Measures of Central Tendency
I. Introduction.
II. Requirements of good measures.
III. Mean Definition.
IV. Properties
V. Merits and Demerits.
VI. Examples
VII. Weighted Arithmetic Mean
VIII. MedianI. Introduction
Everybody is familiar with the word Average. and everybody are used the word average in daily life as, average marks, average of bike, average speed etc. In real life the average is used to represent the whole data, or it is a single figure is represent the whole data. the average value is lies around the centre of the data. consider the example if we are interested to measure the height of the all student and remember the heights of all student, in that case there are 2700 students then it is not possible to remember the all 2700 students height so we find out the one value that represent the height of the all 2700 students in college. therefore the single value represent the whole data and it is lies around to centre of the data is called the Measure of central tendency. the central value gives the idea about the data.
in simply the measure of central tendency is a statistical measure used to describe the central value od the data set.
Now we see the objective of average.
1. to compare two different data sets.
2. to obtain the single representative value.
1. To compare the two different data set.:
if we are interested two comparing the one set of data with other, for comparison we calculate the average value for two data set and comparing it with other average value .
2. To obtain the single representative value : the aim is to calculating the average to get the single figure that represent the all Data set.
the most commonly used measure f central tendency are 1. Mean 2. Median. 3.Mode
therefore the another problem is there are 3 measures out of which one measure is good then use the following are the requirements of good measures of central tendency.
II. Requirements of good measures of central tendency
1. Simplicity: it should be clearly and rigidly defined also it is simple to understand.
2. Easy to calculate: in some cases the it is easy to calculate.
3. Based on all observation : an average represent the all data. or the single value. therefore it should be based on the all observation.
4.It is not affected by extreme value:
it is based on all value and it it a central value .
5. Sampling stability: the value of an Measure should have sampling stability.
III. Arithmetic Mean .
Definition: arithmetic mean is defined as the sum of all observation divided by total total number of observation
the arithmetic mean is denotes as x̄
it is given by
x̄ = mean = (x1+ x2+ x3+ ......... +Xn) / n
where x1, x2, x3, ................xn are observations in a data
n - is the total number of observation in data
it is also denoted as x̄ = ∑ (xi) / n
where ∑ (xi) - sum of all observations in a data set or ∑ (xi) = x1+ x2+ x3+ ......... +Xn
n - total number of observations
∑ - is a greek letter used for denoting summation of all observation.
IV. Properties of Arithmetic Mean:
Arithmetic Mean is also known as simple mean or average. it is a common measure of central tendency in statistics. it is calculated as summing the all observation in data and divided by total number of observation in data set. (Arithmetic mean is said to be best average)
for calculating the Arithmetic Mean of Average or mean we follow the following steps as:
1. we add or summing the all observation in data set.
2. counting the total number of observation in a data set.
3. dividing the sum by total number of observations.
for example.
if the data is 10, 11, 12, 13, 14, 15, 16, 17, 18 . then find the arithmetic mean.
first we add or summing the all observations as 10 + 11 + 12 + 13 + 14 + 15 +16 + 17 + 18 = 126.
the total number of observation is 9 in data set.
Mean = (sum of observation ) / (total number of observations)
Mean = 126/9 = 14
therefore the Arithmetic Mean = 14. and it is a central value in a data set.
Now we see the properties of Arithmetic Mean:
Arithmetic Mean is a important measure of central tendency is has some Properties.
1. The Sum of deviation taken from it mean with sign is always zero.
The some of deviation of observations taken form it mean with sign is always zero, this property is also known as the balance property of arithmetic mean.
For understanding this property ,
we consider as if X1, X2, .........Xn be a n observations and its mean is x̄ then taking the deviation of each value from it mean are X1- x̄, X2-x̄, ......... Xn-x̄, the property of mean is sum of deviation taken from mean is zero we check it as
we know the definition of mean x̄ = (x1+ x2+ x3+ ......... +Xn) / n
we solve this as (x̄ n) = x1+ x2+ x3+ ......... +Xn
x1+ x2+ x3+ ......... +Xn - (x̄ n) = 0
x1+ x2+ x3+ ......... +Xn - ( x̄ +x̄ +......+ x̄) = 0
(X1- x̄) + ( X2-x̄) +......... +( Xn-x̄) = 0
hence the sum of deviation is taken from mean is always zero.
