Topic: 1 2 3

Mode Median QuartilesMEMBERS:

1. AMMARA HASSAN 5. SHAQUFTA 2. HINA SHABBIR 6. FAISAL RAZI3. SAMINA BIBI 7. GHULAM HUSSAIN4. RABIA ASLAM 8. M SHOAIB ALI

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”

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, 205Solution:Mode= 45(occurring most times)

Formula(For Grouped Data):

The exact value of mode in grouped data can be obtained by the following formula.

𝑀𝑜𝑑𝑒=𝑙+𝑓𝑚− 𝑓 1

( 𝑓𝑚− 𝑓 1 )+( 𝑓𝑚− 𝑓 2)𝑋 h

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

Z =1999.5+

Z= Mode=2399.5

Z=1999.5+0.8 ×500= 1999.5+400

Mode=Z =

𝑍=𝟏𝟗𝟗𝟗 .𝟓+8

(8 )+(2)𝑋 500

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.

Disadvantages of Mode :

It is not based on all observations. It is not capable of further mathematical

manipulation. Mode is affected by sampling fluctuations.

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”

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 =

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 22Solution: 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.

Calculation of median(for Group Data):

For calculation of median in a continuous

frequency distribution the following formula will be applied. ,

Median   or  Q 2= l+ hf

 (n2

 − c  )

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 __ 40.541 __ 60 54 47+54=101 40.5 __ 60.561 __ 80 30 101+30=131 60.5 __ 80.581 __ 100 19 131+19=150 80.5 __ 100.5

Total ( f ) =150

Median (M)=40.5+

= 40.5+0.52X20= 40.5+10.37= 50.87

(

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+2854 𝑋 20

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

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.

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

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

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=12/43rd 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=6.5So median isbetween the 6th and 7th value , so we take average of these values.8+8/2= 8

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 setC = the cumulative frequency in the class immediately before the quartile classf = the frequency of the relevant quartile classh = the length of the real class interval of the relevant quartile class

hf

Cn

LQ

41 h

f

Cn

LQ

43

3 h

f

Cn

LQ

22

Example for Quartiles in Group Data:

Maximum Load(short-tons)

Number of Cables

(f)

ClassBoundaries

CumulativeFrequencies

9.3−9.7 2 9.25−9.75 29.8−10.2 5 9.75−10.25 2+3=7

10.3−10.7 12 10.25−10.75 7+12=1910.8−11.2 17 10.75−11.25 19+17=3611.3−11.7 14 11.25−11.75 36+14=5011.8−12.2 6 11.75−12.25 50+6=5612.3−12.7 3 12.25−12.75 56+3=5912.8−13.2 1 12.75−13.25 59+1=60

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  ∴

hf

Cn

LQ

41

hf

Cn

LQ

43

3

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 𝑄𝑢𝑎𝑟𝑡𝑖𝑙𝑒𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛=𝑄3−𝑄1

𝑄3+𝑄1

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.

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.