find mean,median and mode
Answers
Answer:
Mean Median Mode Formula
What is Mean?
The mean of a series of data is the value equal to the sum of the values of all the observations divided by the number of observations. It is the most commonly used measure of central tendency. Also, it is very easy to calculate. We denote Mean by \overline{X}
X
.
Mean Formula
1. Individual Series
\overline{X} = \frac{Sum of all the values of the observations}{No. of observations}
X
=
No.ofobservations
Sumofallthevaluesoftheobservations
2. Discrete Series
a. Direct Method:
\overline{X}= \frac{\sum fx}{\sum f}
X
=
∑f
∑fx
Where,
f Frequency
x Values
b. Assumed Mean or Short-Cut Method:
\overline{X}= A + \frac{\sum fd}{\sum f}
X
=A+
∑f
∑fd
Where,
f Frequency
d X – A
A Assumed Mean
c. Step-Deviation Method:
\overline{X}= A + \frac{\sum fd’}{\sum f} \times C
X
=A+
∑f
∑fd’
×C
Where,
f Frequency
A Assumed Mean
d’ \frac{(X – A)}{C}
C
(X–A)
C Common factor
Frequency Distribution or Continuous Series:
Direct Method:
\overline{X}= \frac{\sum fm}{\sum f}
X
=
∑f
∑fm
Where,
f Frequency
m Mid-Values
Assumed Mean or Short-Cut Method:
\overline{X}= A + \frac{\sum fd}{\sum f}
X
=A+
∑f
∑fd
Where,
f Frequency
d m – A
A Assumed Mean
Step-Deviation Method:
\overline{X}= A + \frac{\sum fd’}{\sum f} \times C
X
=A+
∑f
∑fd’
×C
Where,
f Frequency
A Assumed Mean
d’ \frac{(m – A)}{C}
C
(m–A)
C Common factor
Weighted Arithmetic Mean Formula:
\overline{X}= \frac{\sum WX}{\sum W}
X
=
∑W
∑WX
Where,
X Values
W Weights
What is the Median?
Median is the central or the middle value of a data series. In other words, it is the mid value of a series that divides it into two parts such that one half of the series has the values greater than the Median whereas the other half has values lower than the Median. For the calculation of Median, we need to arrange the data series either in ascending order or descending order.
Individual Series
When the number observations are odd
M = Size of \frac{(N + 1)}{2}^{th} termM=Sizeof
2
(N+1)
th
term
Where,
N Number of observations
When the number observations are even:
M = \frac{\left \{Size of \frac{(N + 1)}{2}^{th} item+ Size of (\frac {N}{2} +1)^{th} item\right \}}{2}M=
2
{Sizeof
2
(N+1)
th
item+Sizeof(
2
N
+1)
th
item}
Where,
N Number of observations
Discrete Series:
Median = Size of \frac{(N + 1)}{2}^{th} termMedian=Sizeof
2
(N+1)
th
term
Where,
N \sum f∑f
In this case, the value corresponding to the cumulative frequency just greater than the value obtained after applying the above formula is the Median of the series.
Frequency Distribution or Continuous Series:
Firstly, we need to calculate the Median class by applying the following formula:
Median class = \frac{N}{2}
2
N
M = \frac{l}{2} + \frac{h}{f} \left [ \frac{N}{2} – c.f. \right ]M=
2
l
+
f
h
[
2
N
–c.f.]
Where,
l Lower limit of the Median class
h Size of the median class
f Frequency of the median class
N Sum of frequencies
c.f. Cumulative frequency of the class just preceding the median class
What is Mode?
Mode refers to the value that occurs a most or the maximum number of times in a data series.
Mode formula
Individual Series:
We find the mode of an individual series by simply inspecting it and finding the item that occurs maximum number of times.
Discrete Series:
The Mode of a discrete series is the value of the item that has the highest frequency.
Frequency Distribution or Continuous Series:
Firstly, we need to find out the Modal class. Modal class is the class with the highest frequency. Then we apply the following formula for calculating the mode:
Mode = l + h \frac{f_1 -f_0}{(2 f_1 -f_0 -f_2)}
(2f
1
−f
0
−f
2
)
f
1
−f
0
Where,
L lower limit of the modal class
f1 Frequency of the modal class
f0 Frequency of the class just preceding the modal class
f2 Frequency of the class just succeeding the modal class
Solved Example
Calculate the Average marks of the following series using Direct Method.
Marks No. of students
0 – 10 10
10 – 20 30
20 – 30 70
30 – 40 50
40 – 50 20
Solution:
Marks Mid-values (m) No. of students (f) fm
0 – 10 5 10 50
10 – 20 15 30 450
20 – 30 25 70 1750
30 – 40 35 50 1750
40 – 50 45 20 900
\sum f = 180∑f=180 \sum fm = 4900∑fm=4900
\overline{X}= \frac{\sum fm}{\sum f}
X
=
∑f
∑fm
\frac{4900}{180}
180
4900
= 27.22