2. The product of mean and total number of observation is equal to sum of all observations. i.e. (x̄ n) = x1+ x2+ x3+ ......... +Xn
Proof :
consider the definition od mean as
x̄ = (x1+ x2+ x3+ ......... +Xn) / n
(x̄ n) = x1+ x2+ x3+ ......... +Xn
Hence The product of mean and total number of observation is equal to sum of all observations.
3. The arithmetic mean of combined group can be determined from the mean of that groups.
if the x̄1 and x̄2 are the two arithmetic means of two groups having n1 and n2 observation respectively.
then the arithmetic mean of combined group is
x̄ = ( x̄1 n1 + x̄2 n2 ) / ( n1 + n2 )
4. The sum of square of deviation of observations from its mean is minimum.
V. Merits and Demerits of Arithmetic mean
Merits:
1. it is clearly and rigidly defined.
2. it is simple to understand.
3. it is easy to calculate.
4. it is based on all observations.
5. it is used for further calculations.
6. it has sampling stability.
Demerits.
1. it is affected by extreme value. if the mean is based on all observation if all values in data are small and one value is large then the mean is close to the larger value.
2. it is not suitable for open end classes.
3. it has up-ward biased: it gives greater importance to large value and less importance to small value.
4. if the mean of two group are same mean does not mean that the values in two group are same.
VI. Arithmetic Mean Examples:
The arithmetic Mean is calculated for different type of data as individual, discrete and continuous data, but the formula are different for a different type of data.
Arithmetic Mean is denoted as A.M.
1. Individual Item: Individual refers to a single unit or observed value. Each item is considered separately.
The formula if Calculating Mean is Arithmetic Mean = ∑( X ) / n
Where X - is individual values of items.
∑ X - is sum of all observations in data.
n -Total Number of observations.
Example 1. Calculate the Arithmetic Mean of following observations.
7, 10, 11, 12, 17, 18, 6, 7, 19, 10, 15, 17, 12, 14. 15.
Solution:
For Calculating the Arithmetic Mean of these Observations we first Adding or summing the all given observations.
first Summing all observation as
∑ X = 7 + 10+12+17+18+6+7+19+10+15+17+12+14+15
= 179
then count the total number of observations.
the total observations are 15
then using the definition of Arithmetic mean
therefore ∑ X = 179 and n = 15
Arithmetic Mean = ∑( X ) / n
Arithmetic Mean = 179 / 15 = 11.9333
therefore the Arithmetic Mean = 11.9333
2. Discrete Distributions : Ungrouped Data : if data takes only Discrete values.
For Calculating the Mean of discrete distribution formula is
Mean = ∑( f X ) / N
where f- is the Frequency of the observation or item.
Frequency mean number of time the particular item or observation repeated in a data set.
e.g. id the data is 1, 4, 1, 2, 2, 5, 1, 1, 2, 4, 2 then the frequency of 1 is 4, frequency of 2 is 4, frequency of 4 is 2 and frequency of 5 is 1 it is expressed as
observations (x) : 1 2 4 5
Frequency (f) : 4 4 2 1
sum of frequency is equal to total number of observations in a data set hence n = ∑ f
∑( f X ) - sum of product of each observation and it corresponding frequency
N - Total number of observation i.e. N = ∑ f
following are the step for calculating mean
i. Multiplying each x by corresponding frequency and summing this values i.e. ∑( f X )
ii. dividing ∑( f X ) by N.
Example 2. Compute the Arithmetic Mean of following data.
