
1. It can not be used if any observation is zero.
2. It gives less weightage to largest observation and
more weightage to lowest observation.

17. Measures of relative position
Quantiles : Quantiles divides the total data into equal
parts after arranging the data in ascending order of
magnitude.
Quartiles divides the total data into 4 equal parts.
Deciles divides the total data into 10 equal parts.
Percentiles divides the total data into 100 equal parts.

18. Quartiles:
Q1, Q2 and Q3 denote 1st, 2nd and 3rd quartiles resp.
If X1, X2, …………Xn are n obs arranged in ascending
order of magnitude then quartiles are defined as
Qj = Xj( n+1)/4 , j = 1, 2, 3
Q1 indicates that 25% of observations are ≤ Q1.
Q2 indicates that 50% of observations are ≤ Q2.
Q3 indicates that 75% of observations are ≤ Q3.
Note: If n is even then Q2 is average of middle two obs.

19. Deciles: Let Dj denotes the jth decile and it is
defined as
Dj = Xj( n+1)/10 j= 1 to 10
For example D2 is given by D2 = X2( n+1)/10
It indicates that 10% of observations are less than
or equal to D2.

20. Percentiles:
Let Pj denotes the jth percentile.
It indicates that j% of observations are less than or
equal to Pj .
It is defined as
Pj = Xj( n+1)/100
For example 30th percentile is given by
P30 = X30( n+1)/100