2. Topic:
1 2 3
Mode Median Quartiles
MEMBERS:
1. AMMARA HASSAN 5. SHAQUFTA
2. HINA SHABBIR 6. FAISAL RAZI
3. SAMINA BIBI 7. GHULAM HUSSAIN
4. RABIA ASLAM 8. M SHOAIB ALI

3. 3. Mode
 Mode is the most frequent value or score in the distribution of Data.
 Highest point of the frequencies distribution curve.(due to maximum
frequency)
 Bimodal -- Data sets that have two modes
 Multimodal -- Data sets that contain more than two modes
Croxton and Cowden(statistician) : defined it as “the mode of a distribution is the value at the
point armed with the item tend to most heavily concentrated. It may be regarded as the most
typical of a series of value”

4. Mode for ungrouped data:
 In ungrouped data we can evaluate Mode by simply observing the data
Simply the Amount Appearing most often is Mode.
Example:
24, 25, 30, 39, 40, 45, 45, 45, 45, 45, 48, 50, 50, 55, 58, 60, 61, 65, 65, 65, 70, 72, 75,
200, 205
Solution:
Mode= 45
(occurring most times)

5. Formula(For Grouped Data):
The exact value of mode in grouped data can be obtained by the following formula.
𝑀𝑜𝑑𝑒 = 𝑙 +
𝑓𝑚 − 𝑓1
𝑓𝑚 − 𝑓1 + (𝑓𝑚 − 𝑓2)
𝑋 ℎ

6. Example: Calculate Mode for the distribution of
monthly rent Paid by Libraries in Karachi?
Monthly rent
(Rs)
Number of
Libraries (f)
Class Bounderies
500-999 5 499.5 __ 999.5
1000-1499 10 9999.5 __1499.5
1500-1999 8 1499.5 __ 1999.5
2000-2499 16 1999.5 __ 2499.5
2500-2999 14 2499.5 __ 2999.5
3000_3499 12 2999.5 __ 3499.5
Total 65

7. Z =1999.5+
Z= Mode=2399.5
Z=1999.5+0.8 ×500= 1999.5+400
Mode=Z =𝟏𝟗𝟗𝟗. 𝟓 +
𝟏𝟔−𝟖
𝟏𝟔−𝟖 +(𝟏𝟔−𝟏𝟒)
𝑿 𝟓𝟎𝟎
𝑍 = 𝟏𝟗𝟗𝟗. 𝟓 +
8
8 + (2)
𝑋 500

8. Advantages of Mode :
 Mode is readily comprehensible and easily calculated
 It is not at all affected by extreme value.
 The value of mode can also be determined graphically.
 It is usually an actual value of an important part of the
series.

9. Disadvantages of Mode :
 It is not based on all observations.
 It is not capable of further mathematical
manipulation.
 Mode is affected by sampling fluctuations.

10. 2.Median:
Median is a central value of the distribution, or the value which
divides the distribution in equal parts, each part containing equal
number of items. Thus it is the central value of the variable,
when the values are arranged in order of magnitude.
Connor has defined as “ The median is that value of the variable
which divides the group into two equal parts, one part
comprising of all values greater, and the other, all values less
than median”

11. Calculation of Median (for ungroup data):
 Arrange the data in ascending or descending order.
 If number of terms is odd, the median is the middle term of the ordered array.
 If number of terms is even, the median is the average of the middle two terms.
 Apply the formula.
Median =
1+𝑁
2
𝑡ℎ 𝑉𝑎𝑙𝑢𝑒

12. Example of median(ungroup Data):
Data of Distance covered by a ball hitted by a child 17 times:
 3 4 5 7 8 9 11 14 15 16 16 17 19 19 20 21 22
Solution:
 There are 17 terms in the ordered array.
 Position of median = (n+1)/2 = (17+1)/2 = 9
3 4 5 7 8 9 11 14 15 16 16 17 19 19 20 21 22
 The median is the 9th term, 15.

13. Calculation of median(for Group Data):
 For calculation of median in a continuous frequency
distribution the following formula will be applied. ,
Median or Q2=l+
h
f
n
2
−c

14. Example: Median of a set Grouped Data in a
Distribution of Respondents by age
Age Group Frequency( f ) Cumulative
Frequencies(C.F)
Class Boundaries
0 __ 20 15 15 - 0.5 __ 20.5
21 __ 40 32 15+32= 47 20.5 __ 40.5
41 __ 60 54 47+54=101 40.5 __ 60.5
61 __ 80 30 101+30=131 60.5 __ 80.5
81 __ 100 19 131+19=150 80.5 __ 100.5
Total ( f ) =150

15. Median (M)=40.5+
= 40.5+0.52X20
= 40.5+10.37
= 50.87
20
54
(
150
2
− 47)
Median or Q2=l+
h
f
n
2
−c
Here
(N/2)th value= 150/2= 75 th value
According to our data 75th value is occurring in
class 41__60 .
Hence this class is our Median class.
l=40.5 (Lower boundary of median class)
H = 20 (interval)
f = 54 (frequency of Median Class)
n = 150 (total Number of observations)
c = 47 ( Cumulative Frequency before Median Class)
= 40.5 +
28
54
𝑋 20

16. Advantages of Median:
 Median can be calculated in all distributions.
 Median can be understood even by common people.
 It can be located graphically
 It is most useful dealing with qualitative data

17. Disadvantages of Median:
 It is not based on all the values.
 It is not capable of further mathematical treatment.
 It is affected fluctuation of sampling.
 In case of even no. of values it may not the value from
the data.