Marks (x): 15 20 25 30 35 40
No. of Students (f) : 2 10 14 15 16 7
Solution: for calculating the ∑( f X ) preparing a table as
Marks (x) | No. of Students (f) | fx |
15 | 2 | 30 |
20 | 10 | 200 |
25 | 14 | 350 |
30 | 15 | 450 |
35 | 16 | 560 |
40 | 7 | 280 |
Total | 64 | 1870 |
N = ∑ f = 64 and ∑ (f X) = 1870
Arithmetic Mean = ∑( f X ) / N
Arithmetic Mean = 1870 / 64
Arithmetic Mean = 29.21875
Therefore the Arithmetic Mean of given data is 29.21875 Marks
3. Continuous distribution: Grouped Data: if the data are divided or classified into groups with corresponding frequency's, then we consider frequency corresponding to the middle value. therefore the formula is
Arithmetic Mean = ∑( f m ) / N
where f - is the Frequency of the observation or item.
m - is the mid-point of the class.
m= mid-point = (Upper Limit + Lower Limit ) / 2
N - Total number of observation i.e. N = ∑ f
following are the step for calculating mean
i. Firstly we calculating the mid0pint foe each class.
ii. Multiplying each m ( mid-point ) by corresponding frequency f and summing this values i.e. ∑( f m)
ii. dividing ∑( f m ) by N.
Example 3. Calculate the mean for following data.
Class | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 | 80-90 | 90-100 |
Frequency (f) : | 7 | 5 | 4 | 8 | 10 | 15 | 7 | 2 | 1 |
Solution: for calculating the m (mid-point) and ∑( f m ) we prepare a table as
Class | Frequency (f) | mid-point m | fm |
10-20 | 7 | 15 | 105 |
20-30 | 5 | 25 | 125 |
30-40 | 4 | 35 | 140 |
40-50 | 8 | 45 | 360 |
50-60 | 10 | 55 | 550 |
60-70 | 15 | 65 | 975 |
70-80 | 7 | 75 | 525 |
80-90 | 2 | 85 | 170 |
90-100 | 1 | 95 | 95 |
Total | 59 | | 3045 |
we calculate mid-points as
m= mid-point = (Upper Limit + Lower Limit ) / 2
m = (10+30)/2 = 15
and fm = 7 x 15 = 105
in table ∑( f m ) = 3045 and ∑ f = N = 59
Arithmetic Mean = ∑( f m ) / N
Arithmetic Mean = 3045 / 59
Arithmetic Mean = 51.61017
Therefore the mean of the given data set is 51.61017
4. Calculate Arithmetic
mean
|
Wages in (RS) |
10-50 |
50-90 |
90-130 |
130-170 |
170-210 |
210-250 |
|
No. of Workers |
110 |
19 |
24 |
37 |
27 |
16 |
|
Class |
Frequency
(f) |
mid-point
m |
fm |
|
10-50 |
10 |
30 |
300 |
|
50-90 |
19 |
70 |
1330 |
|
90-130 |
24 |
110 |
2640 |
|
130-170 |
37 |
150 |
5550 |
|
170-210 |
27 |
190 |
5130 |
|
210-250 |
16 |
230 |
3680 |
|
Total |
133 |
|
18630 |
Arithmetic Mean = ∑( f m ) / N
Arithmetic Mean = 18630 / 133
VII. Weighted Arithmetic Mean:
we see in the Arithmetic Mean, the arithmetic mean is based on the all observations in data set, therefore it gives the equal importance to each items. this is the one of the drawback of the arithmetic mean, because in some cases mean is not proper representative of the whole data set. e.g. in our daily life the importance of edible oil is not same as the importance of hair oil. and the importance of rice in meal is not same the importance of sugar. in that case we are interested to finding the average price of that item. ( in that case for finding the average arithmetic mean is not used, because here different item have different importance). for this type of example we use special type of average which consider the relative importance of the item. the figure of value they represent the relative importance of the item is called "weight" of that item. and calculating the average of items considering with there weights is called "weighted arithmetic Mean" or Weighted Average".
Definition of weighted mean :
The weighted Arithmetic Mean is defined as the sum of product of the weights and the values in data set and divided by total number of observations in data set. And the weighted average is denoted as x̄w .
and it is formulated as
x̄w = (w1* x1+ w2* x2+………….+ wn* xn) / (w1+ w2+……+ wn)
x̄w = ∑(wi* xi)/∑ wi
where x̄w - is the weighted mean
w - is the weights of item.
x - is value of item.
e.g. The price of rice is Rs 10per kg that of sugar is Rs 7 per kg and that of oil is Rs 12 per kg. and the weights of rice, sugar and oil is 2, 10, 5, respectively then find the weighted average of prIce.
Here the item are Rice - x1= 10, sugar - x2 = 7, oil - x3 = 12, with corresponding weights are Rice - w1= 2, sugar - w2 = 10, oil - w3 = 5. then the weighted mean is x̄w
x̄w = (w1* x1+ w2* x2+ w3* x3) / (w1+ w2+ w3)
x̄w = [(10 x 2) + (7 x 10 ) + ( 12 x 5)] / (2+10+5)
x̄w = 150 / 17 = 8.8235 per kg.
Hence the average price of items per kg is 8.8235 .
use of weighted arithmetic mean.
i. In any problem of calculating average of different items have different importance.
ii. weighted arithmetic mean used for calculating the index number.
The weighted arithmetic mean is applied in the situation where certain data points carry more weight than other, such as in financial analysis. in education system teachers assign different weights to various assessments such as Homework, projects, unit tests etc. the weighted mean is used to calculating the final grade, based on all assessments. using the weighted mean we calculating more accurately the final grade.
hence it useful to finding the averages by considering the relative weights assigned to different data point.
Example:
Example 1 . Calculate the Average salary of workers in a company. they gives number of workers and salary is 155, 45, 125, 75, 100 and 100, 20, 150, 75, 200 respectively.
Solution: here number of workers is w weights and the salary is x variable
Weighted mean is calculated as
x̄w = (w1* x1+ w2* x2+ w3* x3+ w4* x4+ w5* x5) / (w1+ w2+ w3+ w4+w5)
x̄w = [(155*100)+(45*20)+(125*150)+(75*75)+(100*200)] / (155+45+125+75+100)
x̄w = 60775 / 500
x̄w =121.55
Hence the average salary of workers is 121.55 .
VIII. Median:
The Median is the measure of central tendency, along with mean used to describe the typical or central value in a data set. the median is a statistical measure that represent the middle value of the data set, it is arranged in ascending or descending order.
It is most preferable measure of location for asymmetric distribution it is also called as positional average. the median is defined as the value or observation which lies at the center if all observations arranged in ascending or descending order of magnitude.
The median Divides the data set into two equal part, with 50% data above to the median value and 50 % data below to the median value in a data set. it is particularly useful when dealing with skewed distribution or data can contain outliers or extreme value. the median is not affected by extreme value as compared with mean.
For calculating the Median of the data we firstly sorted or arranging data in ascending or descending order. the calculation of median is dependence on total number of observation in a data set, means if the data set has total number of observations is odd then median is the middle value in a data set. (if total number of observations is odd there is one observation at the center hence the median value is the middle observation) e.g. if the observations are 14, 20, 17, 18, 15. then firstly we arranging in ascending order as 14, 15, 17, 18, 20. here total number of observation is 5 and it is odd then at the middle only one value is median = 17.
If the total number of observations is even then median is the average of the two middle observations (i.e. if the total number of observation is even then data set contain two value at the middle then we take average of two middle value and consider as median.) e.g. the observations are 20, 100, 25, 14, 16, 10. here total observations are 6 it is even then the arranging in ascending order as 10, 14, 16, 20, 25, 100 in this case at the middle two values 15 and 20 then we take its average as (16+20)/2 = 18
hence the median is 18. in above example we the value are 10, 14, 16, 20, 25, 100 and it median is 18, but in this data we change the last value 100 as 1000 then the median is same 18 for the data 10, 14, 16, 220, 25, 1000 because median is positional average it is not affected by extreme values. it is widely used in various fields including statistics , economics, and social sciences. advantage of median is that it is useful in quantitative data, median is determined as graphically.