18. Quartiles
 The Quartiles of a r set of data values are the three points that divide the data set
into four equal groups, each group comprising a quarter of the data..
Quartiles divide data into four equal parts
 First quartile—Q1
 25% of observations are below Q1 and 75% above Q1
 Also called the lower quartile
 Second quartile—Q2
 50% of observations are below Q2 and 50% above Q2
 This is also the median
 Third quartile—Q3
 75% of observations are below Q3 and 25% above Q3
 Also called the upper quartile

19. Calculating Quartiles(for ungroup Data):
The general Formula for Quartiles for Ungroup data is :
Q1= (N/4) th value.
Q2 = (N/2) th value.
And if n is even Q2 = (1+N/2) th value and then Take average of two
median Values.
Q3 = (3XN/4) th Value.
Q4= Q3-Q1/2

20. Example for Ungroup Data:
 Example 1: Find the median and quartiles for the data below.
12, 6, 4, 9, 8, 4, 9, 8, 5, 9, 8, 10
4, 4, 5, 6, 8, 8, 8, 9, 9, 9, 10, 12
Order the data
Lower
Quartile
n/4=12/4
3rd value= 5
Q1
Median
= 8
Q2
Upper
Quartile
3n/4=3x12/4
9th value= 9
Q3
Median:
(N is Even)
1+N/2 =
1+12/2=6.5
So median is
between the
6th and 7th
value ,
so we take
average of
these values.
8+8/2= 8

21. Calculating Quartiles(for Grouped data):
 Calculation of the quartiles from a grouped frequency distribution
 The class interval that contains the relevant quartile is called the quartile class
where:
L = the real lower limit of the quartile class (containing Q1 or Q3)
n = Σf = the total number of observations in the entire data set
C = the cumulative frequency in the class immediately before the quartile class
f = the frequency of the relevant quartile class
h = the length of the real class interval of the relevant quartile class
 h
f
C
n
LQ








4
1  h
f
C
n
LQ








4
3
3
 h
f
C
n
LQ








2
2

22. Example for Quartiles in Group Data:
Maximum Load
(short-tons)
Number of Cables
(f)
Class
Boundaries
Cumulative
Frequencies
9.3−9.7 2 9.25−9.75 2
9.8−10.2 5 9.75−10.25 2+3=7
10.3−10.7 12 10.25−10.75 7+12=19
10.8−11.2 17 10.75−11.25 19+17=36
11.3−11.7 14 11.25−11.75 36+14=50
11.8−12.2 6 11.75−12.25 50+6=56
12.3−12.7 3 12.25−12.75 56+3=59
12.8−13.2 1 12.75−13.25 59+1=60

23. Q1=Value of (n/4)th item =Value of (60/4)th item = 15th item
Q1 lies in the class (10.25−10.75). Hence this is our Model Class
∴
Where l=10.25, h=0.5, f=12, n/4=15 and c=7
∴ Q1=10.25+0.512(15−7)=10.25+0.33=10.58
Q3=Value of (3n4)th item =Value of (3×604)th item = 45th item
Q3 lies in the class 11.25−11.75. Hence it is our Model Class
Where l =11.25, h=0.5, f=14, 3n/4=45 and c=36
Q3=11.25+0.514(45−36)=11.25+0.32=11.57
∴
 h
f
C
n
LQ








4
1
 h
f
C
n
LQ








4
3
3

24. Quartile Deviation and Co.officient of
Quartile Deviation:
 Q.D=Q3−Q1=11.57−10.582=0.992=0.495
 Coefficient of Quartile Deviation:
= 11.57−10.58 / 11.57+10.58
 Coefficient of Quartile Deviation= 0.99 - 22.15 =0.045
𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑄𝑢𝑎𝑟𝑡𝑖𝑙𝑒 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 =
𝑄3 − 𝑄1
𝑄3 + 𝑄1

25. Advantages of Quartiles:
 It can be easily calculated and simply understood.
 It does not involve much mathematical difficulties.
 As it takes middle 50% terms hence it is a measure better than Range and
percentile Range.
 Quartile Deviation also provides a short cut method to calculate Standard
Deviation using the formula 6 Q.D. = 5 M.D. = 4 S.D.
 In case we are to deal with the center half of a series this is the best measure to
use.

26. Disadvantages of Quartiles:
 As Q1 and Q3 are both positional measures hence are not capable of further algebraic
treatment.
 Calculation are much more, but the result obtained is not of much importance.
 It is too much affected by fluctuations of samples.
 50% terms play no role; first and last 25% items ignored may not give reliable result.
 If the values are irregular, then result is affected badly.
 We can’t call it a measure of dispersion as it does not show the scatter-ness around
any average.
 The value of quartile may be same for two or more series or Q.D. is not affected by the
distribution of terms between Q1 and Q3 or outside these positions.