Merits
i. It is rigidly defined and simple to understand.
ii. it is easy to calculate.
iii. it is not affected by the end values
ii. it is easy to calculate.
iv. it is suitable for open-end classes.
v. it is useful for the quantitative data.
vi. the median is determined graphically.
Demerit
i. it is not representative, if items are few and and variation is large.
ii. the median has not sampling stability.
iii. median is not based on all observations.
Calculating the Median
i. Individual data :- the observation are individual then we arranging the data in ascending order and apply the definition of median is
Median = size of { (N+1) / 2}th item.
Where N is the total number of observations in a data.
Example 1. Calculate the median from the following values 40, 42, 45, 47, 58, 50, 60, 47, 48, 50, 54, .
Solution : For calculating the Median we arranging the data in ascending order as
40, 42, 45, 47, 47, 48, 50, 50, 54, 58, 60
here N = 11
Median = size of { (N+1) / 2}th item. or observation
Median = size of { (11+1) / 2}th item.
Median = size of { 12 / 2}th item.
Median = size of { (6 }th observation.
Median = 48
therefore in data we count the six number observation is 48.
Hence median is 48
OR
the total number of observation is odd then middle value is median hence 48 is meddle value then median is 48.
ii. Discrete Distribution: Ungrouped Data:
if the data take the discrete value, we firstly arranging the data into ascending order then calculating cumulative frequency (c.f.).
the following step are follow to find median.
i. arranging the data in ascending order.
ii. calculating the cumulative frequency and preparing column of c.f.
iii. use formula of the median
Median = size of { (N+1) / 2}th item. or observation
Example 2. Calculate the Median for the following data.
X | 50 | 51 | 52 | 54 | 56 | 55 | 53 | 49 |
F | 10 | 8 | 7 | 15 | 19 | 22 | 14 | 16 |
Solution :
Firstly arranging the data in ascending order (i.e. arranging value of x and corresponding f in ascending order)
X | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 |
F | 16 | 10 | 8 | 7 | 14 | 15 | 22 | 19 |
now calculating cumulative frequency and preparing new column in table.
X | F | C.F. |
49 | 16 | =16 |
50 | 10 | =16+10=26 |
51 | 8 | =26+8=34 |
52 | 7 | =34+7= 41 |
53 | 14 | =41+14=55 |
54 | 15 | =55+15=70 |
55 | 22 | =70+22=92 |
56 | 19 | =92+19=111 |
Total | 111 |
Median = size of { (N+1) / 2}th item. or observation
Median = size of { (111+1) / 2}th item. or observation
Median = size of { (112) / 2}th item. or observation
Median = size of {56}th item. or observation
the column of c.f. shows the order of each observation where the first c.f. 16 represent the value 49 is repeated 16 times. means first 16 observation are 49 and then observation from 17 to 26 is 50 means 50 is repeated 10 times from 17 to 26. and 54 is repeated 15 times from 56 to 70. observation 55 is from 71 to 92 repeated. 22 time from 71 to 92 similarly.
hence 56 th observation is 54
therefore the median is 54 of the data.
because the value of x = 54 is repeated 15 time from 56 to 70.
Hence median is 54
ii. Continuous Distribution: Grouped Data:
if the data Contain class and frequency, we firstly arranging the data into ascending order then calculating cumulative frequency (c.f.).
the following step are follow to find median.
i. arranging the data in ascending order.
ii. calculating the cumulative frequency and preparing column of c.f.
iii. use formula of the median
Median = size of { N/ 2}th item. or observation to find median class i.e. the class in which the median or value of median is lies and then use Median formula
Median = L + (N/2) - c.f. x c f
Where, L - lower limit of the median class.
c. f. - cumulative frequency of the previous class.
f - frequency of the median class
c - class interval of the median class
Example 3: Calculate the median from the following data.
|
Wages in Rs. (X) |
100-200 |
200-300 |
300-400 |
400-500 |
500-600 |
600-700 |
|
No. Of Workers (f) |
12 |
18 |
25 |
40 |
15 |
10 |
Solution:
calculating cumulative frequency and preparing new column in table.
|
Wages in Rs. (X) |
No. Of Workers (f) |
c. f. |
|
100-200 |
10 |
10 |
|
200-300 |
20 |
30 |
|
300-400 |
25 |
55 |
|
400-500 |
40 |
95 |
|
500-600 |
15 |
110 |
|
600-700 |
10 |
120 |
|
Total |
120 |
|
Median = size of { N/ 2}th observation
Median = size of (120/2)th observation.
Median = size of 60 th observation
therefore the median class is 400-500
now L = 400, N/2 = 60, f = 40, c.f. 55, c = 100 i.e. 500-400 = 100
now we use median formula as
Median = L + (N/2) - c.f. x c f
Median = 400 + (60) - 55 x 100 40
Median = 400 + 5 x 100 40
Median = 400 + 0.125 x 100
Median = 400 + 12.5
Median = 412.5
The median of given data set is = 412.5
x | f | c.f |
50 | 10 | 10 |
51 | 10 | 20 |
52 | 21 | 41 |
53 | 8 | 49 |
54 | 12 | 61 |
55 | 24 | 85 |
56 | 9 | 94 |
57 | 10 | 104 |
Total | 101 | |
we use median formula Median = Size of [(N+1)/2] th observation
Median = size of [(104+1)/2] th observation
Median = size of (102/2) th observation
Median = size of 51 th observation
Median = 54 because 51 th observation is 54
the column of cumulative frequency shows the order of each item.
therefore median is 54.
Foe calculating quartile
Second quartile = size of [(n+1)/2] th item = median
Second quartile = median = size of [(101+1) / 2] th item
Second quartile = median = 54
n is total number of items
Median For Continuous Data.
For continuous distribution we first arranging the data in ascending order or descending order and then creating the column of cumulative frequency, and finding the value of (n/2) and the class in which the value of (n/2) item lies. and this value appear in that class is called median class. and the median is formulated as
first finding median class as finding this value. Median = size of (n/2) th items.
Median = L+ {[(n/2)-c.f.] / f} x i
Where L is the lower limit of the median class
c.f. - Cumulative frequency of the previous class of median class.
f - frequency of the median class.
i - class interval of the median class.
Example 5. Calculating the median fir the following data.
Class | 10 - 20 | 20 - 30 | 30 - 40 | 40 - 50 | 50 - 60 | 60 - 70 | 70 - 80 |
F | 10 | 12 | 14 | 17 | 15 | 20 | 4 |
Solution : first we prepare the table with cumulative frequency column.
Class | F | Cumulative Frequency |
20-10 | 10 | 10 |
20 - 30 | 12 | 22 |
30 - 40 | 14 | 36 |
40 - 50 | 17 | 53 |
50 - 60 | 15 | 68 |
60 - 70 | 20 | 88 |
70 - 80 | 4 | 92 |
for finding the median class we calculating (n/2)
Median = size of (n/2) th item.
Median = size of (92/2) th item.
Median = 46 th item
therefore the 46 th item is lies in the class 40 - 50 it is median class 40 - 50.
the median is
Median = L+ {[(n/2)-c.f.] / f} x i
Where L is the lower limit of the median class = 40 i.e. L = 40
c.f. - Cumulative frequency of the previous class of median class. = 36, i.e. = 36
f - frequency of the median class. = 17 i.e. f = 17.
i - class interval of the median class. is = 50 - 40 = 10 i.e. 10
Median = 40 + {[ (92/2) - 36] / 17} x 10
Median = 40 + (10/17) x 10
Median = 40 + 0.5882 x 10
Median = 45.8824
Therefore the median of the data id 45.8824
IX. Quartiles:
We seen that the median is the value that divides the data into two equal parts. the median is lies at the centre of data set when arrange the data in ascending or descending order. in the same way we divide the data into four equal parts it is clear that the three values that divides the data into four equal part that lies at as first quartile ( first quartile which is lies above to the median and it is called as lower quartile), Second Quartile is the median of the data that divided the data into two equal parts. 3rd quartile is lies below the median. and the three quartiles are denotes as the three quartiles as first quartile Q1, second quartile Q2, and 3rd quartile Q3 .
Quartiles are useful t understanding the central tendency and variability of data set, identifying the outliers, it is used to draw Box Plot to visualise the distribution of data set.
the formula calculating quartiles as
i. First Quartile: Q1: Size of {(n+1)/4) th item.
where n is the total number of observations
ii. Second Quartile: Q2: Size of {(n+1)/2) th item.
Q2: is median
iii. Third Quartile: Q3: Size of 3{(n+ 1)/4} th item.
The Quartile are calculated as
i. Individual Data.
if the each item or observation given in separately then firstly arranging the data in ascending order and then applying the formula of quartiles.
Example 1. Calculate the first and third quartile for the following data.
41, 45, 47, 42, 48, 49, 50, 44, 46, 44, 52.
Solution :
first we arranging data in ascending order
41, 42, 44, 44, 45, 46, 47, 48, 49, 50, 52
now First Quartile: Q1: Size of {(n+1)/4) th item.
where n is the total number of observations = 11
therefore Q1: Size of {(11+1)/4) th item.
Q1: Size of {12/4) th item.
Q1: 3 rd observation
Q1 = 44
first quartile is 44.
third Quartile: Q3: Size of 3{(n+ 1)/4} th item.
Q3: Size of 3{(11+ 1)/4} th item.
Q3: Size of 3{12/4) th item.
Q3= 3 x 3 th item
Q3= 9 th observation
therefore Q3 = 49.
therefore the first quartile is 44 and third quartile is 49
Example 2. Calculate the First and third quartile for following data.
Class : 0 - 10 10 - 20 20 - 30 30-40 40-50 50 - 60
Frequency : 25 75 100 175 81 44
Solution: First we calculating cumulative frequency (c.f)
Class (x) | Frequency (f) | c . f. |
0 – 10 | 25 | 25 |
10 – 20 | 75 | 100 |
20 – 30 | 100 | 200 |
30 – 40 | 175 | 375 |
40 – 50 | 81 | 456 |
50 – 60 | 44 | 500 |
Total | 500 | |
the first quartile is = size of [N/4] th item
the First quartile is calculated for continuous data as
First Quartile = L+ {[(n/4)-c.f.] / f} x i
Where L is the lower limit of the median class = 20 i.e. L = 20
c.f. - Cumulative frequency of the previous class. = 100, i.e. = 100
f - frequency of the class. = 100i.e. f = 100.
i - class interval of the median class. is = 50 - 40 = 10 i.e. 10
First Quartile = 20 + {[ (500/4) - 100] / 100} x 10
First Quartile = 20 + {[ 125 - 100] / 100} x 10Third quartile = size of 3[N/4] th item
third quartile = size of 3[500/4] th item
third quartile = size of [125 * 3] th item
Third quartile = 375 th item.
Therefore the 375 th item is lies in the class 30 - 40 it is Quartile class 30-40
the Third quartile is calculated for continuous data as
Third Quartile = L+ {[(n/4)-c.f.] / f} x i
Where L is the lower limit of the median class = 30 i.e. L = 30
c.f. - Cumulative frequency of the previous class. = 200, i.e. = 200
f - frequency of the class. = 175 i.e. f = 175.
i - class interval of the median class. is = 50 - 40 = 10 i.e. 10
Third Quartile = 30 + {[ 3 (500/4) - 200] / 175} x 10
third Quartile = 30 + {[ 3 (125 )- 200] / 175} x 10Therefore the first quartile is 22.5 and third quartile is 40
Use this calculator to check your calculated mean is correct or not.
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